Thursday, 14 March 2024

Pricing Conventions, Discounting, and Arbitrage

Introduction

Bond pricing relies heavily on the concept of present value, where future cash flows are discounted back to their present value using an appropriate discount rate. This discount rate is often based on the required rate of return for similar securities, adjusted for factors such as risk and time preferences. The idea of arbitrage is crucial in the efficient pricing of securities. If two securities with identical future cash flows are priced differently, a riskless arbitrage opportunity exists. Investors can buy the cheaper security and sell the more expensive one, thereby profiting from the price discrepancy until equilibrium is restored and prices converge. This principle is fundamental to financial markets and helps ensure that securities are priced fairly based on their underlying characteristics.

The Value of a Bond:

A bond is a financial instrument that represents a loan made by an investor to a borrower (typically a corporation or government).
The borrower agrees to pay back the principal amount (the face value of the bond) at a future date, known as the maturity date, and usually makes periodic interest payments (coupon payments) to the bondholder until then.

Present Value of Cash Flows:

The value of a bond can be calculated by discounting its future cash flows back to the present.
This means that the amount of money the investor expects to receive in the future (both the coupon payments and the principal repayment) is adjusted to reflect its current value, considering factors such as the time value of money and the risk associated with those cash flows.

Discount Factors:

Discount factors are used to discount each future cash flow of the bond to its present value.
These factors are derived from the appropriate periodic required return, which is the rate of return an investor demands for holding the bond, taking into account factors such as prevailing interest rates, credit risk, and the bond's maturity.

Coupon Bonds:

Coupon bonds are bonds that pay periodic interest payments (coupons) to the bondholder until maturity, at which point the principal is repaid.
Discount factors are particularly important for pricing coupon bonds because they involve multiple future cash flows.

Determining Fair Pricing:

By calculating the present value of a bond's cash flows using appropriate discount factors, investors can determine whether a bond is trading at a fair price, known as "par value," or if it's trading at a discount (cheap) or premium (rich) relative to its intrinsic value.

Law of One Price:

This principle states that securities with identical future cash flows should sell for the same price.
In other words, if two securities offer the same future returns, they should be priced equivalently in the market.

Arbitrage Opportunity:

If a mispricing occurs, where identical securities are priced differently, it creates an opportunity for riskless arbitrage.
Investors can exploit this mispricing by buying the underpriced security and simultaneously selling the overpriced one, profiting from the price difference until market forces correct the discrepancy.

Understanding bond valuation and the law of one price equips you to make informed investment decisions and potentially identify arbitrage opportunities in the market.

Bonds Vs Securities

Bonds:

Definition:

Bonds are debt securities issued by governments, municipalities, or corporations to raise capital.
When you buy a bond, you are essentially lending money to the issuer in exchange for periodic interest payments (coupons) and the return of the bond's face value (principal) at maturity.

Characteristics:

Issuer: Bonds can be issued by governments (e.g., Treasury bonds), municipalities (municipal bonds), or corporations (corporate bonds).
Coupon Payments: Typically pay periodic interest (coupons) to bondholders, usually semi-annually or annually.
Maturity: Bonds have a specified maturity date when the principal amount is repaid to the bondholder.
Risk Profile: Bonds vary in risk depending on the issuer's creditworthiness. Government bonds are generally considered safer (e.g., U.S. Treasury bonds), while corporate bonds may carry higher risk depending on the issuer's financial health.

Types of Bonds:

Government Bonds: Issued by national governments (e.g., U.S. Treasury bonds, German Bunds).
Municipal Bonds: Issued by local governments to fund public projects.
Corporate Bonds: Issued by companies to finance operations or expansions.

Purpose:

Bonds are used to raise capital for long-term investments or to finance operations.
Investors purchase bonds for steady income (from coupon payments) and preservation of capital.

Securities:

Definition:

Securities are financial instruments that represent ownership (equity securities) or debt (debt securities) in an entity (e.g., company, government). Securities are generally tradable and can be bought and sold on financial markets.

Types of Securities:

Equity Securities:Represent ownership in a company, such as stocks (common or preferred shares). Investors in equity securities participate in the company's profits through dividends and capital gains.
Debt Securities: Include bonds (as discussed) and other instruments like treasury bills (T-bills), notes, and commercial paper. Debt securities represent loans that investors make to issuers, with the expectation of repayment with interest.

Marketability:

Securities can be traded on organized exchanges (stock exchanges) or over-the-counter (OTC) markets. They provide liquidity to investors who wish to buy or sell them.

Purpose:

Securities serve various purposes for investors and issuers, including raising capital, managing risk, and providing investment opportunities with varying levels of risk and return.

Key Differences:

Nature:

Bonds are a specific type of debt security, representing loans to issuers with fixed interest payments and maturity dates.
Securities encompass a broader range, including both debt and equity instruments.

Income Generation:

Bonds provide fixed or floating interest income to bondholders.
Whereas equity securities (like stocks) offer potential dividends and capital gains based on company performance.

Risk Profile:

Bonds generally have lower risk compared to stocks (equity securities), but their risk varies based on the issuer's creditworthiness.
Securities, especially equity securities, can fluctuate in value based on market conditions and company performance.

In essence, while bonds are a subset of securities focused on debt instruments with fixed terms, securities encompass a broader range of financial instruments including both debt and equity investments traded in financial markets.

Fundamentals of Bond Valuation

Bond valuation is the process of determining the fair or theoretical market price of a bond. It's crucial for investors as it helps them decide:

If a bond is a good investment: By comparing the calculated value to the market price, you can see if it's undervalued (potentially offering a higher return) or overpriced.
Expected return on a bond: Bond valuation helps estimate the yield to maturity (YTM), which is the internal rate of return an investor gets if they hold the bond until maturity and reinvest all coupon payments at a specific rate.

The process of valuing fixed-income securities, or any security for that matter, typically involves estimating the present value of all expected cash flows associated with the security.

This principle states that a dollar today is worth more than a dollar tomorrow.
Bond valuation considers this by discounting future cash flows (coupon payments and face value) to their present value.

There are three steps in the bond valuation process:

Step 1: Future Cash Flows

Estimate the cash flows over the life of the security.
Face Value (Par Value): The amount repaid to the bondholder at maturity.
Coupon Rate: The annual interest payment as a percentage of the face value. It's typically paid semi-annually.
Maturity Date: The date the bond matures, and the face value is returned.
Bonds typically promise two types of cash flows:

periodic coupon payments
repayment of the bond's face value at maturity / return of principal

Step 2: Discounting Future Cash Flows

Determine the appropriate discount rate based on the risk of (uncertainty about) the receipt of the estimated cash lows.
The value of a bond is calculated by discounting its future cash flows back to the present using an appropriate discount rate.
The discount rate reflects the required rate of return or the yield investors expect to earn from holding the bond.
It accounts for factors such as prevailing interest rates, credit risk, and the bond's time to maturity.
This is the rate used to discount future cash flows. It reflects the:

Risk-free rate: The return on a risk-free investment (e.g., government bond).
Risk premium: Additional return required to compensate for the risk of the bond compared to risk-free investments.

A higher discount rate reduces the present value of the bond's cash flows, making the bond less valuable.

Step 3: Present Value Calculation

Calculate the present value of the estimated cash lows by multiplying the bond’s expected cash lows by the appropriate discount factors.

The present value of each future cash flow is calculated separately and then summed to find the total present value of the bond.

Formula

Bond Price = (Coupon Payment / (1 + Discount Rate)^1) + (Coupon Payment / (1 + Discount Rate)^2) + ... + (Coupon Payment / (1 + Discount Rate)^n) + (Face Value / (1 + Discount Rate)^n)
Where

n = Number of periods until maturity

Factors Affecting Bond Valuation.

Market Interest Rates:

Bond prices and interest rates have an inverse relationship.
When interest rates rise, existing bonds with lower coupons become less attractive, decreasing their price.
Conversely, falling interest rates increase the value of existing bonds.

Creditworthiness of Issuer:

Bonds issued by governments (considered low risk) generally have lower discount rates than corporate bonds (higher risk).

Liquidity:

More liquid bonds (easier to buy and sell) tend to trade closer to their fair value.

The key differences in discount rates used for valuing Treasury bonds versus non-Treasury securities.

Risk-Free Rate for Treasuries:

Treasury bonds are backed by the U.S. government, which is considered virtually risk-free because the government is unlikely to default on its debt obligations.
The appropriate discount rate for valuing Treasury bonds is the risk-free rate itself.
This rate can be derived from the yield of a Treasury security with a similar maturity to the bond being valued.
Since there is minimal risk associated with Treasury bonds, a single discount rate, equal to the risk-free rate, is used to discount all the future cash flows of the bond, including coupon payments and face value at maturity.
Eg: Suppose a Treasury bond with a face value of $1,000 is issued at $950. The discount rate would be ($1,000 - $950) / $1,000 = 5%. At maturity, the bondholder receives $1,000, resulting in a $50 gain.

Non-Treasury Discount Rates:

Non-Treasury securities, such as corporate bonds, carry additional risks compared to Treasuries.
These bonds carry additional risks compared to Treasuries, such as:

Credit Risk: The risk that the issuer might default on their debt obligation.
Liquidity Risk: Difficulty buying or selling the bond quickly.
Call Risk: The issuer's right to redeem the bond before maturity (at a premium), potentially affecting the investor's planned returns.
Prepayment Risk: The possibility of the issuer repurchasing the bond before maturity (usually at a premium), impacting the expected cash flow stream.

To compensate for these additional risks, investors demand a higher return on non-Treasury bonds.
This additional return is called the risk premium.
The appropriate discount rate for valuing non-Treasury securities is obtained by adding the risk premium to the risk-free rate.

Single Discount Rate vs. Multiple Rates:

There are two main approaches to incorporating the risk premium:
Single Discount Rate:

This method adds the risk premium to the risk-free rate to obtain a single discount rate.
This rate is then used to discount all the cash flows of the non-Treasury bond.
Eg:

Suppose you are considering investing in a project that will yield cash flows over the next 5 years. The project requires an initial investment of $1,000, and you expect the following cash flows:

Year 1: $300
Year 2: $400
Year 3: $500
Year 4: $400
Year 5: $300

To evaluate whether this project is financially viable, you need to discount these future cash flows back to the present value (PV). Let's assume a single discount rate of 10% per year.
Calculation: Discounting Cash Flows:

$300/(1+0.10)^1 +$
$400/(1+0.10)^2 +$
$500/(1+0.10)^3$ +
400/(1+0.10)^4 +
300/(1+0.10)^5

Calculating each term:

Year 1: $\frac{300}{1.1 0^{1}} = \frac{300}{1.10} \approx 272.73$
Year 2: $\frac{400}{1.1 0^{2}} = \frac{400}{1.21} \approx 330.58$
Year 3: $\frac{500}{1.1 0^{3}} = \frac{500}{1.331} \approx 375.94$
Year 4: $\frac{400}{1.1 0^{4}} = \frac{400}{1.4641} \approx 273.22$
Year 5: $\frac{300}{1.1 0^{5}} = \frac{300}{1.6105} \approx 186.34$

Summing these present values gives the Net Present Value (NPV):

Since the NPV (1438.81) is positive, this project would be considered financially viable at a 10% discount rate.

