Properties of Interest Rates

 

Types of Interest Rates

• Interest rates increase as the credit risk of the underlying instrument increases.
• Credit risk: This is the chance that a borrower won't be able to repay a loan. The higher the risk, the more likely it is that the borrower will default.  
• Interest rate: This is the price you pay to borrow money. It's essentially the fee the lender charges for taking on the risk of lending you money.
• So, when the credit risk of a loan or investment increases (becomes riskier for the lender), lenders typically respond by raising the interest rate. 
• This makes the loan more expensive for the borrower, but it also compensates the lender for the greater chance of not getting their money back.
• Here's an analogy: imagine lending money to a friend. If it's your best friend with a steady job, you might be happy to lend them money at a low interest rate. But if it's someone you barely know with a history of financial trouble, you'd probably charge them a higher interest rate to account for the greater risk.
• It's important to note that interest rates are also influenced by other factors, like overall economic conditions and monetary policy set by central banks. But credit risk is definitely a major player.
• Treasury Rate
• Bench mark rate or Treasury rate is generally considered to be risk free rate at government of country borrow in its own currency.
• LIBOR
• London Interbank Offer Rate - LIBOR
• It is based upon the estimations of the certain financial instituitions though larger but handful, which subjected it to potential manipulation.
• They could be biased as they determine the borrowing and lending rates.
• And this is the reason of why LIBOR is being phased out in the Mid of 2023.
• SOFR
• The Secured Overnight Financing Rate (SOFR) is a one-day, repo-based rate that is derived from actual transactions. 
• It is one of the proposed replacements for LIBOR.
• Repo
• The “repo” or repurchase agreement rate is the implied rate on a repurchase agreement.
• In a repo agreement, one party agrees to sell a security to another with the understanding that the selling party will buy it back later at a speciied higher price.
• The interest rate implied by the price differential is the repo rate.
• Suppose there are two banks, Borrower A and Lender B
• Borrower A gives its securities of value $90 to Lender B inorder to get loan of $90.
• Borrower A promises to Lender B to repurchase securities in future @$100.
• To arrive interest rate using repo rate
• Borrower A took loan of $90 gave the securities worth of $90
• Now Borrower A pay $100 and get back the securities
• Rate of Interest or Rate of Return = 100 / 90
• The most common repo is the overnight repurchase agreement.
• Overnight Rate
• The overnight rate is the rate at which large financial institutions borrow from each other in the overnight market, without security.
• In US it is called as Federal Fund Rate and is monitored and influenced by the central bank. 
• If a financial institution borrows (lends) funds at the overnight rate, the rate it pays (earns) during the period is the weighted average of the overnight rates.

• The Bloomberg Short-Term Bank Yield Index (BSBY) and overnight-based reference rates are both used to gauge short-term borrowing costs, but they operate differently and serve distinct purposes. Here’s a comparison of the two:

  Bloomberg Short-Term Bank Yield Index (BSBY)

  1. Nature: BSBY is a forward-looking term rate that reflects the average expected bank borrowing cost over a specified term. It's designed to provide a benchmark for short-term borrowing and lending rates.

  2. Calculation: BSBY is based on a panel of contributing banks' unsecured borrowing costs, which are estimated for various maturities (e.g., 1 month, 3 months). It incorporates market expectations and is intended to reflect current borrowing conditions.

  3. Use Case: BSBY is useful for financial products that need a forward-looking reference rate. It’s commonly used in derivative contracts, loans, and other financial instruments that are priced off term rates.

  4. Transparency: As a term rate, BSBY provides a clear picture of expected borrowing costs over future periods, offering more visibility into the cost of credit over time.

  Overnight-Based Reference Rates

  1. Nature: Overnight-based reference rates, such as the Secured Overnight Financing Rate (SOFR) or the Euro Short-Term Rate (€STR), are backward-looking and reflect the average rate at which banks borrow overnight, usually secured by collateral.

