Thursday, 11 July 2024

Bond Yields and Return Calculations

Realized Returns

A bond’s realized return compares its ending investment value with its beginning value, while factoring in any coupon payment or coupon reinvestment.
Calculating gross realized return

The gross realized return (or simply gross return) of a bond is its end-of-period total value minus its beginning-of-period value divided by its beginning-of-period value, but it does not factor in any financing cost.
Formula:

Rt-1, t = (BVt + Ct - BVt-1) / BVt-1

Here:

BVt–1 => initial bond price
BVt => current bond price at time t
Ct => coupons received during time period t
Rt-1 => realized return for a bond from time period t–1 to t

Example:

What is the gross realized return for a bond that is currently selling for $112 if it was purchased exactly six months ago for $105 and paid a $2 coupon today?
Using the above formula:

Rt-1, t = $112 + $2 - $105 / $105
Rt-1, t = 8.57%

Calculating realized return with reinvested coupons

To compute the realized return for a bond over multiple periods, we must keep track of the rates at which coupons received are reinvested.
When a bondholder receives coupon payments, the investor runs the risk that these cash flows will be reinvested at a rate that is lower than the expected rate.
For example, if interest rates go down across the board, the reinvestment rate will also be lower. This is known as reinvestment risk.
Formula:

Rt-1, t = (BVt + Ct + (Ct-1 * (1 + RI/n)) - BVt-1) / BVt-1

Here:

BVt–1 => initial bond price
BVt => current bond price at time t
Ct => coupons received during time period t
Rt-1 => realized return for a bond from time period t–1 to t
RI => reinvested rate of interest
n => reinvested time period like annually, semiannually, monthly

Example:

What is the realized return for a bond that is currently selling for $112 if it was purchased exactly one year ago for $105, paid a $2 coupon today, and paid a $2 coupon six months ago? Assume the coupon received six months ago was reinvested at an annual rate of 1%
Using the above formula:

Rt-1, t = $112 + $2 + [$2 * (1 + (1%/2))] - $105 / $105
Rt-1, t = 10.49%

Calculating net realized return

The net realized return (or simply net return) of a bond is its gross realized return minus per period financing costs.
Cost of financing would arise from borrowing cash to purchase the bond.
When the bond is fully financed, the initial cash outlay would be zero; however, convention is to use the initial bond price as the beginning-of-period value.
Formula:

Substituting the appropriate values into the realized return equation and then subtracting per period financing costs
Rt-1, t = [ (BVt + Ct - BVt-1) / BVt-1 ] - (FRI / n)
Here:

BVt–1 => initial bond price
BVt => current bond price at time t
Ct => coupons received during time period t
Rt-1 => realized return for a bond from time period t–1 to t
FRI => financing cost
n => financing costs per period like annually, semiannually, monthly

Example:

What is the net realized return for a bond that is currently selling for $112 and paid a $2 coupon today if its purchase price of $105 was entirely financed at an annual rate of 0.6% exactly six months ago?
Using the above formula:

Rt-1, t = $112 + $2 + [$2 * (1 + (1%/2))] - $105 - (0.6/2) / $105
Rt-1, t = 8.57% - 0.3%
Rt-1, t = 8.27%

Calculate the bond’s dirty price

So far, we assumed bonds are valued at coupon dates.
What happens if we buy or sell a bond between coupon dates?
In this case, we need to calculate the bond’s dirty price, which is the quoted price plus accrued interest.
For example, consider a semiannual coupon bond with maturity at time T.
In this case, the bond valuation can be expressed through this formula:

Bond Spreads

The market price of a bond may differ from the computed price of a bond using spot rates or forward rates.
Any difference between bond market price and bond price according to the term structure of interest rates is known as the bond spread.
By deriving the spread, we can identify how much the bond is trading cheap or rich in terms of the bond’s return. Spreads will generally increase with maturity.

The bond market price is simply what the bond is currently trading for in the market. It's influenced by supply and demand among investors.
The bond model price, on the other hand, is what theoretical calculations based on the bond's characteristics (like its coupon rate, maturity, and prevailing interest rates) suggest it should be worth.
The spread refers to the difference between the bond's market price and its model price. It tells us whether the bond is trading at a premium (rich) or a discount (cheap) compared to what the bond model predicts.
If the spread is positive, it means the bond is trading at a higher price than the model suggests (trading rich).
If the spread is negative, it means the bond is trading at a lower price than the model suggests (trading cheap).
Spreads increase with maturity, bonds with longer maturities (time until they are repaid) are generally more sensitive to changes in interest rates. This sensitivity can lead to larger deviations between their market prices and model prices. Therefore, the spread tends to increase with maturity because there's more time for interest rates to change and influence the bond's value.

PV Formula:

PV = C/1+F1 + C/(1+F1)(C/1+F2) ... C/(1+F1)(C/1+F2)..(1+FN-1)(1+FN)

Example:

3yr Bond
FW = 6%, 7%, 8% (Forward Rates)
CR = 5% (Coupon Rate)
FV = 100 (Face Value)
Calculate the PV of the Bond

PV = (5/1.06) + (5/1.06*1.07) + (105/1.06*1.07*1.08)
PV = 94.84 = Value of the bond

But the bond in actual is trading at $93, whereas the computed value is shown as $94.84.
To fix this we would need to add (S) -> which assume somevalue (ideally spread), with the r or FW like below to arrive at $93 instead of 94.84

PV = (5/1.06+S) + (5/1.06+S*1.07+S) + (105/1.06+S*1.07+S*1.08+S)

Lets calcuate the 'S' to do that let us arrive at complex average - YTM:

PV = 5/(1+r) + 5/(1+r)^2 + 5/(1+r)^3 = 94.84
Using calculator:

N = 3
Coupon / PMT = 5
FV = 100
PV = -94.84
CPT - I/Y = 6.96%

Now replace PV from 94.84 to 93 to calculate complex average r:

N = 3
Coupon / PMT = 5
FV = 100
PV = -3
CPT - I/Y = 7.70%

Let us minus above both r to derive S

7.70 - 6.96 = 0.73

Now if we susbitute 0.73 with above S we will get $93

PV = (5/1.06+S) + (5/1.06+S*1.07+S) + (105/1.06+S*1.07+S*1.08+S)

Steps to calculate spread

Calculate YTM for current markete price (93 from our example, YTM = 7.70)
Calculate FV of the bond using spot / forward rate (94.84 from our example)
Calculate YTM for PV arrived from the step2 (YTM = 6.96)
Calculate Step1 - Step3 and we will arrive the spread (0.73)

Yield to Maturity

Yield to Maturity (YTM):

Yield to Maturity (YTM) is a crucial concept in bond valuation.
It represents the total return an investor can expect to earn if they purchase the bond at its current market price and hold it until maturity.
YTM takes into account the bond's:

Current market price,
Coupon payments (interest payments),
Time remaining until maturity.