Multiple Discount Rates:

This method recognizes that risks might not be evenly distributed over time.
Different discount rates are used for each cash flow, with higher rates applied to cash flows further in the future, reflecting potentially greater perceived risk as the maturity date approaches.
Eg:

In contrast, using multiple discount rates involves applying different rates to each period's cash flows based on their specific risks or time preferences. For instance:

Year 1: Apply a discount rate of 8%
Year 2: Apply a discount rate of 9%
Year 3: Apply a discount rate of 10%
Year 4: Apply a discount rate of 9%
Year 5: Apply a discount rate of 8%

Calculation: Discounting Cash Flows with Different Rates:

$300/(1+0.08)^1 +$
$400/(1+0.09)^2 +$
$500/(1+0.10)^3$ +
400/(1+0.09)^4 +
300/(1+0.08)^5

Calculating each term:
- Year 1: $\frac{300}{1.0 8^{1}} = \frac{300}{1.08} \approx 277.78$
- Year 2: $\frac{400}{1.0 9^{2}} = \frac{400}{1.1881} \approx 336.67$
- Year 3: $\frac{500}{1.1 0^{3}} = \frac{500}{1.331} \approx 375.94$
- Year 4: $\frac{400}{1.0 9^{4}} = \frac{400}{1.2950} \approx 308.82$
- Year 5: $\frac{300}{1.0 8^{5}} = \frac{300}{1.4693} \approx 204.32$
Summing these present values:

NPV=277.78+336.67+375.94+308.82+204.32=1503.53

The NPV (1503.53) is again positive, indicating that using multiple discount rates also shows this project as financially viable.

The choice between these methods depends on factors such as the complexity of the bond and the investor's preference for risk assumptions.
Single discount rates are simpler but assume constant risk, while multiple discount rates allow for varying risk levels over time.

Understanding these differences in discount rates helps investors make more informed decisions when valuing and comparing different types of bonds in the market, taking into account their risk profiles and expected returns.

Price Yield Curve

Discount Rate and Bond Value:

Imagine a bond that pays you a certain amount at regular intervals until it matures, with a final payment of the principal amount.
The discount rate is basically the interest rate you use to calculate the present value of all these future cash flows (coupon payments and principal repayment) of the bond.
A higher discount rate means you're using a less attractive interest rate, so the present value of those future cash flows goes down. This, in turn, makes the bond itself less valuable.
Conversely, a lower discount rate makes the present value of the cash flows go up, making the bond more valuable.
Let's take a look at how different discount rates affect bond valuation and who benefits in each scenario.

Example:

Consider a bond with the following features:

Face value: $1,000
Annual coupon payment: $50
Maturity: 5 years

We will calculate the present value of this bond under two different discount rates:

Scenario 1: Lower Discount Rate (5%)
Scenario 2: Higher Discount Rate (10%)

Results:

Discount Rate 5% = Present Value of the Bond $1,000.00
Discount Rate 10% = Present Value of the Bond $810.46

Who Benefits?

Lower Discount Rate (5%)

Advantage: Bond investors. In this scenario, the bond's present value is higher, meaning investors can potentially buy the bond at a premium or closer to its face value.
Disadvantage: Bond issuers. They may have to offer a higher coupon rate to attract investors when interest rates are low.

Higher Discount Rate (10%)

Advantage: Bond issuers. The lower present value allows them to potentially issue the bond at a discount or lower price.
Disadvantage: Bond investors. The lower present value translates to a lower return on their investment.

The discount rate has a significant impact on the value of a bond. It's important to consider these factors when making investment decisions.

Price-Yield Curve:

This curve shows the relationship between a bond's price (on the y-axis) and its yield (on the x-axis).

Credits: https://www.rba.gov.au/education/images/explainers/bonds-and-the-yield-curve/bonds-and-the-yield-curve-02.jpg

The curve is convex towards the origin, which means it looks like a half-smile. Why? Because of the inverse relationship.

For option-free bonds (bonds without embedded options like callable or convertible features), the price-yield curve is typically convex towards the origin.
This means that as yields decrease (moving towards the right on the graph), bond prices increase, but not in a linear fashion.
The increase in bond prices becomes more gradual at lower yields, reflecting diminishing returns in terms of price increase for each unit decrease in yield.
The curve often resembles half of a smile, where it steepens as yields decrease (prices increase) and flattens out as yields approach very low levels.

Visualization

Why is the Curve Curved?

The curve isn't a straight line because the impact of a discount rate change on the present value of cash flows isn't constant. Here's a simplified explanation:

For bonds with a longer maturity, small changes in the discount rate can significantly impact the present value due to the time value of money. This is why the curve bends more for longer-term bonds.
For bonds with a shorter maturity, changes in the discount rate have a smaller impact on the present value.

Understanding the price-yield curve is crucial for bond investors as it helps them assess the attractiveness of a bond based on its current price and prevailing interest rates.

Discount Rate vs Yield To Maturity

Yield to maturity (YTM) and discount rate are closely related concepts in the context of bonds, but they are not exactly the same.

Yield to Maturity (YTM):

YTM is the total return anticipated on a bond if it is held until maturity.
It takes into account the bond's current market price, coupon payments, and the time remaining until maturity.
YTM is essentially the internal rate of return (IRR) of an investment in the bond if all coupons are reinvested at the YTM rate until maturity.
It represents the effective interest rate that an investor will receive by holding the bond until maturity, assuming no default.

Discount Rate:

The discount rate, in the context of bonds, refers to the rate used to discount future cash flows of the bond to their present value.
This rate is typically based on the current market interest rates and the credit risk associated with the bond issuer.
It is used to calculate the present value of both the bond's coupon payments and its principal repayment at maturity.
The discount rate reflects the required rate of return that investors demand for investing in a bond of similar risk.

Relationship:

YTM can be considered as a type of discount rate because it reflects the rate at which future cash flows (coupon payments and principal repayment) are discounted to arrive at the bond's current market price.
However, the discount rate used in bond valuation and pricing is more broadly used to determine the present value of each cash flow (coupon and principal) separately, considering the time value of money and the risk associated with the bond issuer.
YTM, on the other hand, is specifically focused on the total return an investor expects from holding the bond until maturity.

In summary, YTM is a specific application of the concept of discount rate in the context of bond valuation, focusing on the return to maturity if held until the end of the bond's life. The discount rate, while related, is a broader concept used in bond pricing to determine the present value of future cash flows based on current market conditions and risk considerations.

Valuation With a Single Yield (Discount Rate)

For an option-free coupon bond, the coupon payments can be valued as an annuity. In order to take into account the payment of the par value at maturity, we will enter this final payment as the future value. This is the basic difference between valuing a coupon bond and valuing an annuity.

For simplicity, consider a security that will pay $100 per year for 10 years and make a single $1,000 payment at maturity (in 10 years). If the appropriate discount rate is 8% for all the cash lows, the value is:

This is simply the sum of the present values of the future cash lows, $100 per year for 10 years and $1,000 (the principal repayment) to be received at the end of the 10th year, at the same time as the final coupon payment.

Take note of a couple of points here.

The discount rate is entered as a whole number in percent, 8, not 0.08.
The 10 coupon payments of $100 each are taken care of in the N = 10 entry;
The principal repayment is in the FV = 1,000 entry.
The PV is negative—it will be the opposite sign to the sign of the PMT and FV.

The calculator is just “thinking” that if you receive the payments and future value (you own the bond), you must pay the present value of the bond today (you must buy the bond).
That’s why the PV amount is negative—it is a cash outlow to a bond buyer.
Just make sure that you give the payments and future value the same sign, and then you can ignore the sign on the answer (PV).

Valuation With a Single Yield and Semiannual Cash Flows

Let’s calculate the value of the same bond with semiannual payments.

Rather than $100 per year, the security will pay $50 every six months. Adjust the discount rate of 8% per year to 4% per six months. The par value remains $1,000.

Bond Components

STRIPS - Zero-Coupon Treasury Bonds:

STRIPS stands for Separate Trading of Registered Interest and Principal Securities.
These are zero-coupon bonds issued by the U.S. Treasury Department.
They aren't created directly by the Treasury but rather through a process called stripping.

The Stripping Process:

A financial institution takes an existing Treasury coupon bond (a bond that pays periodic interest).
This bond is then "stripped" into its two components:

Coupons (C-STRIPS, TINTs, or INTs):

These represent the individual interest payments that would have been received throughout the original bond's life. C-STRIPS (Coupon STRIPS):
These represent the separate coupon payments of the original Treasury bond.
They are fungible, meaning any C-STRIP of a specific maturity date can be used to fulfill the coupon payment for that date on any reconstituted bond of the same maturity.

Principal (P-STRIPS, TPs, or Ps):

This represents the face value of the bond, which is paid at maturity.
These represent the separate principal payments of the original Treasury bond. Unlike C-STRIPS, P-STRIPS are not fungible across different bonds.
Each P-STRIP is tied to a specific original bond and can only be used to reconstitute that bond's principal payment upon maturity.

Reconstitution of Bonds:

When a bond is reconstituted from its STRIPS:
For coupon payments: Any C-STRIPS of the appropriate maturity can be used, regardless of which original bond they were stripped from.
For principal payments: Only P-STRIPS that were stripped from the specific original bond can be used to reconstitute its principal payment.

Arbitrage Opportunities:

The stripping process creates a secondary market for these individual components (P-STRIPS and C-STRIPS).
In theory, arbitrage opportunities can arise when the relative prices of P-STRIPS and C-STRIPS become misaligned.
An arbitrageur could:
- Buy the undervalued component (e.g., cheap P-STRIPS).
- Sell the overvalued component (e.g., expensive C-STRIPS).
- Lock in a risk-free profit by exploiting the price discrepancy.

Transaction Costs and Practicality:

While arbitrage opportunities might exist, transaction costs associated with buying and selling these securities can be significant.
These costs can include commissions, fees, and bid-ask spreads, which can eat into the potential profit.
Often, these transaction costs outweigh the potential arbitrage gain, making it impractical to exploit these opportunities.

In essence:

STRIPS offer an alternative way to invest in Treasury securities by separating the principal and interest components.While arbitrage is a theoretical possibility, transaction costs often make it unrealistic.

Pricing Conventions Between Coupon Dates

Bonds are frequently purchased between coupon dates. We must account for three items in this situation: accrued interest, fractional period compounding, and the day count convention of the bond.

Clean and Dirty Bond Pricing

Dirty Price

The dirty price is the price that the seller of the bond must be paid to give up ownership.
It includes the present value of the bond plus the accrued interest.
The dirty price of the bond is sometimes referred to as the full price or invoice price.

Clean Price

The clean price is the dirty price less accrued interest:

clean price = dirty price – accrued interest

The clean price of the bond is sometimes referred to as the flat price or quoted price.

Note that the dirty price includes the discounted value of the next coupon so that the method of calculating accrued interest does not matter. As long as the clean price is calculated as dirty price minus accrued interest, the sum of the clean price and accrued interest will equal the dirty price.
The full price changes dramatically over time even when the market is unchanged, including a discontinuous jump on coupon payment dates, while the flat price changes only gradually over time. Therefore, when trading bonds day-to-day, it is more intuitive to track flat prices and negotiate transactions in those terms.
Dynamics of a bond's price and cash flows around the coupon payment date:
- Bond Price Dynamics Within a Coupon Period:
- Effect of Coupon Payment Date:
- Illustrative Example:
- Market Response:
- In summary, the bond's full price increases over time within a coupon period as future cash flows become closer and more certain. However, immediately after a coupon payment is made, the bond's price falls by the amount of the coupon payment because that cash flow is no longer included in the present value of future expected cash flows. This reflects how bond prices adjust based on the timing of coupon payments and the present value of cash flows at any given moment.