  2. Calculation: These rates are based on actual transactions or survey data for overnight borrowing and are updated daily. They capture the cost of short-term borrowing on a very granular, day-to-day basis.

  3. Use Case: Overnight rates are commonly used for products that require a daily, real-time reference. They are especially relevant for markets and products where precision and current data are critical.

  4. Transparency: They offer a very precise view of the cost of borrowing on an overnight basis, making them useful for products where short-term, accurate rates are necessary.

  Key Differences

  • Term vs. Overnight: BSBY provides rates for various terms (e.g., 1 month, 3 months), while overnight rates are specific to borrowing costs over a single day.
  • Forward-Looking vs. Backward-Looking: BSBY is forward-looking, reflecting expectations for future borrowing costs, whereas overnight rates reflect actual borrowing costs from the previous day.
  • Application: BSBY is used for financial instruments that need a term structure of rates, while overnight rates are more suitable for products requiring precise daily rates.

  In summary, BSBY and overnight-based reference rates serve different needs in the financial markets, with BSBY providing term structure and forward-looking insights, and overnight rates offering precision and current data.

• Longer-term agreements are called term repos.
• Depending on the parties and structure involved, there is some credit risk with repurchase agreements.
• OIS
• Overnight Index Swap, is an interest rate swap is an exchange contract generally fixed vs floating.
• Fixed vs. Floating Rate Swap: 
• In an OIS, one party agrees to pay a fixed interest rate (the OIS rate) for a certain period.
• The other party agrees to pay a floating rate based on the geometric average of overnight interest rates (typically federal funds rate) over the same period.
• Geometric Average:
• The floating rate is calculated as the geometric average, not the arithmetic average, of the daily overnight rates.
• This means it considers the compounding effect of interest rates over the period.
• Payment Determination: 
• The party that agreed to the fixed rate makes a payment if the OIS rate is higher than the geometric average.
• Conversely, the floating rate party pays if the geometric average is higher.
• This essentially determines who benefits from changes in short-term interest rates during the OIS term.
• Risk Management: 
• OIS are commonly used for managing interest rate risk. 
• By locking in a fixed rate, one party can protect themselves from rising interest rates, while the other party can benefit if rates go down.  
• Market Benchmark: 
• OIS rates are often seen as a benchmark for short-term interest rate expectations. 
• The spread between OIS rates and other rates, like LIBOR, can indicate market sentiment about future interest rate movements.  
• Participants: 
• OIS are primarily traded between banks and other financial institutions.
• However, they can also be used by asset managers and hedge funds for interest rate speculation.
• Treasury rates, such as those for T-bills (short-term) and T-bonds (long-term), are frequently regarded as benchmarks for nominal risk-free rates in financial markets. These rates are considered risk-free because they are backed by the full faith and credit of the government issuing them (in this case, the US government). However, there are nuances in how these rates are perceived by different market participants.
• Derivative traders, in particular, often find Treasury rates to be lower than what they consider truly risk-free. This perception arises because demand for Treasuries is not solely driven by market forces but also by regulatory requirements. For instance, banks and financial institutions often hold Treasuries as part of their regulatory capital requirements or liquidity buffers. This regulatory demand creates artificial buying pressure for Treasuries, driving their prices up and their yields (or rates) down.
• Example Scenario:
• Let's consider the current market conditions where:
• T-bill Rate: 0.1% (annualized rate for a 3-month T-bill)
• OIS Rate: 0.5% (overnight indexed swap rate for the same period)
• In this scenario:
• T-bill Rate (0.1%):
• This rate is considered the risk-free rate for short-term borrowing or lending in theory, as it reflects the yield on a short-term US Treasury security. 
• However, derivative traders might argue that this rate is artificially low due to the regulatory demand for Treasuries. 
• They believe this rate does not adequately reflect the true opportunity cost of capital in the market because it is influenced by factors other than purely market supply and demand dynamics.
• OIS Rate (0.5%):
• The overnight indexed swap rate represents the market's expectation for the overnight rate over a specified period (like 3 months). 
• Unlike Treasury rates, OIS rates are influenced primarily by market forces and are less affected by regulatory demand for Treasuries. 
• Derivative traders often prefer to use OIS rates as a proxy for the risk-free rate in short-term derivative pricing because they believe OIS rates better reflect the true cost of capital in the market without distortions caused by regulatory requirements.
• Why OIS Rates?
• Reflecting Opportunity Cost: 
• OIS rates are seen as reflecting a trader's true opportunity cost of capital because they are based on actual market transactions rather than regulatory-driven demand.
• Market Dynamics:
• Traders use OIS rates to price derivatives because they believe these rates are more indicative of the market's consensus on short-term risk-free rates, accounting for supply and demand dynamics in the interbank lending market.
• In summary, while Treasury rates are considered nominal risk-free rates due to government backing, derivative traders often prefer to use OIS rates as a more accurate reflection of the true risk-free rate for short-term transactions. This preference stems from the belief that OIS rates better capture the opportunity cost of capital in the market, considering market dynamics rather than regulatory influences on Treasury prices.