The YTM is like the interest rate you earn on a bond if you buy it today and hold it until it matures (until it pays back all its principal).
YTM is essentially the internal rate of return (IRR) of a bond, considering all cash flows (coupon payments and the face value at maturity) and the bond's current price.

Equivalent to Internal Rate of Return:

It's similar to the concept of the internal rate of return (IRR) in investments.
YTM is the discount rate (interest rate) that makes the present value of all future cash flows (like interest payments and the bond's final repayment) equal to the bond's current price.

Impact on Bond Price:

Premium: If the YTM is less than the bond's coupon rate (the fixed interest it pays annually), the bond is more attractive because it pays higher interest than similar bonds. Investors will pay more for it, so it trades at a premium (higher than its face value).
Discount: If the YTM is higher than the coupon rate, the bond pays less interest compared to other options. Investors won't pay as much for it, so it trades at a discount (lower than its face value).
Example:

Face Value (Par Value): $1,000
Coupon Rate: 5% per annum (annual interest payment)
Yield to Maturity (YTM): 4%
Coupon Rate:

This is the fixed annual interest rate that the bond issuer promises to pay the bondholder, based on the bond's face value.
In our example, the coupon rate is 5%, which means the bond pays $50 in interest annually (5% of $1,000).
When the bond is issued, the coupon rate is set based on prevailing interest rates and the creditworthiness of the issuer. However, market conditions can cause the bond's price to fluctuate after issuance.

Yield to Maturity (YTM):

This represents the total return an investor can expect to earn if the bond is held until maturity.
YTM takes into account the bond's current market price, its coupon payments, and the time remaining until maturity.
In our example, the YTM is 4%.

Premium Bond:

A bond is considered to be trading at a premium when its YTM is lower than its coupon rate. Also could be said as a bond is considered to trade at a premium when its coupon rate is higher than the prevailing YTM in the market.
In our example:

Coupon Rate = 5%
YTM = 4%

Since the YTM (4%) is lower than the bond's coupon rate (5%), this bond is attractive to investors seeking higher interest income. Investors are willing to pay more than the bond's face value ($1,000) to obtain this higher interest rate.
Why does this happen? When a bond's coupon rate is higher than the current YTM, it means the bond pays a higher interest rate compared to what investors can get from new bonds being issued or other similar bonds in the market.
Investor Behavior: Investors seeking higher income are willing to pay more than the face value of the bond to secure those higher coupon payments. This pushes the bond's price above its face value.
Calculation:

To determine the bond's price when it's trading at a premium, we compare the coupon rate and the YTM. If the YTM is lower, investors will pay a premium above the face value to capture the higher coupon payments.
Suppose a bond has a face value of $1,000, a coupon rate of 5% ($50 annual interest), but the current YTM in the market is 4%. Investors recognize that the 5% coupon rate offers better returns than the current market rate of 4%, so they bid up the price of the bond above $1,000 to capture that higher yield.

Discount Bond:

A bond is considered to be trading at a discount when its YTM is higher than its coupon rate.
For example, if the YTM were 6% instead of 4%:

Coupon Rate = 5%
YTM = 6%

Here, the YTM (6%) is higher than the bond's coupon rate (5%). This indicates that the bond's coupon payments are lower compared to other available options in the market. To compensate for the lower coupon payments, investors will pay less than the bond's face value ($1,000).
Why does this happen? When a bond's coupon rate is lower than the YTM, it means the bond pays less interest compared to what investors can get from new bonds or other similar bonds in the market.
Investor Behavior: Investors seeking higher yields will only buy the bond if its price is lower to compensate for the lower coupon payments relative to market rates. Thus, the bond's price falls below its face value.
Calculation:

When a bond trades at a discount, its market price is below the face value because the YTM is higher than the coupon rate, making it less attractive to investors seeking higher yields.
If a bond has a face value of $1,000, a coupon rate of 5% ($50 annual interest), but the current YTM in the market is 6%, investors recognize they can get a higher yield elsewhere. They will only purchase this bond at a price below $1,000 to achieve a yield closer to the prevailing 6% YTM.

Conclusion:

Premium Bond: YTM < Coupon Rate. Investors pay more than the face value because the bond offers higher interest payments relative to current market rates.
Discount Bond: YTM > Coupon Rate. Investors pay less than the face value because the bond offers lower interest payments relative to current market rates.

In both cases, the bond's price adjusts in the market to align with investor expectations of yield compared to alternative investments. This dynamic pricing mechanism helps maintain equilibrium between bond prices and market interest rates.

Par: When the YTM equals the coupon rate, the bond trades at par, meaning it's selling for its face value.

Par refers to the face value of a bond. Imagine you have a $1,000 bond. That $1,000 is its par value. When people say a bond is trading "at par," it means it's selling for its exact face value—$1,000 in this case.
Par Value: This is the amount of money the bond issuer promises to repay the bondholder when the bond matures. It's like the principal amount.
Trading at Par: If a bond is trading at par, buyers pay $1,000 to own it because that's what it's worth according to its face value and the current market conditions.
So, "at par" simply means the bond is selling for its original value, not more or less. It's straightforward and doesn't involve any premium (higher than face value) or discount (lower than face value) pricing.

In simple terms, YTM tells you the effective interest rate you'll earn if you buy a bond and hold it until it matures. Whether a bond trades at a premium, discount, or par depends on how its coupon rate compares to the prevailing YTM.
YTM

For a security that pays a series of known annual cash lows, the computation of yield uses the following:

Formula:

Example:

Suppose a fixed-income instrument offers annual payments in the amount of $100 for 10 years. The current value for this instrument is $700. Compute the YTM on this security.
$700 = $100/(1+y)^1 + $100/(1+y)^2 + ... + $100/(1+y)^10
Using calculator:
N = 10; PMT = 100; PV = –700; CPT ⟶ I/Y = 7.07%

Periodic yield and YTM

If cash lows occur more frequently than annually, the previous equation can be repurposed as following:
Formula:

Example:

Suppose now that the security in the previous example pays the $100 semiannually for five years. Compute the periodic yield and the YTM on this security.
Using a financial calculator:

N = 10; PMT = 100; PV = –700; CPT ⟶ I/Y = 7.07%.