Accrued Interest:

When bonds are purchased between coupon dates, the buyer owes the seller the accrued interest for the period from the last coupon payment up to the settlement date of the transaction.
Accrued interest is calculated based on the fraction of the coupon period that has elapsed since the last coupon payment.
The formula for accrued interest depends on the bond's coupon rate, the number of days since the last coupon payment, and the total days in the coupon period.
Eg: Consider a $100 par value bond that pays 3% coupon semiannually. This means a coupon of $1.50 is paid every six months. If the bond is sold (and settles) 41 days after the last coupon, the buyer will need to pay the seller $1.50 * 41 / 182 = $0.3379 for every $100 purchased.
Eg: A $1,000 par value U.S. corporate bond pays a semiannual 10% coupon. Assume the last coupon was paid 90 days ago and there are 30 days in each month. Accrued interest is computed as follows:

AI = $50 ( 90 / 180 ) = $25
Here,

$50 = semiannual value 10% coupon for $1000 par value annually is $100/2
180 = 30 * 6 months

Eg: A EUR 100,000 par value French corporate bond pays 3.5% coupon with a semiannual frequency. Assume the last coupon was paid 75 days ago and there are 30 days in each month. The accrued interest is closest to:

When calculating the discount factor, the amount of the accrued interest needs to be added to both the bid and ask quotes before calculating the midpoint.

Fractional Period Compounding:

Bonds typically pay coupons semi-annually (every six months), but when bonds are purchased between these dates, the actual period to the next coupon payment may not be exactly six months.
To account for this, the interest accrued during the fractional period is calculated based on the actual number of days in that period, rather than assuming a full six-month period.

Day Count Convention:

The day count convention determines how interest accruals are calculated based on the number of days in the period.
Several day count conventions are used in practice in the bond markets. The day count used will depend on the type of security. Common day count conventions include:

30/360: Assumes every month has 30 days and every year has 360 days.

U.S. corporate and municipal bonds pay semiannual interest with a 30/360 day count.

Actual/360: Uses the actual number of days in a month and assumes a 360-day year.

In money markets, for discount securities and floating legs of interest rate swaps, the actual/360 day-count convention is used.

Actual/Actual: Uses the actual number of days in both the numerator and the denominator.

U.S. government bonds pay coupons semiannually and have an actual/actual day count.

Actual/365: Uses the actual number of days in a year (365 days).

An actual/365 day count is typically used for money market securities in Canada, New Zealand, and Australia.

The day count convention affects how accrued interest and yield calculations are performed.
We need to modify the bond pricing formula to incorporate the appropriate day count convention. Specifically, the bond pricing equation becomes:

Formula

When expressing w in the preceding equation, the number of days to use for the coupon period is determined by the appropriate day count convention.
For example, the denominator is 180 for semiannual bonds that use the 30/360 convention. This equation computes the dirty price the bond because it includes the discounted value of the first full coupon payment even though the accrued interest belongs to the seller of the bond.
Eg:

Ronam Ltd. invests in semi-annual US Treasury bonds with face values of USD 1,000 of 15 August 2020. A bond made a coupon payment of USD 40 on February 15, 2017. The next coupon is due on August 15, 2017. If the quoted price for the bond for delivery on June 15, 2017, is USD 1001-16, then what is the full price of the bond?

Previous coupon date (Feb 15, 2017) -> 120 days
Settlement date (June 15, 2017) -> 61 days
Next coupon payment date (Aug 15,2017) -> 120+61 -> 181 days
Accrued interest using actual/actual day-count- convention:

Accrued interest = USD 40 * (120/181) days = USD 26.5193
Full price = Quoted price + Accrued price = 1,001.50 + 26.5193 = USD 1028.0193
Note: 1001-16 = 1,001 + 16/32 = 1,001.5

A $1,000 par value U.S. corporate bond pays a semiannual 10% coupon. Assume the last coupon was paid 100 days ago and there are 30 days in each month. The accrued interest is closest to:
Coupon Rate per Period:

The bond pays a 10% coupon semiannually, so each coupon payment is 10% / 2 = 5% of the face value.
Since the face value is $1,000, the coupon payment amount is 5% * $1,000 = $50.

Days in Coupon Period:

We're assuming a 30/360 day count convention (30 days per month, 360 days per year).
A semiannual coupon period consists of 360 days / 2 = 180 days. Proportion of

Coupon Period Elapsed:

The last coupon was paid 100 days ago.
So, the proportion of the coupon period elapsed is 100 days / 180 days = 5/9.

Accrued Interest:

The accrued interest is the coupon payment amount multiplied by the proportion of the coupon period that has elapsed.
Accrued Interest = $50 * (5/9) = $27.78 (rounded to two decimal places). Therefore, the accrued interest on the bond is closest to $27.78.

In summary, when purchasing bonds between coupon dates, one must account for accrued interest, calculate interest for the fractional period correctly, and use the appropriate day count convention to ensure accurate calculations of interest payments and yields.
These considerations are crucial for both buyers and sellers to understand the exact financial obligations and returns associated with bond transactions.

Discount Factors for Treasury Bills

Treasury bills are securities that mature within one year and are issued by governments to inance their short-term funding needs.
The cash price paid for a Treasury bill is a function of the maturity price (e.g., 100), quoted price (Q), and number of calendar days until maturity (n).
This can be expressed as:

cash price = 100 * Qn / 360

If the Treasury bill matures in one year and n = 360, then the price paid by a buyer would be 100 – Q. In other words, the buyer would pay 100 – Q today and receive 100 in 360 days.
Note that the quoted price, Q, is essentially the annualized discount of the Treasury bill.
The quoted price is referred to as the clean price and does not include accrued interest. The cash price is referred to as the dirty price and includes accrued interest.
Formula for calculating the discount factor:

Discount Factor = 1 / (1 + discount rate * time to maturity)
Where:

Discount rate: The interest rate on the T-bill (expressed as a decimal).
Time to maturity: The time remaining until the T-bill matures, typically measured in years or fractions of a year.

Let's say you're considering a 182-day T-bill with a face value of $10,000 and a discount rate of 5%.

Convert the time to maturity to years: 182 days / 365 days/year = 0.5 years
Calculate the discount factor: Discount Factor = 1 / (1 + 0.05 * 0.5) = 0.9757
Multiply the discount factor by the face value to get the purchase price:

Purchase price = $10,000 x 0.9757 = $9,757

In this example, you would pay $9,757 for the T-bill and receive $10,000 at maturity. The difference of $243 represents your earned interest.

We could calculate a cash price based on both the bid and ask quotes for the bill. The midpoint of these two values is the discount factor.

The bid price is the highest price an investor is willing to pay for a security.
The ask price is the lowest price a seller is willing to accept for a security.
The discount factor is a value between 0 and 1 that reflects the present value of a future cash flow (face value) received at maturity. The midpoint between the bid and ask price is considered the discount factor in this context.

If the cash price based on the bid for a security that matures in 80 days is calculated as 99.60 and the cash price based on the ask is 99.65, the midpoint is 99.625, or 0.99625.
A security worth $100,000 in 80 days would be priced at $99,625 today. Therefore, the 0.99625 is the discount factor for this maturity date.
If a security was priced at $100,000 today, it would be worth $100,000 / 0.99625 = $100,376.41 in 80 days based on the midmarket discount factor.

Discount Factors for Treasury Bonds

Treasury bonds are securities issued by governments to finance their mid- or long-term needs.
They promise a stream of future cash lows, and are therefore deined by their face value, cash low (coupons), and maturity.
A series of Treasury bond (T-bond) prices can be used to generate the discount function.
Eg:

Below selected T-bond prices for semiannual coupon $100 face value bonds. Settlement is T + 1. Generate discount factors for the dates indicated.

Bond 1: When this bond matures on November 15, 2021, it will repay its principal of 100 and will make its last interest payment of:

(0.0425 / 2 * 100) = 2.125
The current cash price of the bond is 101.50. Therefore:

d(1) = 101.50 / (100 + 2.125) = 0.9939

Moving farther out on the curve, the function becomes slightly more complex, as each point of the curve must be included.

For example, to solve for Bond 2, we must include both d(1) and d(2).
Bond 2: The coupon payment at Time 1 is 7.25 / 2 = 3.625. The inal cash flow at Time 2 is 100 + 3.625 = 103.625. These two cash flows discounted back to present value using the discount function should equal the price of the bond:

[3.625 × d(1)] + [103.625 × d(2)] = 105.98
Since it’s already known that d(1) = 0.9939:

(3.625 × 0.9939) + [103.625 × d(2)] = 105.98
d(2) = 0.9880

Using the same methodology for Bonds 3, 4, and 5:

Bond 3: [1.0 × d(1)] + [1.0 × d(2)] + [101 × d(3)] = 101.22
Thus: d(3) = 0.9825
Bonds 4 and 5: d(4) = 0.9731 d(5) = 0.9633

Summary of Discount Factors

Determining Value Using Discount Functions - Law of one price

The law of one price is an economic principle that states that identical goods should sell for the same price in different markets when there are no transportation costs and no differential taxes applied to the goods in those markets.
This concept is rooted in the idea of arbitrage, where any price difference between identical goods in different markets would quickly be eliminated by market forces.
If investors are able to exploit a mispricing because of the law of one price, it is referred to as an arbitrage opportunity. To take advantage of an arbitrage opportunity, investors should short sell the more expensive instrument/portfolio and buy the cheaper instrument/portfolio. Since both provide identical future cash lows, the investor can then generate a proit.
Example of the Law of One Price: Let's consider a hypothetical example with gold:

Identical Good: Gold bars of the same purity (e.g., 99.99% pure gold).
Different Markets: Suppose there are two markets, Market A and Market B, in different countries.
No Transportation Costs or Trade Barriers: Assume there are no transportation costs, tariffs, or other barriers to trade between these markets.
Application of the Law:

In Market A, the current price of a 1-ounce gold bar is $2,000.
In Market B, the current price of a 1-ounce gold bar is $1,950.

According to the law of one price, these two prices should converge because the gold bars are identical and there are no obstacles to arbitrage. Here’s how the market would react:

Arbitrage Opportunity: Traders could buy gold bars in Market B at $1,950 and sell them in Market A for $2,000, making a profit of $50 per ounce (excluding transaction costs).
Market Adjustment: As traders exploit this price difference, the increased demand for gold in Market B would push prices up, while the increased supply in Market A would push prices down, eventually leading to the prices in both markets converging towards the equilibrium price, likely somewhere between $1,950 and $2,000.
Elimination of Price Difference: Once the prices stabilize, the law of one price suggests that the price of gold bars of identical quality should be the same in both Market A and Market B, barring any new factors that could affect the price.

In essence, the law of one price highlights the tendency of markets to equalize prices for identical goods across different locations when market conditions allow for free trade and arbitrage. Also in other words if the law of one price is not present it will allow people to do arbitrage, excluding the cost involved.
Arbitrage means a transaction, in which there is zero net investment and positive payoff. Considered to be risk less investment.

Short positions identify arbitrage opportunities

Short positions:

Short positions are important considerations for arbitrage.
Short positions involve selling securities the investor does not own, with the expectation that the security price declines and the investor can repurchase the security at a lower price.
If the security pays income (coupon for bonds or dividends for equities), the investor must pay this income to the lender of the security.
The risk of short positions is that the security price moves up, or that the security can no longer be borrowed and the investor needs to buy back the security (potentially at a loss).