Compounding Frequencies

• If we have an initial investment of A that earns an annual rate R, compounded m times a year for n years, then it has a future value of:
• FV1 = A ( 1 + R / m ) ^ m*n
• If our same investment is continuously compounded over that period, it has a future value of:
• FV2 = Ae ^ R*n
• For any rate, R, FV2 will always be greater than FV1. The difference will decrease as m increases. In fact, as m becomes ininitely large, the difference goes to zero.
• In most circumstances, rates are discretely compounded, so we need to use the continuously compounded rate that gives the same future value. Using the previous two equations, the goal is to solve the following:
• A ( 1 + R / m ) ^ m*n = Ae ^ Rc*n
• R = discreate compounded rate
• Rc = continuously compounded rate
•  We can solve for Rc as:
• Rc = ln((1 + R/m)^m)
• We can solve for R as:
• R = m ((e^Rc/m) -1)
• EXAMPLE: Computing continuous rates 
• Suppose we have a 5% rate that is compounded semiannually. Compute the corresponding continuous rate. Repeat this for quarterly, monthly, weekly, and daily compounding.
• Semi annually
• Rc = 2ln ( 1 + 0.05/2 ) = 0.049385
• Quarterly
• m = 4
• Rc = 4ln ( 1 + 0.05/4 ) = 0.049690
• Monthly
• m = 12
• R = 12ln ( 1 + 0.05/12 ) = 0.049896
• Weekly
• m = 52
• R = 52ln ( 1 + 0.05/52 ) = 0.049976
• Daily
• m = 365
• R = 365ln ( 1 + 0.05/365 ) = 0.049995
• Notice that as m increases, the difference between the rates decreases.
• EXAMPLE: Discrete compounding rate 
• A loan is quoted at 12% annually with continuous compounding. Interest is paid monthly. Calculate the equivalent rate with monthly compounding.
• R = 12 (e^0.12/12 -1) = 12.06%
• EXAMPLE: 
• What is the continuously compounded rate of return for an investment that has a value today of $86.50 and will have a future value of $100 in one year?
• FV = Ae ^ R*n
• 100 = 86.50 * e ^ R*1
• 100 / 86.50 = e^R
• 1.156069 = e^R
• ln(1.156069) = R
• 14.50% = R

Spot Rates

• Spot rates are the rates that correspond to zero-coupon bond yields. 
• They are the appropriate discount rates for a single cash low at a particular future time or maturity. 
• Spot rates are also often called zero rates. 
• Most interest rates that are observed in the market, such as coupon bond yields, are not spot rates.