To compute the annual YTM, we must multiply the periodic yield by the number of periods per year, m, which in this case is equal to 2.
This produces a YTM of 14.14%.

Reinvestment risk

Reinvestment risk is a major threat to the bond’s computed YTM, as it is assumed in such calculations that the coupon cash lows can be reinvested at a rate of return that’s equal to the computed yield (e.g., if the computed yield is 8%, it is assumed the investor will be able to reinvest all coupons at 8%).
If the average reinvestment rate is below the YTM, the realized yield will be below the YTM.
For this reason, it is often stated that the yield to maturity assumes cash flows will be reinvested at the YTM and assumes that the bond will be held until maturity.

Annuity and Perpetuity

Present value of an annuity

We can easily calculate the price of cash lows (annuities) if given the YTM and cash flows.
Example

Suppose a fixed-income instrument offers annual payments in the amount of $100 for 10 years. The YTM for this instrument is 10%. Compute the price (PV) of this security.
Using a financial calculator, the price equals $614.46:

N = 10; PMT = 100; I/Y = 10; CPT PV = $614.46

Price of a perpetuity

The perpetuity formula is straightforward and does not require an iterative process.
Formula:

Example:

Suppose we have a security paying $1,000 annually into perpetuity. The interest rate is 10%. Calculate the price of the perpetuity.
The price of the perpetuity is simply $10,000:

PV = $1000 / 0.10 = $10,000

Japanese Yields

The yield convention in Japan differs from the U.S. yield convention.
Japanese bond yields are typically quoted on a simple yield basis without factoring in compounding differing from U.S.
Formula:

( C / P ) + (FV - P) / P * N

C = Coupon
P = Purchase Price
FV = Face Value
N = Number of Years
C/P = Returns of Coupons
(FV - P) / P * N = Yield at end per yield capital gain

Example:

Consider a Japanese bond with a 3% coupon and 6 years to maturity, with a price of 98:
yield = 3 / 98 + (100 - 98) / 98*6 = 0.034 or 3.4%

The Relationships

Spot Rates and YTM

Spot Rates and YTM:

YTM (Yield to Maturity) is the total return anticipated on a bond if it is held until it matures.
Spot rates are the interest rates for specific periods in the future.

Impact of Coupon Size:

The coupon is the fixed annual interest rate paid by the bond issuer to the bondholder. When the coupon is larger:
Early Spot Rates More Important:

If a bond has a large coupon, the early spot rates (interest rates for short-term periods) have a bigger influence on calculating the YTM.

Upward Sloping Term Structure:

This means interest rates increase with time. Here’s how it affects YTM:
Early Spot Rates are Lower:

The interest rate for early payments (near the beginning of the bond’s life) is lower compared to the rate for the final payment (when the bond matures).

YTM Declines with Higher Coupon:

As the coupon rate (annual interest rate) increases, the YTM (total return on the bond) decreases.
This is because higher coupon payments mean more of the bond's return comes from those early, lower spot rates.

Understanding the Term Structure of Interest Rates:

The term structure of interest rates describes the relationship between interest rates (or yields) and the time to maturity of debt securities. In a typical upward-sloping term structure:

Short-term interest rates are lower than long-term interest rates.
This implies that longer-term bonds usually have higher yields (YTM) compared to shorter-term bonds.

Relationship between Coupon Rate and YTM:

Coupon Rate: This is the fixed annual interest rate that the bond issuer pays to the bondholder based on the bond's face value.
Yield to Maturity (YTM): This represents the total return an investor can expect if they hold the bond until maturity, taking into account its current market price, coupon payments, and the time remaining until maturity.

Intuition behind the Statement:

Higher Coupon Rate, Lower YTM: When a bond offers a higher coupon rate, it means the bond pays more in annual interest relative to its face value. For example, a bond with a $1,000 face value and a 6% coupon rate pays $60 in annual interest.
Effect on YTM: The YTM takes into account all expected future cash flows (coupon payments and the face value repayment at maturity) discounted back to the present value. In an upward-sloping yield curve environment, longer-term interest rates (YTM) are higher than shorter-term rates.
Impact of Higher Coupons: Higher coupon payments mean that a larger portion of the bond's total return comes from these early, higher coupon payments. This can effectively lower the YTM because:

The bondholder receives more interest income upfront, which reduces the present value of future cash flows (coupon payments and face value).
Lowering the present value of future cash flows increases the bond's price, which decreases the YTM because YTM is inversely related to bond price.

Example Illustration:

Imagine two bonds:

Bond A has a 5% coupon rate.
Bond B has an 8% coupon rate.

Assume both bonds have the same face value ($1,000), and the YTM for both bonds is initially set based on prevailing market rates.

Bond A (5% Coupon): Pays $50 annually. If the YTM in the market is 6%, Bond A's price would be lower to reflect its lower coupon payments relative to the market rate.
Bond B (8% Coupon): Pays $80 annually. Given the same YTM of 6%, Bond B's price would be higher than Bond A's because its higher coupon payments make it more attractive. Investors are willing to pay more upfront for Bond B to capture those higher coupon payments.

Conclusion:

In an upward-sloping term structure of interest rates:

Bonds with higher coupon rates tend to have lower YTMs because the higher coupon payments provide more immediate income relative to the prevailing market yield.
Investors adjust the bond's price to reflect these differences in coupon payments, influencing the bond's YTM.

This relationship underscores how bond prices adjust in response to changes in coupon rates and market interest rates, reflecting investor preferences for current income versus future potential returns.

Flat Term Structure:

This means interest rates are similar across different time periods.
Constant Spot Rate:

All spot rates, including early ones, are the same as the rate for the final maturity.

Downward Sloping Term Structure:

This means interest rates decrease with time.
YTM Increases with Higher Coupon:

As the coupon rate increases, the YTM also increases.
This happens because higher coupons mean more of the bond’s return comes from early payments, which have lower rates in a downward sloping curve.

In a downward-sloping term structure of interest rates, the relationship between a bond's coupon rate and its yield to maturity (YTM) operates differently compared to an upward-sloping term structure.
Understanding a Downward-Sloping Term Structure:

In a downward-sloping term structure of interest rates:

Short-term interest rates are higher than long-term interest rates.
This means longer-term bonds typically have lower yields (YTMs) compared to shorter-term bonds.