Example:

Suppose you observe the annual coupon bonds. The 2-year spot rate is 12%. Is there an arbitrage opportunity? If so, describe the trades necessary to exploit the arbitrage opportunity.
Input:

Here:

CR = Coupon Rate
YTM = Yield to Maturity
FV = Face Value
CMP = Current Market Price

Spot Rate for 2yrs = 12%

Spot Rate Definition:

The spot rate, also known as the zero-coupon yield or the spot yield, is the yield or interest rate on a bond that pays no periodic interest payments (coupon payments).
Instead, it is issued at a discount to its face value and pays the face value at maturity.
The spot rate is the rate of return an investor would earn if they bought the bond today and held it until maturity.

Spot Rate for 2 years = 12%:

This statement specifically indicates that the annualized yield to maturity on a theoretical zero-coupon bond that matures in 2 years is 12%.

Sell = Get Cash

Buy = Pay Cash get Security

Step 1: Buy Bond3 for $1 million of the 2-year, 10% coupon bonds.
Step 2: Short sell Bond1 $10,000 of the 1-year, zero-coupon bonds at 95.23.
Step 3: Short sell Bond2 $110,000 of the 2-year, zero-coupon bonds at 82.64.

The result is receiving positive income today in return for no future obligation, which is an arbitrage opportunity.

Constructing a Replicating Portfolio

Constructing a replicating portfolio involves using a combination of different fixed-income securities to replicate the cash flows of a given fixed-income security.
Here this is another way of identifying arbitrage opportunities.
Example:

Suppose a 5-year fixed-income security exists with $1000 face value and a 20% coupon rate. The coupons are paid on a semiannual basis, and the security’s DR is assumed to be 10%. The present value of this bond, Bond 1, and its cash lows are calculated as follows:
Original Bond1:

Face Value - FV: $1000
Coupon Rate - CR: 20%
Maturity - n: 5years
Discount Rate - DR: 10%
Present Value - PV or Current Market Price - CMP: $1379.08 (using calculator)

If this bond is determined to be trading cheap, then a trader can conduct an arbitrage trade by purchasing the undervalued bond and shorting a replicating portfolio that mimics the bond’s cash lows.
To demonstrate the creation of a replicating portfolio, assume the following four fixed-income securities exist in addition to the bond we are trying to replicate.
Other Bonds:

The goals is to use the above bonds and try to replicate the Bond1. Combinations of these five bonds can help in matching the cash flow which is the same characteristics. As the bonds with same cash flow should be priced same in the market.
First lets try to match the cash flow and then we can look into the pricing arbitrages.

Calculate the difference price of original bond vs synthetic bond/bond portfolio (replicated version)

Original Bond value is underpriced, here the arbitrage could be leveraged. Buy original bond and short/sell synthetic bond/bond portfolio.

Credits and References

https://www.spellbrand.com/wp-content/uploads/2014/11/value-based-pricing-strategy-1.jpg

https://gemini.google.com/

https://chat.openai.com/

SchweserNotes and BionicTurtle Notes

#Reading55

Thursday, 22 February 2024

Corporate Bonds

Introduction

The term "bond" refers to a variety of assets that offer a wide range of interest rate payments from fixed cash payments, to accruals without cash where if a bondholder is entitled to receive interest on a bond, they will record that interest income as it accrues, even if the cash hasn't been received yet, to payments in the additional securities.

When an organization needs to raise funds, it typically has a few options, with equity issuance, debt financing from banks, or issuing bonds being among the most common.

Equity Issuance:

This involves selling ownership stakes in the company, often in the form of stocks or shares.
When an organization issues equity, it sells a portion of itself to investors in exchange for cash.
This can be done through initial public offerings (IPOs) or private placements.
Equity issuance dilutes existing ownership stakes but does not require repayment of funds.

Debt Financing from Banks:

Organizations can borrow money directly from banks or other financial institutions.
This usually involves taking out loans that need to be repaid over a specified period, along with accrued interest.
These loans can be secured (backed by collateral) or unsecured, depending on the terms negotiated between the borrower and the lender.
Debt financing requires repayment with interest but does not dilute ownership.

Issuing Bonds:

Bonds are debt securities issued by corporations, governments, or other entities to raise capital.
When an organization issues bonds, it essentially borrows money from investors who purchase the bonds.
Bonds typically have a fixed interest rate (coupon rate) and a maturity date at which the principal amount must be repaid.
Interest payments are made periodically (usually semiannually) until the bond matures.
Bonds provide a way to raise large amounts of capital upfront but also involve regular interest payments and repayment of principal at maturity.

The choice between these options depends on factors such as the organization's financial situation, risk tolerance, and strategic goals.

Bond Trading

Publicly traded bonds are usually traded in the over-the-counter (OTC) markets as opposed to exchanges.
Dealers exist in the bond market to buy and sell bonds and earn profit through the bid-ask spread (e.g. buy low, sell high).
Bond pricing is based on the laws of supply and demand.

If demand > supply then prises rises
If demand < supply then prises falls

Corporate bond yield

A corporate bond yield is a function of the risk free return + a credit spread to reflect the risk of default.
A corporate bond yield curve is a graphical representation of the relationship between the yield (interest rate) offered by corporate bonds of different maturities (the time until the bond matures and the investor gets their money back).
It essentially shows you how much return you can expect on your investment based on how long you're willing to lend your money.
Yield:

This is the annual return an investor receives by holding a bond until maturity.
It's typically expressed as a percentage.

Maturity:

This is the length of time until a bond reaches its maturity date and the issuer needs to repay the principal amount borrowed.
Bonds can have maturities ranging from a few months to several decades.

Corporate Bonds:

These are debt instruments issued by corporations to raise capital.
Investors who buy corporate bonds essentially loan money to the company in exchange for a fixed interest rate payout over time and the return of the principal amount at maturity.
The Y-axis of the corporate bond yield curve represents the yield (interest rate), and the X-axis represents the maturity of the bond.
By plotting yields of corporate bonds with similar credit quality but different maturities, we can see the shape of the curve.

There are three main shapes a corporate bond yield curve can take:

Upward Sloping Curve:

This is the most common scenario.
It indicates that investors typically demand higher yields for lending money for longer periods.
This can reflect expectations of rising interest rates in the future or a risk premium associated with longer-term investments.

Downward Sloping Curve:

This is less common and suggests that investors are willing to accept lower yields for longer maturities.
This might occur when there's an expectation of falling interest rates or a flight to safety during economic uncertainty, where investors prioritize security over higher returns.

Flat Curve:

This indicates minimal yield difference between short-term and long-term bonds.
It can be a sign of an economy with uncertain future interest rate movements.

Inverse Relationship between Bond Prices and Yields:

When bond yields rise, bond prices fall, and vice versa.
This fundamental relationship is known as the bond pricing rule.
It occurs because as yields increase, newly issued bonds offer higher interest payments, making existing bonds with lower yields less attractive to investors.
To compensate for the lower interest payments, the prices of existing bonds must decrease to bring their yields in line with the market rate.

Relationship between Bond Liquidity and Investor Demand:

Higher bond liquidity generally means that a bond can be bought or sold more easily without significantly impacting its price.
Bonds with higher liquidity tend to have lower bid-ask spreads and higher trading volumes.
However, higher liquidity may also indicate lower demand by investors.
This can occur when investors perceive lower risks associated with the bond, leading to less urgency in buying or selling it.
Conversely, bonds with lower liquidity may have higher demand from investors seeking higher returns, but they may also come with higher transaction costs and greater price volatility.
Price of bond is the function of demand and supply impacted by interest rate (coupons), market interest rate and time.

Credits: https://www.investopedia.com/thmb/7zB480fOW2FxdfuK_PaSC5apASc=/1500x0/filters:no_upscale():max_bytes(150000):strip_icc()/CorporateBonds_CreditRisk22-8c12f1dbc1494f28b3629d456fb4fa63.png

Bond Indenture and Corporate Trustee

A bond indenture and a corporate trustee play essential roles in facilitating the issuance and management of the bonds.
The bond indenture defines the terms of a corporate bond issue, while the corporate trustee acts as a guardian of bondholders' interests and ensures compliance with those terms.
Together, they help to facilitate smooth and transparent bond issuance and management processes.
Bond Indenture

A bond indenture is a legal contract between the issuer of the bonds (the corporation) and the bondholders.
The document that provides clarity and certainty to both the issuer and the bondholders regarding their rights, obligations, and recourse in various scenarios.
It is usually a detailed document filled with legal language.
Principal Amount:

The amount of money borrowed by the corporation, which will be repaid to the bondholders at maturity.

Coupon Rate:

The interest rate paid to bondholders, typically expressed as a percentage of the bond's face value and paid at regular intervals (e.g., annually or semi-annually).

Maturity Date:

The date when the principal amount of the bond becomes due and payable to the bondholders.

Call Provisions:

Terms specifying whether the issuer has the right to redeem the bonds before their maturity date, and under what conditions.

Covenants:

Restrictions or requirements imposed on the issuer to protect the interests of bondholders, such as limitations on additional debt issuance or requirements for maintaining certain financial ratios.
The trustee would monitor corporation's activities to make sure the issuer abides by the indenture's covenants.

Negative or restrictive

What company should not do
E.g. limited additional debt financing, dividend declarations

Positive

What company have to do
E.g. to produce financial statements, maintain insurance

Financial

Financials donts and dos
E.g. maintaining key ratios above/below a given number

Default and Remedies:

Procedures and remedies in case of default by the issuer, including potential acceleration of repayment or appointment of a trustee to act on behalf of bondholders.

Corporate Trustee:

A corporate trustee is a financial institution or trust company appointed to represent the interests of bondholders and ensure compliance with the terms of the bond indenture.
One of the roles of the corporate trustee is to interpret the legal language and represent the interests of the bond holders.
By serving as an independent third party, the corporate trustee helps to enhance transparency, accountability, and trust in the bond issuance process, benefiting both issuers and investors.
Requirements are explicitly stated in the indenture, and the trustee only needs to meet those requirements and no more.
The indenture would specify how and the frequency with which the trustee would make reports to bondholders and what to do if the issuer failes to pay interest or principal.
Typically a corporate trustee is a bank, financial institution, or a highly reputable individual.
Safeguarding Bondholder Interests:

The trustee ensures that the issuer complies with the terms of the bond indenture and protects the interests of bondholders.

Payment Administration:

The trustee typically receives interest and principal payments from the issuer and distributes them to bondholders in accordance with the bond terms.
Making sure the number does not exceed the limit specified in the indenture.

Enforcement of Rights:

In the event of default or other breaches of the bond indenture, the trustee may take legal action on behalf of bondholders to enforce their rights and seek remedies.

Record Keeping:

The trustee maintains records of bond ownership, transactions, and communications with bondholders.