Bond Pricing

• A coupon bond makes a series of cash flows. 
• Each cash low considered in isolation is equivalent to a zero-coupon bond. 
• Using this interpretation, a coupon bond is a series of zero-coupon bonds.
• Formula for Non Continous:
• PV =  (( CR / t ) / (1 + (r1/t)^1 ) + (( CR / t ) / (1 + (r2/t)^2 ) + .. + (( CR / t ) / (1 + (rn/t)^n )
• CR = Coupon Rate
• t = Frequency eg: Semiannually, Annually, Quarterly
• r = Bond equivalent spot rate that corresponds to n periods
• n = Maturity in years
• Formula for Continous:
• PV =  (( CR / t ) e^-(r1/t)*1 ) + (( CR / t ) e^-(r2/t)*2 ) + .. + (( CR / t ) e^-(rn/t)*n )
• Notice that the two discounting approaches will produce a similar result.

Bond Yield

• Bond yield is a return an investor expects to receive on their investment in a bond. It essentially reflects the annualized interest you'll earn on a bond if you hold it until maturity (when the principal amount is repaid).
• The yield of a bond is the single discount rate determined based on its current market price and the present value of its future cash flows (coupon payments and principal repayment).
• Yield of a Bond:
• The yield of a bond, often referred to as the yield to maturity (YTM), is the total return an investor can expect to earn if the bond is held until maturity.
• It represents the annualized return on investment considering both the periodic coupon payments and any gain or loss upon maturity if the bond is purchased at its current market price.
• Single Discount Rate:
• The "single discount rate" mentioned in the line is the yield to maturity (YTM).
• It is the discount rate that, when applied to all future cash flows (coupon payments and principal repayment), equates their present value to the current market price of the bond.
• In other words, it's the rate at which the sum of the present values of all future cash flows equals the bond's current price.
• Equates the Present Value:
• The present value of a bond's cash flows is calculated by discounting each cash flow (coupon payments and principal repayment) at the yield to maturity. 
• When you discount all these future cash flows at the yield to maturity, the sum of these present values should equal the current market price of the bond. 
• Implication:
• This relationship is crucial in bond pricing and valuation. 
• If the bond is priced lower than its face value (at a discount), the yield to maturity will be higher than the coupon rate because investors will earn more on their initial investment due to the bond's appreciation to par value at maturity. 
• Conversely, if the bond is priced higher than its face value (at a premium), the yield to maturity will be lower than the coupon rate because investors will receive less than they paid at maturity.
• Example:
• Compute the yield for the bond.
• FV = $100
• N = 4 which is 2 years
• PV = -102.14
• CR = 4% semiannual
• PMT = 2 which is calculated from FV and CR and its semiannaul hence div by 2 = 100 * 4% / 2 
• Answer
• CPT using calculator -> I/Y = 1.446%
• YTM = 1.446% * 2 = 2.89%
• The bond’s par yield is the rate that makes the price of a bond equal to its par value. When the bond is trading at par, the coupon will be equal to the bond’s yield.
• Bond's Par Yield:  
• The par yield of a bond is the coupon rate (annual interest rate) that makes the bond's price equal to its par value. 
• Par value, also known as face value, is the nominal value of a bond that is typically repaid to the bondholder at maturity. 
• When the bond's price in the market is exactly equal to its par value, the coupon rate (par yield) is the rate at which the annual coupon payments (interest payments) are exactly equal to the interest yield that investors receive based on the bond's current market price. 
• Bond Trading at Par:
• When a bond is trading at par, it means the market price of the bond equals its par value. 
• For example, if a bond has a par value of $1,000 and is trading at $1,000, it is trading at par. 
• In this scenario, the coupon rate (par yield) is the same as the bond's current yield, which is the annual coupon payment divided by the bond's current market price (expressed as a percentage). 
• Implication:
• When the bond trades at par, the coupon rate (par yield) determines the rate of return for investors who buy the bond at that price.
• The coupon payments received by the investor over the bond's life, when discounted at the bond's yield to maturity (YTM), will exactly equal the bond's current market price. 
• Investors who purchase the bond at par will receive coupon payments that match the yield implied by the bond's market price, making the bond's total return consistent with its coupon rate when it is trading at par. 
• In summary, the statement explains that the par yield of a bond is the coupon rate that aligns its market price with its par value.
• When the bond is trading at par, the coupon rate is exactly equal to the bond's yield, ensuring that the bond's price reflects its nominal value and the investor's return matches the coupon payments received.