Relationship between Coupon Rate and YTM:

Coupon Rate: This is the fixed annual interest rate that the bond issuer pays to the bondholder based on the bond's face value.
Yield to Maturity (YTM): This represents the total return an investor can expect if they hold the bond until maturity, considering its current market price, coupon payments, and the time remaining until maturity.

Intuition behind the Relationship in a Downward-Sloping Term Structure:

Higher Coupon Rate, Higher Attractiveness: In a downward-sloping yield curve environment, longer-term bonds have lower yields (YTMs). Bonds with higher coupon rates become relatively more attractive because they offer higher annual interest payments compared to the prevailing lower market rates.
Effect on YTM: Despite the downward slope of the yield curve:

Bonds with higher coupon rates still provide higher current income compared to similar bonds with lower coupon rates.
Investors are willing to pay more for bonds with higher coupon rates to capture these higher income streams, which can lead to higher bond prices and lower YTMs.

Example Illustration:

Consider two bonds:

Bond X has a 3% coupon rate.
Bond Y has a 6% coupon rate.

Assume both bonds have the same face value ($1,000), and the YTMs for both bonds are initially set based on prevailing market rates.

Bond X (3% Coupon): Pays $30 annually. In a downward-sloping yield curve where the YTM for similar bonds might be 2%, Bond X would be relatively less attractive because its coupon payments are lower compared to the market rate. Investors might bid the price of Bond X down to increase its YTM closer to market rates.
Bond Y (6% Coupon): Pays $60 annually. Despite the lower YTMs in the market, Bond Y is more attractive because it offers a higher coupon rate. Investors may bid the price of Bond Y up to capture these higher coupon payments, resulting in a lower YTM for Bond Y compared to Bond X.

Conclusion:

In a downward-sloping term structure of interest rates:

Bonds with higher coupon rates tend to have lower YTMs because their higher coupon payments provide more immediate income relative to the prevailing lower market yields.
Investors bid up the price of bonds with higher coupon rates to benefit from these higher income streams, thereby lowering the effective YTM of these bonds.

This dynamic reflects investor preferences for higher current income when market interest rates are expected to decrease over time, leading to adjustments in bond prices and YTMs based on their coupon rates.

In essence:

For bonds with higher coupons and an upward sloping term structure, early lower spot rates lead to a lower YTM.
In a flat term structure, all spot rates are the same, maintaining a consistent YTM.
In a downward sloping term structure, higher coupons lead to a higher YTM due to increased reliance on early, lower rates.

These relationships illustrate how different factors like coupon size and term structure impact the Yield to Maturity of bonds.

When the YTM is less than the coupon rate, the bond will trade at a premium.
When the YTM is greater than the coupon rate, the bond will trade at a discount.
When the YTM equals the coupon rate, the bond trades at par.

Example:

Premium Bond (YTM < Coupon Rate):

Scenario:

Coupon Rate: 5% (Annual interest rate the bond pays based on its face value)
YTM (Yield to Maturity): 4% (Overall return an investor can expect considering the bond's price and future payments)

Why Premium?:

When the bond's coupon rate (5%) is higher than the YTM (4%), it means the bond offers a higher interest rate compared to what new bonds or other similar bonds in the market are offering.
Investors find this bond attractive because they get more interest income relative to the price they pay.

Outcome:

Investors are willing to pay more than the bond's face value (let's say $1,000) to secure these higher interest payments.
This pushes the bond's price above $1,000, resulting in a premium.

Intuition

Think of buying something at a premium price because it's better or more desirable.
In this case, investors pay a premium for the bond because it offers a higher interest rate than what's typically available in the market.

Discount Bond (YTM > Coupon Rate):

Scenario:

Coupon Rate: 5%
YTM: 6%

Why Discount?:

When the bond's YTM (6%) is higher than its coupon rate (5%), it means the bond pays less interest compared to what new bonds or similar bonds in the market are paying.
Investors consider this bond less attractive because they could earn more interest elsewhere.

Outcome:

To compensate for the lower interest payments relative to market rates, investors will only buy the bond if its price is discounted below the face value (e.g., less than $1,000).
This results in the bond trading at a discount.

Intuition

Imagine buying something on discount because it's not as desirable or valuable.
Here, investors pay less for the bond because its interest rate is lower than what's available in the market, making it less attractive.

Par Bond (YTM = Coupon Rate):

Scenario:

Coupon Rate: 5%
YTM: 5%

Why Par?:

When the bond's YTM is equal to its coupon rate (5%), it means the bond's interest payments are in line with what other similar bonds or new bonds in the market are offering.
There's no extra incentive or disadvantage compared to other bonds.

Outcome:

The bond trades at its face value (e.g., exactly $1,000).
Investors are indifferent between buying this bond or another bond with similar characteristics because the return matches market expectations.

Intuition:

It's like buying something at its regular price.
The bond trades at par when its interest rate matches market rates, neither gaining nor losing attractiveness compared to other bonds.

These concepts help investors and analysts understand how bond prices adjust based on prevailing interest rates and the attractiveness of bond payments relative to those rates.

YTM, Coupon Rate and Price

The coupon effect describes a scenario where two bonds with identical maturities but different coupons will have different yields to maturity.
If two bonds are identical in all respects except their coupon, the bond with the smaller coupon will be more sensitive to interest rate changes.
That is, for any given change in yield, the smaller-coupon bond will experience a bigger percentage change in price than the larger-coupon bond.
All else being equal:

the lower the coupon rate, the greater the interest rate risk
the higher the coupon rate, the lower the interest rate risk

Imagine two bonds that mature in 10 years, both issued by the same company and with the same credit rating. The only difference between these bonds is their coupon rate.

1. Bond A: Has a lower coupon rate of 3%.
2. Bond B: Has a higher coupon rate of 6%.

Both bonds pay interest semi-annually (every six months), and their face value (the amount you get when the bond matures) is $1,000.
Understanding Interest Rate Risk:

Interest rate risk, refers to how sensitive a bond's price is to changes in interest rates.
Lower Coupon Bond (Bond A, 3% coupon):

This bond pays a lower fixed interest rate of 3% per year on its face value of $1,000.
If interest rates in the market rise after Bond A is issued, new bonds being issued will offer higher interest rates.
Investors holding Bond A, which pays a lower fixed rate, will find it less attractive compared to new bonds offering higher rates. As a result, the price of Bond A will decrease in the secondary market to make up for the difference in interest payments.
Conversely, if interest rates fall, Bond A becomes more attractive because its fixed rate of 3% is higher than what new bonds are offering. This would increase the price of Bond A.