Bond Issuers

Bond issuers come from various sectors, each with its own characteristics and reasons for issuing bonds.
There are five general groups of bond issuers:
Utilities
Utilities include companies that provide essential services such as electricity, water, and natural gas.
These companies often have stable cash flows and predictable revenue streams, making them attractive candidates for issuing bonds.
Utilities may issue bonds to finance infrastructure projects, upgrade facilities, or refinance existing debt.
Transportation companies
Transportation companies encompass a wide range of entities involved in transporting goods and people, including airlines, railroads, shipping companies, and logistics firms.
These companies may issue bonds to fund capital expenditures, expand their fleets, or improve infrastructure.
Bond investors may be attracted to transportation bonds based on factors such as the stability of the industry, economic growth projections, and government regulations.
Insdustrials
Industrial companies span various sectors, including manufacturing, construction, technology, and consumer goods.
These companies may issue bonds for purposes such as financing expansion projects, acquiring new equipment, or restructuring debt.
Bond investors assess industrial bonds based on factors such as the company's financial health, competitive position, and industry outlook.
Financial institutions
Financial institutions include banks, insurance companies, and other financial intermediaries.
These entities may issue bonds as a means of raising capital to support lending activities, meet regulatory requirements, or manage liquidity.
Bond investors evaluate financial institution bonds based on factors such as the institution's creditworthiness, regulatory environment, and interest rate risk.
Internationals
International organizations such as the World Bank and the International Monetary Fund (IMF) issue bonds to raise funds for development projects, provide financial assistance to member countries, or support global economic stability.
These bonds, often referred to as sovereign or supranational bonds, are typically backed by the issuing organization's creditworthiness and may carry concessional terms for certain projects or regions.
Each group of bond issuers has its own risk profile, financial characteristics, and market dynamics.
Investors consider factors such as credit quality, industry trends, economic conditions, and geopolitical risks when evaluating bonds issued by these entities.
Diversifying across different types of issuers and sectors can help investors manage risk and achieve their investment objectives.

Bond Maturities

Short term bond notes maturities from 1 to 5 years.
Medium term bond notes have maturities from 5 to 12 years.
Long term bond notes have maturities from greater than 12 years.
Tenor of 0-1 year is not called as bond but called as bill.

Bonds Types basis on Interest Payment

Fixed-rate bonds

Fixed-rate bonds, as the name suggests, have a predetermined interest rate that remains constant throughout the life of the bond.
Fixed Interest Rate:

Fixed-rate bonds pay a specified interest rate, known as the coupon rate, at regular intervals (such as annually or semi-annually) until the bond matures.
This interest rate is determined at the time of issuance and remains unchanged, regardless of fluctuations in market interest rates.

Interest Payments:

The issuer of the fixed-rate bond is obligated to make periodic interest payments to bondholders based on the fixed coupon rate.
These payments provide a predictable income stream for investors.

Maturity:

At maturity, the issuer repays the principal amount (face value) of the bond to the bondholders.
Fixed-rate bonds typically have a specified maturity date, at which point the bondholder receives the final interest payment along with the repayment of the principal.

Foreign Currency Payments:

In some cases, fixed-rate bonds may offer interest payments in a foreign currency.
This feature is known as a foreign currency bond.
For example, a U.S. based investor may purchase a bond issued by a European company that pays interest in euros rather than U.S. dollars.
Foreign currency payments introduce currency exchange rate risk for investors.
Fluctuations in exchange rates between the foreign currency and the investor's home currency can affect the value of interest payments received.
Investors need to consider this risk when investing in foreign currency bonds.

Fixed-rate bonds are popular among investors seeking stable income streams and predictable returns.
They provide a level of certainty regarding future cash flows, which can be advantageous for income-oriented investors, pension funds, and institutional investors.
However, investors should carefully assess the credit quality of the issuer, prevailing market conditions, and currency risk when evaluating fixed-rate bonds for investment.

Floating-rate bonds

Floating rate bonds, also known as variable rate bonds, are bonds whose interest rates fluctuate over time based on changes in a specified benchmark interest rate or reference rate.
Interest Rate Structure:

Unlike fixed-rate bonds where the interest rate remains constant, floating rate bonds have variable interest rates that adjust periodically according to a predetermined formula.
This formula typically ties the bond's interest rate to a benchmark rate or a reference rate, such as LIBOR (London Interbank Offered Rate) plus a fixed spread or a government bond yield or PLR (Primary Lending Rate).

While both LIBOR (London Interbank Offered Rate) and the primary lending rate, often referred to as the prime rate, are interest rate benchmarks, they serve different purposes and are used in different contexts.
The prime rate primarily applies to domestic short-term lending within a specific country, while LIBOR is used in global financial markets for various currency denominations and maturities.

Benchmark Rate:

The benchmark rate serves as the reference point for determining the bond's interest rate.
Coupon rate changes as benchmark rate changes.
Common benchmark rates used for floating rate bonds include short-term interbank lending rates or government bond yields.
For example, a floating rate bond may pay interest at a rate equal to LIBOR plus a specified spread.

Interest Adjustment Frequency:

The interest rate (coupon) to be paid is determined at the beginning of the period and the interest is paid at the end of the period.
Floating rate bonds typically have predefined intervals at which the interest rate adjusts. This could be monthly, quarterly, semi-annually, or annually, depending on the terms of the bond.

Interest Rate Floor and Ceiling:

Some floating rate bonds include provisions that set a floor and/or a ceiling on the interest rate adjustments. The floor ensures that the bond's interest rate does not fall below a certain level, while the ceiling caps the maximum interest rate that can be paid on the bond.

Investor Protection Against Interest Rate Risk:

Floating rate bonds offer investors protection against interest rate risk, as the interest payments adjust in response to changes in prevailing market interest rates.
When interest rates rise, the interest payments on floating rate bonds increase, helping to preserve the bond's value.
Conversely, when interest rates fall, the interest payments decrease, but this is typically less of a concern for investors as they still receive higher interest payments relative to fixed-rate bonds.

Floating rate bonds are attractive to investors, particularly during periods of rising interest rates, as they offer the potential for higher income compared to fixed-rate bonds.
They are commonly issued by governments, financial institutions, and corporations seeking to manage interest rate risk while still accessing the bond market for financing.
However, investors should carefully consider the credit quality of the issuer, the terms of the bond, and prevailing market conditions before investing in floating rate bonds.

Zero-coupon bonds (ZCB)

Zero-coupon bonds, also known as discount bonds or deep discount bonds, are a type of fixed-income security with some unique characteristics.
Key features of zero-coupon bonds:

No periodic interest payments:

Unlike traditional bonds, zero-coupon bonds don't make any regular coupon payments (interest payments) throughout their lifespan.

Sold at a discount:

These bonds are issued and sold at a significant discount to their face value (maturity value).
This discount represents the investor's return on investment.

Profit at maturity:

When the bond reaches its maturity date, the investor receives the full face value, essentially pocketing the difference between the discounted purchase price and the face value.

Implied Yield:

The yield to maturity (YTM) of a zero-coupon bond is the annualized rate of return that investors earn if they hold the bond until maturity.
Since zero-coupon bonds do not make periodic interest payments, their yield is based on the difference between the purchase price and the maturity value, compounded over the holding period.

Price Volatility:

Zero-coupon bonds are more sensitive to changes in interest rates compared to coupon-paying bonds.
This is because their entire return is derived from the difference between the purchase price and the face value, so any change in interest rates can have a magnified impact on their price.

Bankruptcy and Bondholder Rights:

Zero-coupon bonds do have an implicit interest component, and understanding how bankruptcy impacts them is essential.
Implicit Interest in Zero-Coupon Bonds:

Even though there are no regular coupon payments, a zero-coupon bond's value increases year after year.
This growth reflects the implicit interest earned on the investment.
The discount you receive at purchase represents the compounded interest you would have earned on a traditional bond with the same face value and maturity.

Impact of Bankruptcy on Zero-Coupon Bonds:

Unfortunately, zero-coupon bondholders don't enjoy the same level of protection as traditional bondholders in case of issuer bankruptcy.
Since they don't receive regular interest payments, their claim on the issuer's assets is limited.
If the company goes bankrupt before maturity, bondholders are typically only entitled to:

The original discounted purchase price they paid for the bond.
Accrued interest up to the bankruptcy date. This accrued interest is calculated based on the implicit interest and not any actual coupon payments.
In the event of issuer bankruptcy before the bond matures, bondholders may be entitled to receive the accrued interest up to the date of the bankruptcy filing, in addition to the return of the bond's original issue price. This accrued interest reflects the compensation that the bondholders have earned for holding the bond until the issuer's default, based on the implicit interest embedded in the bond's appreciation in value.

Here's an analogy:

Think of buying a zero-coupon bond like buying a discounted train ticket.
The discount represents the total fare you would have paid if you bought regular tickets with included seat reservations (coupons).
With a zero-coupon bond, you get the discounted upfront price, but if the train company goes bankrupt before your trip (maturity), you might only get a refund for the initial discounted price, not the full face value of the ticket (maturity value).

Key takeaway:

While zero-coupon bonds offer potential benefits like guaranteed returns and interest rate stability, their vulnerability in bankruptcy situations is a crucial consideration for investors.
It's important to weigh the risks and rewards before adding them to your portfolio.

Example:

Let's say you buy a 10-year zero-coupon bond with a face value of $10,000 for $6,000.
You hold the bond for 10 years and receive no interest payments during that time. However, at maturity, you get the full $10,000 face value.
Your profit is the difference between the purchase price and the maturity value, which is $4,000 ($10,000 - $6,000).
This $4,000 represents your effective return on investment over the 10 years.

Benefits of zero-coupon bonds:

Reinvesting interest:

Unlike coupon (coupon means interest) bonds, the zero coupon bondholders does not have to make an effort to reinvest cash interest payments or worry about the available rates in which to reinvest them.
Can be considered both as both has advantage and disadvantage associated with it, like seeing the glass half full or half empty.

Guaranteed return:

As long as you hold the bond until maturity, you are guaranteed to receive the face value, locking in your return at the time of purchase.

Compounding:

While you don't receive regular interest payments, the effective return from the discount can be thought of as compounding over time.

Interest rate stability:

Zero-coupon bonds can be attractive for investors seeking protection from fluctuating interest rates because the return is locked in at purchase.

Tax jurisdictions:

Zero-coupon bonds can be the potential tax benefit of converting interest income to a capital gain.
Traditional Bond Taxation:

Coupon bonds typically have regular interest payments that are taxed as ordinary income in most jurisdictions.
This means investors pay income tax on the interest they receive each year.

Zero-Coupon Bonds and Tax Advantages:

With zero-coupon bonds, there are no regular interest payments.
The investor's return comes from the difference between the discounted purchase price and the face value received at maturity.
In some tax jurisdictions, this difference may be considered a capital gain rather than ordinary income.

Capital Gains vs. Ordinary Income:

Capital gains tax rates are often lower than ordinary income tax rates in many countries.
This means that investors might pay less tax on their overall return from a zero-coupon bond compared to a traditional coupon bond with the same yield if taxed as ordinary income.

Important Considerations:

Tax laws and treatment of zero-coupon bonds can vary significantly between countries and even states or provinces within a country.
It's crucial to consult with a tax advisor to understand the specific tax implications of zero-coupon bonds in your jurisdiction.
Even if capital gains are taxed favorably, other factors like the bond's creditworthiness, liquidity, and interest rate sensitivity should also be considered before investing.

Drawbacks of zero-coupon bonds:

Reinvesting interest:

You don't receive any cash flow until maturity, so you can't reinvest the interest payments to potentially grow your returns faster.

Price volatility:

The price of zero-coupon bonds fluctuates more than traditional bonds with coupon payments due to changes in interest rates.

Tax implications:

Even though you don't receive any cash interest, the IRS may consider a portion of the increasing value as taxable income each year (accrued interest).

Who should consider zero-coupon bonds?

Long-term investors:

These bonds are suitable for investors with a long-term investment horizon who can hold the bond until maturity.

Investors seeking predictable returns:

They can be appealing to investors who prioritize a guaranteed return at maturity over regular cash flow.

Those planning for a future event:

They can be useful for planning for a specific future event, such as a child's education, where you know you'll need the money at a certain time.