Bootstrapping Spot Rates

• Bootstrapping spot rates refers to a method used in finance to derive the zero-coupon yield curve from the prices of fixed-income securities, such as bonds or swaps, with varying maturities. 
• Steps in Bootstrapping Spot Rates:
• Understanding Spot Rates:
• Spot rates (or zero-coupon rates) are the interest rates for a specific maturity date, which can be derived from the prices of bonds that provide cash flows at various points in time.
• Starting Point:
• Begin with the prices of bonds or other fixed-income securities available in the market. 
• These securities will have different maturity dates and corresponding market prices.
• Identifying Cash Flows:
• For each bond or security, identify the cash flows it promises over its lifetime.
• This typically includes periodic coupon payments and the principal repayment at maturity.
• Reverse Engineering:
• To bootstrap the spot rates, work backward from the securities with the shortest maturities to those with longer maturities.
• Start with the shortest maturity instrument, often a cash deposit or a very short-term bond, which effectively has only one cash flow (the principal and possibly a single coupon payment).
• Calculation Process:
• Calculate the spot rate for the first maturity (the shortest) by solving for the rate that equates the present value of its cash flows to its market price.
• This rate is often referred to as the zero-coupon rate for that maturity.
• Use this spot rate to discount the cash flows of the next longer maturity instrument (which typically has more than one cash flow). This will provide the implied spot rate for the next maturity.
• Continue this process iteratively, using each newly derived spot rate to value the cash flows of the next longer maturity instrument.
• Iterative Adjustment:
• Each step involves adjusting the spot rate until the present value of all cash flows matches the observed market price of the bond.
• This iterative process ensures that the spot rates derived are consistent with the market prices of the bonds being used as inputs.
• Yield Curve Construction:
• After bootstrapping all spot rates for various maturities, you obtain a yield curve that plots these spot rates against their respective maturities.
• This yield curve is crucial in finance for pricing other financial instruments, such as swaps, futures contracts, and options, as well as for making investment decisions and risk management.
• In conclusion, bootstrapping spot rates is a fundamental technique in finance for deriving the term structure of interest rates from market prices of bonds and other fixed-income securities, enabling precise valuation and risk assessment in financial markets.

FR and FRA

Forward Rate

• Forward rates are interest rates implied by the spot curve for a speciied future period. 
• Recall that spot rates are the appropriate rates that an investor should expect to realize for various maturities.
• Suppose an investor is faced with the following two investments, which are based on the spot curve.   
• Invest for two years at 2.915%.
• Invest for a year at 2.136%, and then roll over that investment for another year at the forward rate.
• It does not matter which investment is chosen if they both offer the same return at the end of two years. 
• This is the same as stating that both strategies give the same future value at the end of two years. 
• Formula : Equating the two future values
• e^( (r2/t)*(n2*t) ) = (e^( (r1/t)*(n2*t) )) * (e^FR/t*n)
• r1 and r2 = spot rate of years 1 and 2 respectively
• FR = forward rate
• n = number of years
• t = frequence
• Formula : Calculate FR by using the following equation (which assumes continuously compounded rates)
• FR = R2T2 - R1T1 / T2 - T1