Higher Coupon Bond (Bond B, 6% coupon):

This bond pays a higher fixed interest rate of 6% per year on its face value of $1,000.
If interest rates rise, new bonds will pay higher interest rates, making Bond B relatively more attractive because it pays a higher fixed rate. Therefore, its price won’t decrease as much as Bond A's.
If interest rates fall, Bond B becomes less attractive because its fixed rate of 6% is higher than what new bonds are offering, potentially causing its price to decrease.

Intuition

The intuition behind the interest rate risk differences between a lower coupon bond (Bond A) and a higher coupon bond (Bond B) lies in their respective cash flows and how these cash flows are valued in the market in relation to prevailing interest rates.
Lower Coupon Bond (Bond A):

Cash Flow Structure:

Bond A pays a lower coupon rate relative to its face value. This means its periodic interest payments to bondholders are smaller compared to Bond B.

Price Sensitivity:

Since Bond A has lower coupon payments, the majority of its return comes from the final principal repayment at maturity. Therefore, its price is more sensitive to changes in interest rates because:
When interest rates rise, newly issued bonds offer higher coupon payments, making existing lower coupon bonds less attractive. This lowers the demand for existing lower coupon bonds, causing their prices to decrease.
Conversely, when interest rates fall, lower coupon bonds become more attractive because they offer higher relative yields compared to newly issued bonds. This increases demand for existing lower coupon bonds, causing their prices to rise.

Duration Sensitivity:

Lower coupon bonds tend to have longer durations. Duration measures the sensitivity of a bond's price to changes in interest rates. Longer duration means greater price sensitivity to interest rate changes.

Higher Coupon Bond (Bond B):

Cash Flow Structure:

Bond B pays a higher coupon rate relative to its face value. This results in larger periodic interest payments to bondholders compared to Bond A.

Price Sensitivity:

Bond B is less sensitive to changes in interest rates because:
Higher coupon payments provide a greater cushion against changes in prevailing interest rates. This means that the impact of interest rate changes on the bond's price is mitigated by the higher cash flows received from coupon payments.
Investors may be less inclined to sell higher coupon bonds when interest rates rise because they are already receiving attractive interest income relative to prevailing rates.

Duration Sensitivity:

Higher coupon bonds tend to have shorter durations compared to lower coupon bonds. Shorter duration implies less sensitivity of the bond's price to changes in interest rates.

Intuitive Understanding:

Lower Coupon Bond (Bond A): Imagine a bond that pays a minimal coupon. Most of its attractiveness comes from the eventual return of its principal. Therefore, if market interest rates rise, new bonds issued with higher coupons will be more appealing, reducing demand for the lower coupon bond and causing its price to fall more significantly.
Higher Coupon Bond (Bond B): Picture a bond with a generous coupon payment relative to market rates. This bond provides a substantial income stream through its coupons, which makes it less dependent on changes in market interest rates for its attractiveness. Therefore, if interest rates rise, the impact on its price is muted because investors continue to receive attractive coupon payments despite the rise in rates.

In essence, the interest rate risk of a bond is influenced by how its cash flows (coupons and principal repayment) compare to prevailing interest rates. Bonds with lower coupons (Bond A) are more sensitive to changes in interest rates because their returns are heavily reliant on price appreciation (or depreciation) due to changes in market rates. Bonds with higher coupons (Bond B) are less sensitive because their higher cash flows provide a stronger buffer against interest rate fluctuations.

Bond Price Reactions to Changes in Yield

Below example summarizes the relationship between bond price sensitivity and coupon size. The bonds have equal maturities but different coupons. Assume semiannual coupons for both bonds.

For the same change in interest rates, the 20-year, 8% bond experiences a greater change in price than the 20-year, 12% bond. This suggests that bonds with similar maturities, but different coupon rates, can have different yields to maturity.

As a summary, investors demand higher yields (which means lower prices) for bonds with lower coupon rates when interest rates rise, because these bonds are less competitive compared to newly issued bonds with higher coupon rates. Therefore, the lower the coupon rate, the greater the risk that changes in interest rates will impact the bond's price. Conversely, higher coupon bonds are less affected by changes in interest rates, hence they have lower interest rate risk.

Bond Return Decomposition

Return decomposition for a bond breaks down bond proit and loss (P&L) into component parts.
This decomposition of P&L helps bond investors understand how their investments are making or losing money.
A bond’s proitability or loss is generated through price appreciation and explicit cash flows (e.g., cash-carry), such as coupons and financing costs.
The change in the bond’s price can be broken down into three component parts for price effect analysis:

carry roll-down:

The carry roll-down is the estimated return from bond price movements and coupon payment assuming no change to interest rate expectations.

Coupon payments:

Imagine you buy a bond that pays you a fixed amount of money every year (like a regular allowance). This fixed payment is the carry.

Maturity gain:

Think of the bond as a discounted bus ticket. You buy it cheap now, but it lets you ride the bus (get your full money back) at its maturity date (end of the ticket). As the date gets closer, the discounted ticket price should slowly increase to match its full value. This gradual price rise is the roll-down.

In other words, the expected forward rates are realized and become the spot rates.

This essentially describes how forward rates, which are predicted future interest rates agreed upon today for transactions that will occur in the future, eventually become actual spot rates.
Forward Rates: These are interest rates agreed upon now for transactions (like borrowing or lending) that will take place in the future. They are based on expectations of future interest rate movements.
Spot Rates: These are the current interest rates for immediate transactions, typically for loans or investments with maturities of less than one year.
Realization Process: The process of forward rates becoming spot rates occurs over time as those future rates eventually become the current spot rates when the agreed-upon time period arrives.
Explanation:

Initial Agreement: Suppose today, two parties agree on a forward interest rate for a loan that will be disbursed in one year. This forward rate is based on their expectations of what the market interest rates will be in one year.
Time Passes: Over the course of the year, expectations about future interest rates may change due to economic conditions, central bank policies, inflation expectations, etc.
Future Becomes Present: When the agreed-upon time period (one year in our example) elapses, the forward rate that was agreed upon one year ago now becomes the spot rate applicable for loans or investments of that maturity today. In other words, what was once a future expectation (forward rate) is now the current reality (spot rate).

Example:

Today's spot rate for a 1-year loan is 4%.
However, two parties agree today on a forward rate of 5% for a loan to be made in one year (starting from today).
One year later:

Economic conditions unfold as expected or differently, leading to changes in expectations.
When the loan is actually taken out one year later, the agreed-upon forward rate of 5% becomes the actual spot rate applicable for 1-year loans at that time.