Remember: Zero-coupon bonds are a specific type of investment with their own set of advantages and disadvantages. It's crucial to carefully consider your investment goals, risk tolerance, and investment timeframe before deciding if zero-coupon bonds are a good fit for your portfolio.

Bond Types based on Collateral

Corporate bonds can have collateral, such as real property, underlying the issue.
The collateral may be useful if a defaulting firm will be liquidating because the sale proceeds from the collateral will be paid first to the bondholders who have a collateral position. This serves as a form of protection for investors in case of default.
If the defaulting firm is reorganized, then the bondholders with collateral will have better negotiating powers. The collateral serves as a bargaining chip, as the company would need your consent to sell or use it for other purposes. This can give you a stronger voice in the restructuring process and potentially lead to a more favorable outcome.
Bonds can be classified into two main types based on whether they are secured by collateral or not:
Secured Bonds:
These bonds are issued with a specific asset pledged as collateral.
This collateral acts as security for the investor in case the issuer defaults on the bond.
If a default occurs, the lender can seize and sell the collateral to recoup their losses. Secured bonds typically offer lower interest rates to investors compared to unsecured bonds because they are considered less risky.
Collateral based bonds: Mortgage bonds, Collateral trust bonds, Equipment trust certificates
Unsecured Bonds:
These bonds are not backed by any specific collateral.
Instead, they rely solely on the creditworthiness of the issuer to repay the debt.
Unsecured bonds, also known as debentures, generally offer higher interest rates to investors to compensate for the increased risk.
Mortgage bonds:
Mortgage bonds have supporting collateral that can be sold to pay off the bondholders if there is a default.
Restricting future bond issues
Restricting future bond issues, commonly known as a negative pledge covenant.
This is a common covenant included in mortgage bond indentures that restricts the issuer (typically a bank) from issuing new debt secured by the same pool of mortgages.
By doing so, it safeguards the interests of existing mortgage bondholders by ensuring that the collateral pool remains intact and isn't diluted by additional debt obligations.
Analogy, would be like saying your friend can't use their existing car (or any other car they own in the future) as collateral for other loans without your permission.
After-acquired clause
An after-acquired clause could be used to restrict any assets acquired after the bond issuance to be used as collateral only for the existing bonds (and not new bond issues), thereby safeguarding the interests of existing bondholders.
This prevents the issuer from pledging newly acquired assets to secure additional debt, thereby maintaining the value of the collateral backing existing bonds.
Analogy, imagine you lend a friend money to buy a specific car (original collateral). An after-acquired clause would be like saying they can't use any future cars they buy (after-acquired assets) as collateral for other loans without your permission.
In summary, while both provisions aim to protect the interests of bondholders, the negative pledge covenant specifically restricts the issuer from using the same pool of assets for additional debt issuances, whereas the after-acquired clause restricts the use of newly acquired assets as collateral for future bonds.
Collateral trust bonds
Collateral trust bonds serve as a form of debt financing where the bonds are backed by various assets such as stocks, notes, bonds, or other similar obligations owned by the issuing company. These underlying assets, referred to as collateral or personal property, provide security for the bondholders in case of default.
Collateral Trust Bond Structure:
Collateral: Backed by a pool of financial assets such as stocks, bonds, or notes owned by the issuing company. These assets act as security (similar to personal property in a traditional secured loan).
Issuer: Typically holding companies, which use claims on their subsidiaries (essentially, IOUs from their own companies) as collateral.
Trustee: A third-party entity that holds the collateral on behalf of the bondholders(and not the shareholders), ensuring their rights are protected.
Issuer Considerations:
Voting Rights:
The issuer might retain voting rights for the stock used as collateral, as long as they are not in default.
This allows them to maintain some control over their subsidiaries.
Indenture Provisions:
The bond indenture may specify actions if the value of the collateral falls below a certain threshold (compared to the loan value).
For example, the issuer may have to contribute additional securities to back the bonds to maintain the collateral's value.
Overall, collateral trust bonds offer a unique way for companies to raise capital by leveraging their existing assets. However, the complex structure involving a trustee and potential limitations on voting rights require careful consideration by both issuers and investors.
Equipment trust certificates (ETCs)
ETCs share some similarities with mortgage bonds but are designed specifically for financing specific equipment.
Similarities to Mortgage Bonds:
Secured Debt:
Both ETCs and mortgage bonds are secured by a specific asset (equipment for ETCs, real estate for mortgages).
This collateral provides security to investors in case of default.
Pass-through Structure:
Similar to mortgage bonds, the proceeds from leasing the equipment (or property payments in mortgages) are passed through to the investors who hold the ETCs.
Differences from Mortgage Bonds:
Underlying Asset:
ETCs are backed by a single piece of equipment, while mortgage bonds are backed by a pool of mortgages.
Ownership Structure:
The usual arrangement is that the borrower does not actually purchase the equipment. Instead, the trustee purchases the equipment and leases it to the user of the equipment (the effective borrower), who pays rent on the equipment, and that rent is passed through to the holders of the ETCs.
With mortgage bonds, the borrower typically already owns the property and grants a lien on it.
Transfer of Title:
Upon full payment, the title for the equipment is transferred to the lessee (borrower) in an ETC.
This rarely happens with mortgage bonds, as the homeowner usually keeps ownership after the mortgage is paid off.
Resale Potential:
It is especially attractive if the equipment is standardized, like aircraft financed through ETCs can be easily leased to another borrower if needed.
Mortgaged properties are less fungible and resale may take longer.
Additional Points:
Tax Benefits:
ETCs can offer tax advantages, particularly in North America.
Since the lessee doesn't own the equipment initially, they may not have to pay property taxes on it until the lease is complete.
Focus on Specific Industries:
ETCs are commonly used in industries where equipment is a significant expense, such as airlines financing airplanes or railroads financing locomotives.
Overall, ETCs offer a unique financing option for companies that require specific equipment. The secured nature and potential for resale of standardized equipment make them an attractive option for both investors and borrowers.
Debentures:
Debentures are unsecured bonds, meaning they are not backed by specific collateral.
Because of this lack of security, debentures generally carry higher interest rates compared to secured bonds, like mortgage bonds or collateral trust bonds.
In the event of default, debenture holders rank below secured bondholders in terms of priority for repayment.
To manage risk and protect the interests of debenture holders, certain provisions are often included in debenture agreements:
Restriction on Additional Issues:
If the issuer already has secured debt, there may be restrictions on issuing additional debentures.
This restriction helps maintain the integrity of the issuer's debt structure and prevents overleveraging.
Negative Pledge Clause:
In cases where there is no existing secured debt, a negative pledge clause may be included in the debenture agreement.
This clause stipulates that if the company issues secured bonds in the future, the debentures will be secured equally with the newly issued secured bonds.
Essentially, it ensures that debenture holders are not disadvantaged if the issuer decides to secure future debt.
These provisions provide a degree of protection for debenture holders, helping to mitigate some of the risks associated with investing in unsecured debt.
Subordinated debenture bonds:
Subordinated debenture bonds:
Low Ranking in Default:
Subordinated debenture bonds have a claim that is at the bottom of the list of creditors if the issuer goes into default.
If a company defaults, subordinated debenture holders are only paid after all senior debt holders (including other unsecured bonds with higher claims) are satisfied.
This makes them riskier for investors.
Unsecured and Higher Interest:
Since they are unsecured by any collateral and have another unsecured bond with a higher claim above them.
This means that the issuer has to offer a higher interest rate on the subordinated debentures as compensation for the additional risk.
Guaranteed Bonds:
Guarantor's Backing:
These bonds come with a guarantee from another entity, often a parent company or a stronger financial institution.
This promises to fulfill the debt obligation if the original issuer fails.
Not Risk-Free:
The guarantee itself isn't a guarantee of eliminating default risk.
The issuing entity's ability to meet the obligation ultimately depends on the guarantor's financial health.
Correlation Impact:
The value of the guarantee is influenced by the correlation between the issuer's and guarantor's profitability.
A negative correlation (where one goes up as the other goes down) strengthens the guarantee's value. Conversely, a positive correlation weakens it.
Analogy:
Subordinated Debenture Bonds: Imagine you're a lender and give out small personal loans to friends. A subordinated debenture bond would be like loaning money to a friend with a shaky credit history. You charge a higher interest rate to compensate for the higher risk of not getting repaid.
Guaranteed Bonds: It would be like your friend's parent guaranteeing the loan. This adds some security, but only if the parent has good financial standing. If both your friend and friend's parent lose your jobs (positively correlated), the guarantee becomes less valuable.
Overall, subordinated debenture bonds and guaranteed bonds offer different risk-return profiles for investors. Subordinated debentures provide higher returns but come with significant default risk. Guaranteed bonds can offer some comfort, but the guarantor's creditworthiness is a crucial factor.

Methods for Retiring Bonds

Bond indentures often outline various methods for retiring debt, each serving different purposes and offering different mechanisms for repayment.
Some are included in the bond’s indenture while others are not included.
The indenture would not include fixed-spread tender offers.
The indenture would include the call provisions, sinking funds, maintenance and replacement funds, and redemption through sale of assets.
Call Provisions:

Call provisions allow the issuer right to buy back (redeem) the bonds before their maturity date, typically at a predetermined price (often face value) known as the call fixed price either in whole or in part.
This gives issuers flexibility in managing their debt obligations based on prevailing market conditions. Especially if interest rates have fallen since the bonds were issued, as they can refinance the debt at a lower cost.
Benefits for Issuers:

Reduce Interest Costs:

Call provisions allow companies to call back high-coupon debt and reissue new debt with a lower coupon rate if interest rates fall.
This saves the company money on interest payments ultimately increasing shareholder value.

Improve Financial Flexibility:

Call provisions offer flexibility by allowing the issuer to adjust their debt structure based on market conditions.
They can react to falling interest rates or changing financial needs by calling back existing debt.

Alter Capital Structure (Indirectly):

While not directly changing the capital structure, calling back debt allows the issuer to potentially reissue new debt with different terms, which can indirectly affect the debt-to-equity ratio.

Eliminate Restrictive Covenants (Limited Impact):

When a company issues bonds or takes out loans, lenders often impose restrictions known as restrictive covenants.
These covenants outline certain actions the company can or cannot take, aiming to protect the lender's interests.
However, there are instances where a company might want to eliminate these restrictions, as they could limit its flexibility in conducting business.
One way to potentially eliminate these restrictive covenants is through call provisions, which are clauses in bond contracts that allow the issuer to redeem or "call back" the bonds before their maturity date.
Sometimes, call provisions might be structured to trigger the removal of specific restrictive covenants upon exercising the call option.
However, this isn't always the case. It's important to note that while call provisions can provide an opportunity to eliminate restrictive covenants, it's not a guaranteed outcome.
Additionally, restrictive covenants are more commonly associated with loan agreements rather than bond contracts.
So, even if call provisions are exercised, it may not necessarily result in the removal of these restrictions, especially if they are tied to loan agreements rather than bonds.

Call Provision Types:

Fixed-Price Call:

A fixed-price call provision is a type of call provision found in bond contracts that allows the issuer to redeem the bonds at a predetermined specific prices that can vary over the life of the bonds, regardless of prevailing market conditions.
This price is predetermined and does not change, hence the term "fixed-price."
The fixed price at which the bonds can be called back is usually set at a premium to the bond's face value. This premium compensates bondholders for the early redemption of their bonds.
Fixed-price call provisions offer issuers flexibility and control over their debt obligations.
They allow issuers to redeem bonds if it becomes advantageous for them to do so, such as when interest rates decline or if they want to eliminate debt ahead of schedule.
For bondholders, fixed-price call provisions introduce a degree of uncertainty, as they may have to reinvest the proceeds from the redeemed bonds at potentially lower interest rates.
Variable call prices: The call price, typically starting high and declining towards the face value (par value), is specified in the bond indenture, a legal document outlining the terms of the bond issuance.
Call protection period: Most bonds do have a call protection period in the initial years, preventing the issuer from calling them back right away. This protects investors from early redemption and ensures they receive the promised interest payments for a certain timeframe.