Forward Rate Agreements

• A forward rate agreement (FRA) is a forward contract obligating two parties to agree that a certain interest rate will apply to a principal amount during a specified future time. 
• Obviously, forward rates play a crucial role in the valuation of FRAs. 
• The T2 cash flow of an FRA that promises the receipt or payment of Rk is:
• cash flow (if receiving Rk) = L * (Rk - R) * (T2 - T1)
• cash flow (if paying Rk) = L * (R - Rk) * (T2 - T1)
• here:
• L = principal
• Rk = annualized fixed rate, expressed with compounding period T2 - T1
• R = annualized floating rate, expressed with compounding period T2 - T1
• Ti = time i, expressed in years
• Example:
• Suppose an investor has entered into an FRA where he has contracted to pay a fixed rate of 3% on $1 million based on the quarterly rate in three months. Assume that rates are compounded quarterly. Compute the payoff from the FRA if the quarterly rate is 1% in three months.
• cash flow (if paying Rk) = L * (R - Rk) * (T2 - T1)
• Rk = 3%
• L = $1million
• R = 1%
• $1,000,000(0.01 - 0.03)(.25)
• For this FRA, the payoff will take place in six months. The net payoff will be the difference between the fixed-rate payment and the floating rate receipt. If the floating rate is 1% in three months, the payoff at the end of the sixth month will be $5000.
• The value of an FRA if receiving or paying the fixed interest rate is:

• Example:
• Suppose the three-month and six-month floating rates are 4% and 5%, respectively (continuously compounded rates). An investor enters into an FRA in which she will receive 8% (assuming quarterly compounding) on a principal of $5,000,000 between Months 3 and 6. Calculate the payoff from the FRA.
• 3 months quarter floating rates (continously compounded rates) = 4%
• 6 months quarter floating rates (continously compounded rates) = 5%
• L = $5,000,000
• Fixed rates (quarterly compounding) = 8%
• FR = 0.05 + (0.05 - 0.04) * (1/ 2 -1) = 0.06 = 6%
• FR (quarterly compounding) = 4 * ((e ^ 0.06/4) - 1) = 0.060452 = 6.05%
• payoff = 
• $51,000,000(0.08 - 0.060452)(.50 - .25) / 1 + 0.05 * (0.50 - 0.25)
• $24,074

Term Structure Theory

• Market segmentation theory
• The market segmentation theory states that the bond market is segmented into different maturity sectors and that supply and demand for bonds in each maturity range dictate rates in that maturity range. 
• The market segmentation theory does not fully make sense because many investors are more likely to move between the maturity sector based on the attractiveness of the available yields.
• Expectations theory
• The expectations theory suggests that forward rates correspond to expected future spot rates. 
• That is, forward rates are good predictors of expected future spot rates. 
• An expectation of rising (falling) interest rates would suggest an upward-sloping (downward-sloping) yield curve. 
• In reality, the expectations theory may be in doubt because upward-sloping yield curves occur far more frequently than downward-sloping and a logical expectation would be for upward- and downward-sloping curves to occur with equal frequency. Here the people expectations would be logical 50/50 chances.
• Liquidity preference theory
• In general behaviour,
• borrower would want money for longer duration
• but lender would want money money back in short duration
• in this theory as the lender is taking the risk he/she would be paid higher interest of longer than shorter, which leads to upward slope.
• The liquidity preference theory attempts to clear up the doubt with the expectations theory. 
• Liquidity preference suggests that most depositors prefer short-term liquid deposits to meet current needs.
• In order to coax them to lend/invest longer term, the intermediary will raise longer-term rates by adding a liquidity premium.

Duration of a Bond

• Duration means on average during how much time the bond holder gets his money back.
• Average time taken by the bond holder to receive his money back
• For zero coupon bond the duration is simply the time to maturity
• For coupon bond its duration will be necesarily shorter than its maturity
• Formula to calculate duration is:

• The usefulness of the duration measure lies in the fact that the approximate change in a bond’s price, B, for a parallel shift in the yield curve of Δy is:
• ΔB / B = -duration * Δy
• The change in yield is often expressed as a basis point change. One basis point is equivalent to 0.01%. So a 100 basis point change is a change of 1% in the yield. 
• When yields are continuously compounded, the provided duration measure is known as Macaulay duration.
• Modified duration is used when the yield given is something other than a continuously compounded rate. When the yield is expressed as a semiannually compounded rate, for example, modified duration = duration / (1 + y/2).