Therefore, "the expected forward rates are realized and become the spot rates" means that over time, the forward rates agreed upon for future transactions eventually align with the actual spot rates when those transactions occur. It underscores the dynamic nature of interest rate forecasting and the transition of expectations into reality in financial markets.

This component does not account for spread changes.

It doesn't consider changes in the spread (the difference between the bond's market price and its estimated value).

So, carry roll-down is a simplified way to estimate your total return from holding a bond until it matures, assuming there are no surprises with interest rates.

rate change:

The rate change is the realized return when this realized return is different from what was assumed under the carry roll-down.
Similar to carry roll-down, this component does not account for spread changes.

spread change effects:

The spread change component accounts for price changes due to changes in the bond’s spread relative to other bonds.
Expected changes in the spread are frequently the subject of investments for traders who are betting that a security is trading either cheap or rich.

Dividing each component return by the bond’s price will give us components of the gross return.
For example, lets consider the information for a bond with a 4% coupon paid semiannually.

Details:

Bond initial price = 102.65
Carry roll down = 0.85
Rate changes = 0.30
Spread changes = 0.08
Bond final value = 101.88
Cash and carry = 2.00

Observations:

The gain on the bond is 101.88 + 2.00 – 102.65 = 1.23
The total gain can be broken down into its component yields:

1.23 = 0.85 (carry roll- down) + 0.30 (impact of rate changes) + 0.08 (impact of spread change)

The gross return is 1.23 / 102.65 = 1.198%

There are two extensions to P&L analysis:

Consider the impact of financing. If financing is considered, the cost of financing should be added as a fourth component of the P&L, and both gross and net returns would be calculated.

Profit and Loss Statement Components:

Revenue:

This is the total income generated from sales or services rendered.

Expenses:

These are the costs incurred to generate revenue.
Expenses include costs of goods sold, operating expenses (like salaries, rent, utilities), interest expenses, taxes, depreciation, and other expenses.

Net Income (or Net Loss):

This is the bottom-line figure after subtracting all expenses from revenue.
It represents the profit (positive figure) or loss (negative figure) generated by the business operations.

Impact of Financing:

When financing is considered in the context of a business's operations, especially when using borrowed funds or capital:
Cost of Financing:

This includes interest payments on loans, fees associated with obtaining financing (like origination fees), and any other financing costs.

Gross Returns:

This refers to the total revenue generated from business operations before deducting any expenses, including financing costs.

Net Returns:

This refers to the revenue remaining after deducting all expenses, including financing costs.

Adding Financing Costs to P&L:

To accurately reflect the impact of financing on the financial performance of a business, financing costs should be included as a separate component in the Profit and Loss statement:
Revenue: Total income generated from sales or services.
Expenses: Including:

Cost of Goods Sold: Direct costs associated with producing goods or services.
Operating Expenses: Costs related to running the business.
Interest Expenses: Interest paid on borrowed funds.
Other Financing Costs: Fees and charges related to financing activities.

Net Income (or Net Loss): Calculated as Revenue minus all Expenses, including financing costs.

Example Scenario:

Let's say a company generates $1,000,000 in revenue from sales. They incur $600,000 in operating expenses and $50,000 in interest expenses on loans obtained for business operations.

Gross Returns: $1,000,000 (Revenue)
Net Returns: $1,000,000 - $600,000 - $50,000 = $350,000

In this example:

The $50,000 in interest expenses is added as a separate line item under expenses on the P&L statement.
Net returns of $350,000 reflect the actual profit after deducting all expenses, including the cost of financing.

In summary, considering the impact of financing involves adding financing costs as a component of the Profit and Loss statement, alongside revenue and other expenses. This approach provides a comprehensive view of the financial health and profitability of the business.

Consider accrued interest on both the initial and final valuation dates. So far, we looked at returns simply between two coupon dates. However, both the initial and final bond valuations could be between coupon dates. In this case, it is necessary to add a fourth component for the impact of accrued interest.

Accrued interest is an important concept in bond valuation, especially when valuations occur between coupon payment dates.
Understanding Accrued Interest in Bond Valuation

Coupon Payments and Accrued Interest:

Bonds typically pay periodic interest (coupons) to bondholders, often semi-annually or annually.
The amount of interest accrued between two coupon payment dates needs to be considered when valuing a bond outside these payment dates.

Initial and Final Bond Valuations:

Bond valuation calculations are usually based on present value principles, where future cash flows (coupon payments and principal repayment) are discounted back to their present values.
If you're valuing a bond on a date that is not a coupon payment date (i.e., between coupon dates), you need to account for the interest that has accrued up to that point.

Components of Bond Valuation:

Coupon Payments: These are the regular interest payments made to bondholders.
Principal Repayment: The final repayment of the bond's face value at maturity.
Discount/Premium: This accounts for the difference between the bond's price and its face value, reflecting current market interest rates.
Accrued Interest: This is the interest that has accumulated on the bond since the last coupon payment up to the valuation date.

Impact of Accrued Interest:

When a bond is valued between coupon dates, accrued interest affects the actual purchase price a buyer would pay or a seller would receive.
The buyer typically compensates the seller for the accrued interest since the seller has held the bond and is entitled to the interest that has accrued since the last coupon payment.
Therefore, in the context of bond valuation between coupon dates, accrued interest is added to the quoted price to arrive at the full purchase price.

Example Scenario:

Suppose a bond pays semi-annual coupons of $50 each, and the current coupon period started 45 days ago. The bond is being valued today, which is 30 days after the last coupon payment.
Accrued interest for these 30 days needs to be added to the bond's quoted price to determine the full purchase price.
If the quoted price is $1,020, and accrued interest for the 30 days amounts to $10, the actual purchase price would be $1,030 ($1,020 + $10).

Accrued interest is crucial in bond transactions occurring between coupon payment dates because it compensates the seller for interest earned during their ownership period. It's an integral part of bond valuation, ensuring that the buyer pays the correct amount reflecting both the quoted price and the accrued interest up to the valuation date. This consideration maintains fairness in pricing and accurately reflects the bond's market value at any point in time.