Overall, fixed-price call provisions are a tool used by bond issuers to manage their debt effectively, providing them with the option to retire bonds early under specific conditions.

Make-Whole Call:

A "make-whole call" provision is a type of call provision found in some bond contracts. Unlike traditional fixed-price call provisions where the redemption price remains constant, a make-whole call provision calculates the call price based on current market conditions.
Determining the Call Price: The call price under a make-whole call provision is calculated as the present value of the bond's remaining cash flows. This means taking into account all future coupon payments and the bond's principal repayment. The present value is calculated using a discount rate based on the yield of comparable-maturity Treasury securities, commonly referred to as the Treasury yield.
Floor Price: The call price cannot fall below a certain threshold, known as the floor price. In this case, the floor price is set equal to the bond's par value. This ensures that bondholders will receive at least the par value of their bonds if the issuer decides to call them back.
Discount Rate: The discount rate used to calculate the present value typically consists of the yield of comparable-maturity Treasury securities plus a premium. This premium accounts for additional risk factors associated with the bond, such as credit risk or liquidity risk.
Market Rates: Since the call price is determined based on current market rates, it can fluctuate over time as interest rates change. If market interest rates rise, the present value of the bond's future cash flows decreases, resulting in a higher call price. Conversely, if market interest rates fall, the present value increases, leading to a lower call price.
Purpose: The purpose of a make-whole call provision is to compensate bondholders for the early redemption of their bonds by ensuring that they receive fair value based on prevailing market rates. By using a make-whole call provision, issuers can redeem bonds early without unduly disadvantaging bondholders.

Conversion or Convertible Bonds:

An alternate form for bond retirement is to allow the bonds to be converted to common shares at a predetermined rate.
Conversion Option: This refers to a feature in some bonds that allows bondholders to convert their bonds into a predetermined number of common shares of the issuing company. For example, a bond may offer the option to convert each $1,000 bond into 100 shares of common stock.
Potential Benefit to the Issuer (Call Option): The call option benefits the issuer because it allows them to retire the bonds early if it becomes advantageous for them to do so, such as if interest rates decrease or if they want to eliminate debt ahead of schedule.
Potential Benefit to the Investor (Conversion Option): The conversion option benefits the investor because it provides the opportunity to convert bonds into common shares if the stock price rises above a certain level, potentially allowing them to participate in any increase in the value of the company's stock.
The fact that a call option (potential benefit to the issuer) is often combined with the conversion option (potential benefit to investor) may incentivize the investor to exercise the conversion option earlier (e.g., before the stock price has risen too much).
Incentive for Early Conversion: When a bond offers both a call option and a conversion option, it creates an incentive for bondholders to exercise the conversion option earlier, especially if they believe the stock price will rise further. This is because if the issuer sees a significant increase in the stock price, they may be more likely to call the bonds to avoid issuing shares at a lower predetermined price through conversion.
Issuer's Perspective: From the issuer's perspective, if the stock price rises significantly, it may become more expensive for them to issue shares through conversion than to call the bonds at the predetermined call price. In such a scenario, the issuer is more likely to call the bonds to avoid the potential dilution of issuing shares at a relatively low predetermined price.

Sinking Funds:

Sinking funds require the issuer to set aside a portion of funds regularly to retire the bond principal gradually.
This ensures that funds are available for repayment at maturity and can provide investors with greater confidence in the issuer's ability to meet its obligations.
The bonds can either be retired by use of a lottery where the owners of the selected bonds must redeem them, or the bonds are purchased in the open market.

Bond call lottery:

This method involves randomly selecting a certain number of bonds for repurchase. Bondholders whose bonds are chosen are obligated to sell them back to the company at the call price.
For bondholders: If your bond gets picked in the lottery, you must sell it back to the company at the predetermined call price (usually the face value).
For the company: This method is advantageous because it ensures a specific number of bonds are retired. It can be a good option if they have a limited amount of money available for buybacks.

Open market repurchase:

Here, the company goes into the open market and buys back its own bonds from willing sellers.
The price is negotiated and may be at a premium depending on market conditions. Think of it as the company going shopping for its own bonds in the open market.
This method offers more flexibility than a lottery but can be more expensive depending on market conditions.

A sinking fund can be used to fund either a bond call lottery or open market repurchases, depending on the terms of the bond issuance. The issuer decides how they will use the accumulated funds to retire the bonds.
Bond call lotteries and open market repurchases are specific methods for executing a call provision on a bond. A sinking fund doesn't directly trigger a call provision, but it provides the financial resources to do so.
Sinking-fund provisions also make sense when the value of the collateral goes down with time; therefore, the provisions would reduce the borrowings at the same time. Alternatively, if it is desired not to reduce the borrowing levels, then additional collateral can be provided to offset the potential decline in the value of existing collateral.

Sinking-Fund Provisions and Collateral Value: Imagine a scenario where a company has issued bonds backed by collateral, such as real estate or other assets. Over time, the value of this collateral may decrease due to various factors like depreciation, market fluctuations, or wear and tear.
Reducing Borrowings: When the value of the collateral decreases, it affects the overall financial position of the company. Sinking-fund provisions come into play here as they provide a structured way for the company to retire or pay off a portion of its debt each year. By using the sinking fund to buy back some bonds, the company effectively reduces its borrowings.
Offsetting Decline in Collateral Value: However, if the company wishes to maintain its borrowing levels steady despite the declining collateral value, it has another option. Instead of reducing the debt through the sinking fund, the company can provide additional collateral to compensate for the potential decline in the value of existing collateral.
Maintaining Financial Stability: By either reducing borrowings through the sinking fund or providing additional collateral, the company aims to maintain a stable financial position. This ensures that it meets its obligations to bondholders while also managing any risks associated with the changing value of collateral.

Maintenance and Replacement Funds:

Maintenance and replacement funds are similar to sinking funds but are specifically designated for the maintenance or replacement of certain assets that serve as collateral for the bonds.
This helps ensure that the collateral remains in good condition, thereby protecting the interests of bondholders.
The key differences between sinking funds and maintenance and replacement fund provisions.

Sinking Funds:

Simpler: Generally, sinking funds involve setting aside money periodically to eventually retire debt. The valuation of underlying assets isn't a direct concern.

Maintenance and Replacement Fund Provisions:

More Complex: These provisions require a more proactive approach. The fund must be sufficient to maintain the value of the underlying assets, which often necessitates valuation formulas.

Analogy:

Home Mortgage: Just like a homeowner needs to maintain their property value, a maintenance and replacement fund ensures the underlying assets used to secure debt (like equipment or buildings) don't depreciate excessively.

Fulfilling the Provision:

There are two ways mentioned to satisfy the provision:
Cash Acquisition: The company can accumulate enough cash within the fund to maintain the overall financial health of the firm. This cash can then be used strategically, like retiring debt, which can improve the company's financial standing.
Collateral Sale: The company can sell some of the collateral associated with the debt. However, the proceeds from this sale must typically be used to retire the bonds early, fulfilling the purpose of the maintenance and replacement fund.

Sinking funds and maintenance and replacement funds serve similar goals (debt management), but they differ in complexity and approach. Sinking funds are simpler, focusing on periodic contributions for debt repayment. Maintenance and replacement funds require ongoing efforts to ensure the value of underlying assets is maintained.

Redemption Through Sale of Assets:

Some bond indentures may allow for the redemption of bonds through the sale of specific assets.
This can be particularly relevant for asset-backed securities, where the proceeds from the sale of underlying assets are used to repay bondholders.

Tender Offers:

Tender offers are usually a means for retiring debt for most firms.
The firm openly indicates an interest in buying back a certain dollar amount of bonds or, more often, all of the bonds at a set price.
Firms can also announce that they will buy back bonds at an amount calculated as the present value of future cash lows based on a speciic discount rate (e.g., the yield to maturity on a comparable-maturity Treasury plus a spread).
Fixed-spread tender offers involve the issuer making a tender offer to repurchase bonds at a predetermined spread above a benchmark, such as a government bond yield.
This method is not typically included in the bond indenture and is instead executed through separate tender offer documents.

While call provisions, sinking funds, maintenance and replacement funds, and redemption through asset sales are commonly included in bond indentures to manage debt repayment, fixed-spread tender offers are usually executed through separate procedures outside of the indenture.
Each method offers issuers and investors different avenues for retiring debt and managing bond obligations.

Credit Risk

Credit risk includes credit default risk and credit spread risk.
Credit default risk

Credit default risk is the uncertainty concerning the issuer’s making timely payments of interest and principal as prescribed by the bond’s indenture.
The most widely used indicators of this risk are bond ratings that major rating agencies assign when those agencies perform credit analysis of a firm.
Bond ratings act like credit scores for bonds, assigned by major rating agencies to assess the creditworthiness of a bond issuer (company or government) and the risk of them defaulting on their debt.
Fitch Ratings, Moody’s, and Standard & Poor’s are the main rating agencies in the United States.
The agencies assign a symbol associated with the rating (e.g., AAA or Aaa for the corporate debt with the least credit default risk).
The rating can be interpreted as a probability of default within some time period, as well as the probability of a change in a rating within some time period.

Credit spread risk

Credit spread risk focuses on the difference between a corporate bond’s yield and a yield of risk-free bond (usually government bonds like Treasuries). It reflects the additional risk investors demand for holding a riskier asset (corporate bond) compared to a safe haven (government bond).
This difference is known as the credit spread.
It should be noted that other factors such as embedded options and liquidity factors can affect this spread; therefore, it is not only a function of credit risk.
Credit spread risk increases when the economy deteriorates (e.g., moves through the business cycle).

When the economy weakens, investors generally become more risk-averse.
They seek the safety of government bonds, which are perceived as having a very low chance of default.
This increased demand for Treasuries drives their prices up, pushing their yields down.
In contrast, corporate bonds become less attractive during economic downturns.
Investors perceive a higher risk of default by companies due to factors like:

Lower profits
Increased difficulty repaying debt
Potential for bankruptcies

To compensate for this higher perceived risk, investors demand a higher yield on corporate bonds.
This means the price of corporate bonds falls (as yield and price have an inverse relationship) to reflect the increased risk premium.
As government bond yields go down and corporate bond yields go up, the credit spread widens. This widening reflects the increased risk premium investors demand for holding corporate bonds in a weak economy.

A method commonly used to evaluate credit spread risk is spread duration. The duration of the spread is the approximate percentage change in a bond’s price for a 100- basis-point change in the credit spread assuming that the Treasury rate is constant. If a bond has a spread duration of 4, for example, a 50-basis-point change in the spread will change the value of the bond by 2%.
Spread duration

It's a measurement that estimates the sensitivity of a bond's price to changes in the credit spread. It tells you how much a bond's price is likely to change (as a percentage) for a given change in the difference between a corporate bond's yield and a government bond's yield (credit spread).
Units and Interpretation:

Spread duration is typically expressed in years.
A higher spread duration indicates a greater sensitivity of the bond's price to changes in the credit spread.
Conversely, a lower spread duration indicates a lesser sensitivity of the price to credit spread fluctuations.