• Just to reiterate, to calculate the approximation of change in a bond's price:
• for non continuously compounding rate is using modified duration
• for continuously compounding rate is using Macaulay duration
• Note that as m goes to infinity (continuous compounding), the two measures are equal and there is no difference between the two.
• Dollar duration is simply modified duration multiplied by the price of the bond.

Convexity

• So far so good, duration is a good approximation of price changes for an option-free bond, but it’s only good for relatively small changes in interest rates.
• As rate changes grow larger, the curvature of the bond price/yield relationship becomes more important, meaning that a linear estimate of price changes, such as duration, will contain errors.
• In fact, the relationship between bond price and yield is not linear (as assumed by duration) but convex. 
• This convexity shows that the difference between actual and estimated prices widens as the yield swings grow. 
• That is, the widening error in the estimated price is due to the curvature of the actual price path. This is known as the degree of convexity.
  
• In order to obtain an estimate of the percentage change in price due to convexity, or the amount of price change that is not explained by duration, the following calculation will need to be made:

• Convexity formula is derived through calculus.
• To calculate the approximation of the change in a bond's price:
• ΔB / B = -duration * Δy + convexity effect

Credits and References

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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:
• PV=
• 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: $\frac{300}{1.10^1}=\frac{300}{1.10}\approx 272.73$
• Year 2: $\frac{400}{1.10^2}=\frac{400}{1.21}\approx 330.58$
• Year 3: $\frac{500}{1.10^3}=\frac{500}{1.331}\approx 375.94$
• Year 4: $\frac{400}{1.10^4}=\frac{400}{1.4641}\approx 273.22$
• Year 5: $\frac{300}{1.10^5}=\frac{300}{1.6105}\approx 186.34$
• Summing these present values gives the Net Present Value (NPV):
• $NPV=272.73+330.58+375.94+273.22+186.34=1438.81$
• 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:
• PV=
• 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: $\frac{300}{1.08^1}=\frac{300}{1.08}\approx 277.78$
  • Year 2: $\frac{400}{1.09^2}=\frac{400}{1.1881}\approx 336.67$
  • Year 3: $\frac{500}{1.10^3}=\frac{500}{1.331}\approx 375.94$
  • Year 4: $\frac{400}{1.09^4}=\frac{400}{1.2950}\approx 308.82$
  • Year 5: $\frac{300}{1.08^5}=\frac{300}{1.4693}\approx 204.32$
• 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:
  • The full price of a bond is the present value of all its expected future cash flows, including coupon payments and the principal repayment at maturity.
  • As time progresses within a coupon period, the present value of the remaining cash flows increases. This is because each subsequent coupon payment becomes closer and more certain, adding value to the bond.
  • Effect of Coupon Payment Date:
  • Just before the coupon payment date, the bond's price is the present value of all future cash flows up to that moment, including the upcoming coupon payment. 
  • The present value calculation incorporates the full coupon payment expected to be received shortly.
  • Immediately after the coupon payment is made, the bond's price drops by the amount of the coupon payment. This occurs because:
  • The coupon payment, which was previously a future cash flow, is now a realized cash flow.
  • The present value of the remaining cash flows is recalculated without including the coupon payment that has just been paid out.
  • Illustrative Example:
  • Suppose a bond pays a $50 coupon semi-annually. Just before the coupon payment date, the bond's price reflects the present value of all future cash flows, including the $50 coupon about to be received.
  • After the coupon is paid, the bond's price immediately drops by $50. This is because the present value calculation now starts anew for the remaining future cash flows, which no longer include the $50 coupon that has just been paid out.
  • Market Response:
  • In the financial markets, this phenomenon is known as the "ex-coupon" effect. 
  • When a bond trades "ex-coupon," its price is typically lower by the amount of the coupon that has just been paid.
  • Investors buying the bond after the ex-coupon date do not receive the upcoming coupon payment, and therefore, they pay a lower price to compensate for this missed cash flow.
  • 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
• Eg:
• 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.
• Eg:
• 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.
• Eg:
• 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

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