Carry Roll Down Scenarios

As mentioned, the carry roll-down is the estimated return from bond price movements and coupon payment assuming no change to interest rate expectations.
Traders make investment return calculations based on their expectations, and many traders will consider scenarios where rates do not change.
Given this expectation, term structure choices for no-change scenarios include realized forward, unchanged term structure, and unchanged yields.
Forward Rates for Realized Forward Scenario

The realized forward scenario assumes that forward rates for future periods remain unchanged as time passes.
This means that as forward rates are realized, they will be equal to the expected future spot rates.
As a result, when the beginning of a forward period is reached, the forward rate becomes the spot rate.
For example, consider a Treasury bond with a 2-year maturity and 2% coupon, currently valued at $100.785.

Under the realized forward scenario, in six months the realized forward rates will be 1.0% for 0–0.5 years, 1.2% for 0.5–1.0 years, and 1.4% for 1.0–1.5 years.
We can therefore value the bond, which now has 1.5 years left to maturity, compounded on a semiannual basis:

1.0 / 1.005 + 1.0 / (1.005)(1.006) + 101.0 / (1.005)(1.006)(1.007) = $101.188

Therefore, if forward rates are realized in six months, the bond’s price is expected to change to $101.188.
Now lets see an alternative way of calculating the bond’s price using the carry roll-down is to assume that the return earned is equal to the currently prevailing 1-period forward rate. In our example, the 6-month forward rate, semiannually compounded, is 0.8%. The carry roll-down can be calculated as:

$100.785 * 0.004 = $0.403
$100.785 + $0.403 = $101.188

Note: forward rates are always quoted on an annual basis.

Forward Rates for Unchanged Term Structure Scenario

Unchanged Term Structure Scenario

The unchanged term structure scenario assumes that the yield curve, or term structure of interest rates, remains constant over the investment horizon. This means that the interest rates for different maturities do not change; they stay the same as they are today.

Implications and Realized Return

In this scenario, the gross realized return on a bond investment will depend significantly on the relationship between the bond's coupon rate and the last forward rate before the bond matures.

Coupon Rate vs. Forward Rates:

The coupon rate of a bond is fixed when the bond is issued and remains constant throughout its life.
Forward rates, on the other hand, represent market expectations of future interest rates. The last forward rate before the bond matures reflects the market's expectation of the interest rate environment at that maturity.

Gross Realized Return:

The gross realized return is influenced by how the bond's coupon rate compares to the prevailing forward rates.
If the bond's coupon rate is higher than the last forward rate (implying lower future interest rates), the bond is considered attractive because it pays a higher rate of interest relative to current market expectations.
Conversely, if the bond's coupon rate is lower than the last forward rate (implying higher future interest rates), the bond may be less attractive because it pays less interest than what is expected in the future.

Risk Premium in Forward Rates

The scenario also implies that there is a risk premium built into forward rates. Here’s how:
Upward Sloping Term Structure: If the term structure is upward sloping and remains unchanged, it means longer-term interest rates are higher than short-term interest rates.
Investor Risk Premium: The shape of the term structure (upward sloping in this case) reflects an investor risk premium that increases over the investment horizon.

Investors typically demand higher yields for locking in their money for longer periods due to uncertainty and perceived risk associated with longer-term investments.
Therefore, the higher forward rates for longer maturities imply a risk premium that investors require for holding longer-term bonds.

Example:

Suppose the current term structure is upward sloping:

Short-term interest rates are lower (say 2% for 1-year Treasury bills).
Long-term interest rates are higher (say 4% for 10-year Treasury bonds).

If an investor buys a 10-year bond with a coupon rate of 3%, and the term structure remains unchanged, the gross realized return will depend on whether the coupon rate of 3% is above or below the last forward rate before maturity (which might be around 4%).
If the last forward rate is 4%, the bond's 3% coupon rate is lower than the expected future interest rate, making the bond less attractive in terms of gross realized return.
This situation reflects that the initial yield curve (upward sloping) compensates investors for holding longer-term bonds due to the risk premium embedded in the forward rates.

In conclusion, the unchanged term structure scenario highlights how the shape of the yield curve affects bond investments and the gross realized return. It underscores the importance of understanding the relationship between a bond's fixed coupon rate and market expectations embedded in forward rates over the investment horizon.
For example, continuing with our previous example of a Treasury bond with a 2-year maturity and 2% coupon, currently valued at $100.785, the unchanged term structure assumption means that the assumed forward rates will materialize.

We can therefore value the bond, which now has 1.5 years left to maturity, compounded on a semiannual basis:

1.0 / 1.004 + 1.0 / (1.004)(1.005) + 101.0 / (1.004)(1.005)(1.006) = $101.487

As the name suggests, the unchanged yields scenario assumes that bond yields remain unchanged over the investment horizon.
This means that the 1-period gross realized return will equal a bond’s yield (i.e., its yield to maturity).
Thus, this scenario assumes that bond coupon payments are reinvested at the YTM.
As stated earlier, there are limitations to this reinvestment assumption because the term structure is unlikely to be flat and remain unchanged.

To compare the concept of forward rates in the context of the unchanged term structure scenario versus the realized forward scenario:

Unchanged Term Structure Scenario

Assumption: The term structure of interest rates (yield curve) is assumed to remain constant over the entire investment horizon.
Implication: Under this scenario, forward rates are derived based on the assumption that current interest rates for different maturities will persist into the future without any change.
Forward Rates Calculation:

Forward rates are calculated based on the current spot rates (yield curve).
For example, the 1-year forward rate ( f(1,1) ) is calculated as ( f(1,1) = r(2) ), where ( r(2) ) is the 2-year spot rate.

Investor Expectations: Investors expect that the interest rates (yield curve) they see today will accurately predict future interest rates. Hence, they use these forward rates to estimate future yields and make investment decisions.
Risk Premium Consideration: In the context of an upward sloping yield curve, the forward rates for longer maturities are typically higher than the spot rates. This reflects a risk premium investors demand for holding longer-term bonds.

Realized Forward Scenario

Assumption: The realized forward scenario considers that the actual future spot rates (interest rates) will evolve differently from today's forward rates.
Implication: This scenario recognizes that forward rates are predictions based on current expectations and may not accurately reflect future spot rates due to economic changes or other factors.
Forward Rates Calculation:

Similar to the unchanged term structure scenario, forward rates are initially calculated based on current spot rates.
However, in the realized forward scenario, actual future spot rates may deviate from these forward rates.

Investor Expectations: Investors recognize the uncertainty in predicting future interest rates accurately. They understand that forward rates are estimates and may adjust their strategies accordingly.
Impact on Investments: If actual future spot rates deviate significantly from the forward rates used for planning, it can impact bond prices and investor returns. For instance, if forward rates predict rising interest rates but actual rates stay the same or decrease, bonds with lower coupon rates might become more valuable.