The Example:

Spread Duration of 4: This means the bond's price is expected to change by approximately 4% for every 1% change in the credit spread (assuming government bond yields remain constant).
50 Basis Point Change: A 50 basis point change is equivalent to 0.5% (50 divided by 100).
2% Price Change: Given the spread duration of 4, a 0.5% widening of the credit spread would translate to an estimated 2% decrease in the bond's price (4 multiplied by 0.5).

Why is Spread Duration Important?

It helps investors understand how their bond portfolio might react to changes in the credit market. Bonds with higher spread durations are more volatile and can experience larger price swings when the credit spread widens or narrows.
Investors can use spread duration to manage their portfolio's risk profile. By choosing bonds with varying spread durations, they can achieve a balance between potential returns and risk exposure to credit spread fluctuations.

Limitations of Spread Duration:

It assumes a parallel shift in the yield curve, meaning government bond yields remain constant. In reality, the yield curve can also change shape, impacting bond prices.
It doesn't account for other factors that can affect bond prices, such as changes in call provisions or embedded options.

Event Risk

Event risk addresses the adverse consequences from possible events involving signiicant increases in leverage, such as mergers, recapitalizations, restructurings, acquisitions, leveraged buyouts, and share repurchases, which may escape being included in the indenture.
Such events can drastically change the irm’s capital structure and reduce the creditworthiness of the bonds and their value.
In order to protect bondholders, a company may include in the indenture a maintenance of net worth clause that can require the company to maintain a minimum equity level.
If that level is breached, then it must repurchase a suficient amount of its debt at par value to reach the minimum equity level.

High Yield Bonds

High-yield bonds (a.k.a. junk bonds) are those bonds rated below investment grade by ratings agencies.
This includes a broad range of ratings below the cutoff, (e.g., Ba1/BB+ down to default).
Over long periods of time, high-yield bonds should offer higher average returns.
However, over shorter periods, the returns will be volatile where large losses are possible.
Types of high-yield bonds:

Rising stars: These are bonds issued by companies with strong growth prospects but that are not yet considered investment-grade due to their limited track record. Companies who issue bonds with a non-investment-grade rating. Such issuers include the below:

Young and growing companies: would not have strong financial statements but have promising prospects.
Companies with consistent cash flows: These companies may have a solid track record of generating cash flow, but their credit rating might not be investment-grade due to other factors. For example, they might have a high debt burden from previous acquisitions or expansion plans. By issuing high-yield bonds, they can access capital at a lower cost than issuing new equity (stocks). However, the interest payments on these bonds are higher compared to investment-grade bonds.

Fallen angels: These are bonds that were originally issued by investment-grade companies but have since been downgraded to high-yield status due to a deterioration in the issuer's creditworthiness.
Cyclicals: These are bonds issued by companies in industries that are sensitive to economic cycles. The value of these bonds can fluctuate significantly depending on the state of the economy.
Distressed bonds: These are bonds issued by companies that are in financial trouble and may be at risk of default. These bonds offer the highest potential returns but also carry the highest risk.
Emerging market bonds: These are bonds issued by companies in developing countries. These bonds can offer higher yields than developed market high-yield bonds, but they also carry additional risks, such as political instability and currency fluctuations.

Types of coupon structures:

Deferred-coupon bonds, which would sell at a discount and not pay any interest for an initial period and then pay the stated coupon afterward.
Step-up bonds, pay a low coupon in the early years and then a higher coupon in later years.
Payment-in-kind bonds, allow the issuer to pay interest in the form of additional bonds over the initial period.
Extendable reset bonds, allow the issuer to reset the coupon as frequently as needed to keep the bond price at a specified level. This means they can adjust the interest rate paid to bondholders to ensure that the bond's price remains close to a specified level, often its par value. The purpose of this feature is to help maintain the stability of the bond's price. If the bond's price starts to deviate significantly from its specified level, the issuer can reset the coupon rate to bring it back in line. For investors, extendable reset bonds offer a degree of stability in terms of the bond's price, as the issuer can adjust the coupon rate to prevent large fluctuations. However, they may also introduce uncertainty about future interest payments, as the coupon rate can change over time.

Default Rate

A default occurs if there are any missed or delayed disbursements of interest and/or principal.
It has been proven that lower credit ratings indicate a higher probability of default, but there are two ways to measure default:

by the raw number of issuers that defaulted
by the dollar amount of issues that defaulted

For each approach in measuring default rates, there are different formulas, which can lead to researchers reporting different default rates for the same data set.
Issuer default rate

Formula:

number of issuers that defaulted over a year / the total number of issuers at the beginning of the year.

It is only a proportion of the number of issuers who do fulill their obligations and does not include a measure of the dollar amount involved.

Dollar default rate

The dollar default rate is the par value of all bonds that defaulted in a given calendar year divided by the total par value of all bonds outstanding during the year.
Formula:

Over a multiyear period, often-used measures are ratios of cumulative dollar value of all defaulted bonds divided by some weighted-average measure of all bonds issued. One such measure attempts to weight the bonds outstanding by the number of years they are in the market:
Formula:

Recovery Rate

The recovery rate is a crucial concept in bond investing, representing the amount investors receive as a proportion of the total obligation after a bond defaults.
Definition of Recovery Rate:

After a bond defaults, investors may not receive the full amount owed to them by the issuer.
The recovery rate quantifies how much of the defaulted bond's value investors are able to recover.
It's typically expressed as a percentage of the bond's face value or the total amount owed.

Complexity of Measurement:

Determining the recovery rate can be complex for several reasons.
Firstly, it involves calculating the present value of the remaining cash flows from the bond at the time of default.
This requires estimating the future cash flows of the bond and discounting them back to their present value, taking into account factors like the timing and probability of receiving these cash flows.

Form of Recovery:

Additionally, the recovery amount may not always be in the form of cash.
Sometimes, investors may receive securities or assets, such as stock in the defaulting company, as part of the recovery process.
This adds another layer of complexity to measuring the recovery rate, as the value of these securities needs to be accounted for.

Moody's Study Findings:

A study by Moody's, a leading credit rating agency, estimated that the average recovery rate for defaulted bonds has been around 38%.
This means that investors typically recover about 38% of the total obligation owed to them after a bond defaults.
It's important to note that this is an average figure and actual recovery rates can vary widely depending on various factors.

Impact of Seniority:

Bonds with higher seniority in the capital structure of a company, such as senior secured bonds, typically have higher recovery rates.
This is because these bonds have priority claims on the company's assets in the event of default, making them more likely to receive a larger portion of the recovery amount compared to junior or subordinated bonds.

In summary, the recovery rate represents the percentage of the total obligation investors are able to recover after a bond defaults. Measuring the recovery rate involves considering factors such as the form of recovery, the present value of remaining cash flows, and the impact of bond seniority.

Expected Return

Earnnings from the bond.
A bond’s expected return is calculated as:

risk-free rate + credit spread − expected loss rate

risk-free rate: provides the baseline return.
credit spread: credit spread is the difference in yield or interest rate between a bond with credit risk and a risk-free investment, typically a government bond.

It represents the additional compensation that investors demand for bearing the risk of default associated with the bond issuer.
In other words, it's the premium investors require for taking on the credit risk of holding the bond.
Compensation for risk:

When investors purchase bonds, they expect to be compensated for the risks they are taking.
Credit risk is one of these risks.
The credit spread compensates investors for the additional risk of holding a bond compared to a risk-free investment like a government bond.

Expected Loss Rate:

The expected loss rate represents the anticipated loss due to default risk.
It takes into account the probability of default and the potential loss severity in the event of default.
This rate is typically estimated based on historical default data, issuer credit ratings, and other relevant factors.

Relationship between Credit Spread and Expected Loss Rate:

When the credit quality of the issuer is higher (meaning the issuer is considered less likely to default), investors perceive less risk associated with holding the bond.
Therefore, they may demand a smaller credit spread as compensation for this lower risk.
Conversely, when the credit quality of the issuer is lower (indicating a higher likelihood of default), investors perceive greater risk and may require a higher credit spread to compensate for this increased risk.

Study Findings:

The study notes that the excess of the credit spread over the expected loss rate tends to be lower (or higher) when the credit quality of the issuer is higher (or lower).
This observation aligns with the general principle that investors adjust their required compensation (credit spread) based on their assessment of the issuer's credit risk and the expected loss rate associated with holding the bond.

expected loss rate: adjusts for the potential loss due to default risk.

expected loss rate here is equal to: probability of default × (1 − expected recovery rate).

The Treasury rate is a widely used benchmark, but it's not always the most appropriate risk-free rate for corporate bonds and a higher rate such as the interbank borrowing rate may be appropriate.

Limitations of Treasury Rate for Corporate Bonds:

While the Treasury rate is a good benchmark for government bonds, it might not be entirely suitable for corporate bonds for a few reasons:
Default Risk: Corporate bonds carry default risk, meaning there's a chance the issuer (company) may not be able to repay the loan. Treasury bonds, on the other hand, are considered risk-free.
Liquidity: Treasury bonds are generally highly liquid, meaning they can be easily bought and sold in the market. Corporate bonds, especially those issued by smaller companies, may be less liquid.

Interbank Borrowing Rate:

The interbank borrowing rate, such as the London Interbank Offered Rate (LIBOR) or the Overnight Indexed Swap (OIS) rate, reflects the interest rate at which banks lend to each other in the interbank market.
This rate is typically higher than the Treasury rate and is influenced by various factors, including credit risk and liquidity risk.

Appropriateness for Corporate Bonds:

Using a higher rate like the interbank borrowing rate as the risk-free rate for corporate bonds may be more appropriate because it better reflects the additional risk inherent in these bonds.
Since the interbank borrowing rate incorporates credit risk premiums, it provides a more accurate measure of the opportunity cost of investing in corporate bonds compared to investing in risk-free assets.

Market-Based Approach:

Some argue that using market-based rates such as interbank borrowing rates aligns better with the principle of opportunity cost, as it reflects the rates at which investors can earn a return in the market adjusted for risk.
This approach acknowledges that investors require a higher return for bearing credit risk when investing in corporate bonds compared to risk-free assets.

Regardless of the risk-free measure, in calculating expected return, investors in corporate bonds expect to earn more than the risk-free rate.

Credits and References

https://www.fisdom.com/wp-content/uploads/2021/07/39-1.jpg

https://gemini.google.com/

https://chat.openai.com/

SchweserNotes and BionicTurtle Notes

#Reading43

HelloWorldEngineer

Thursday, 14 March 2024

Pricing Conventions, Discounting, and Arbitrage

Introduction

Bonds Vs Securities

Fundamentals of Bond Valuation

Price Yield Curve

Discount Rate vs Yield To Maturity

Valuation With a Single Yield (Discount Rate)

Valuation With a Single Yield and Semiannual Cash Flows

Bond Components

Pricing Conventions Between Coupon Dates

Discount Factors for Treasury Bills

Discount Factors for Treasury Bonds

Determining Value Using Discount Functions - Law of one price

Constructing a Replicating Portfolio

Credits and References

Thursday, 22 February 2024

Corporate Bonds

Introduction

Bond Trading

Bond Indenture and Corporate Trustee

Bond Issuers

Bond Maturities

Bonds Types basis on Interest Payment

Bond Types based on Collateral

Methods for Retiring Bonds

Credit Risk

Event Risk

High Yield Bonds

Default Rate

Recovery Rate

Expected Return

Credits and References

Scarcity Brings Efficiency: Python RAM Optimization