Comparison:

Unchanged Term Structure Scenario: Focuses on the assumption of a stable yield curve and uses forward rates derived from current spot rates to make investment decisions. It emphasizes the risk premium embedded in forward rates and assumes these premiums reflect investors' compensation for future uncertainties.
Realized Forward Scenario: Acknowledges the possibility of future interest rates deviating from current expectations (forward rates). It reflects a more dynamic approach where investors may adjust their strategies based on changing economic conditions or new information.

In essence, while both scenarios utilize forward rates as a tool for forecasting future interest rates, the key difference lies in the expectation of whether these rates will remain constant (unchanged term structure) or diverge (realized forward scenario) over time. This difference impacts how investors assess risk and make investment decisions in bond markets.

Credits and References

https://bpcdn.co/images/2010/10/bond-yield-vs-price.png

Chapter 57

Thursday, 27 June 2024

Lessons from Financial Disasters - Part II

Introduction

Due to different risks, we will see a few case studies that have resulted in financial crises - Part II.

Operational Risk

Case Study 1: Barings Bank - Rogue Trading

The collapse of Barings Bank due to rogue trading is a fascinating and cautionary tale in the world of finance.
The venerable British institution, with a history dating back over 200 years, was brought down by the unauthorized trading of a single rogue trader: Nick Leeson.

Background:

Barings Bank, founded in 1762, was one of the oldest and most respected banks in the United Kingdom.
It had a long history of success in international banking and had played a significant role in financing various industries and governments around the world.

Rogue Trader:

In the early 1990s, Barings Bank experienced catastrophic losses as a result of unauthorized speculative trading conducted by one of its employees, Nick Leeson.
Leeson was working as a derivatives trader at Barings' Singapore office and Leeson's role expanded as local head of operations and had been given authority to manage both trading and back-office operations —a critical oversight that allowed him to manipulate the bank's risk controls.

Unauthorized Speculation:

Arbitrage is the simultaneous purchase and sale of the same or similar asset in different markets in order to profit from tiny differences in the asset's listed price. In traditional arbitrage trade involves in taking long position in a potentially underpriced security and a short position in a potentially overpriced one.
However Leeson decided to take the rogue action of speculating in a directional move by taking large directional bets on the movement of one asset without hedging his positions with offsetting trades. Specifically Leeson began making highly speculative trades in the Japanese futures market, primarily in Nikkei 225 index futures and Japanese government bonds.
He believed that the market would move in his favor, allowing him to profit from the anticipated rise in the index. However, by taking such one-sided directional bets without proper risk management or hedging, Leeson exposed Barings Bank to significant losses if the market moved against his positions.
He took increasingly larger positions and concealed his losses by creating fictitious accounts and manipulating trading records.
Unfortunately, what happened—his speculative trades incurred substantial losses, ultimately leading to the collapse of Barings Bank.

Cover-Up:

As losses mounted, Leeson attempted to cover up his activities by fabricating profit reports and misleading Barings' management and auditors.
His actions looked like they were making large returns for Barrings Bank.
The reality was the Leeson controlled the bank-office accounting of this own trades, and he managed the reporting through a hidden reconcilation account that was never reported to the home office. What appeared to be £102 million profit in 1994 was actually a £200 million loss.
He also made desperate attempts to recoup losses by doubling down on his risky trades, which only exacerbated the situation.

Dismissed Red Flags in 1994:

There were warnings about Leeson's activities in 1994, but they were reportedly dismissed by risk managers.
This lack of investigation allowed Leeson's unauthorized trading to continue unchecked.

Unusual Profits in 1995:

In 1995, when Leeson reported a suspiciously high profit of $10 million, it should have been a giant red flag for risk management.
However, instead of raising serious concerns, this anomaly was also overlooked.
These failures by Barings' risk management allowed Leeson to hide his growing losses for far too long.
By the time the truth came out, it was too late for the bank.

Downfall:

Leeson's gamble backfired in early 1995.
A devastating earthquake in Kobe, Japan, triggered a massive sell-off in the Nikkei stock index, on which Leeson had heavily wagered.
The losses became impossible to conceal, and Barings was faced with a staggering £827 million (US$1.4 billion) hole in its finances – more than twice its available capital, effectively bankrupting Barings Bank.

The bank was forced into administration, and its operations were eventually sold to Dutch financial services firm ING Group. 200-year-old banking was decimated and sold for £1 because of rogue trader was permitted to act unchecked.

Aftermath:

Nick Leeson was arrested in Singapore and extradited to the UK, where he faced criminal charges for fraud and forgery.
He was ultimately convicted and sentenced to six and a half years in prison.
The collapse of Barings Bank sent shockwaves through the financial industry and prompted widespread scrutiny of risk management practices and internal controls within banks.

Lesson Learned:

Too Good To Be True: If something appears too good to be true, then it should be questioned. All amazing results should be treated with a degree of healthy skepticism. If profits appear to be either outsized or strangely consistent, then scrutiny is needed to ensure that the books are being kept correctly.
Blind Trust: There might have been an over-reliance on Leeson's apparent success, leading to blind trust in his activities. Trades needs to be judged relative to their ultimate outcome not based on isolated sections of the full holding period. The outcome is what truly defines a trade's success, not cherry-picking moments within the holding period.
Internal Controls: Inadequate internal controls could have made it easier for Leeson to manipulate the accounting system. There should be independent verification mechanism, and the bank's back office should never be controlled by front office (i.e. traders) influencers.
Lack of Communication: Poor communication between traders, risk management, and compliance could have prevented a clear picture of Leeson's activities from emerging.
Be Skeptical of Outsized Promises: Don't get swept away by promises of high returns with low risk. There's always a trade-off between risk and reward.
Do Your Research: Before investing, thoroughly research any financial product or opportunity. Understand the risks involved and don't rely solely on someone else's claims.
Beware of Hidden Fees: Watch out for hidden fees and charges that can eat into your returns.

The Barings Bank collapse serves as a stark reminder of the risks associated with unchecked speculation, inadequate risk controls, and the potential for catastrophic losses resulting from rogue trading. It also highlighted the importance of effective oversight, transparency, and accountability in financial institutions to prevent similar incidents in the future.

Financial Engineering Issues

Financial engineering is a broad field that applies concepts from engineering, mathematics, statistics, and computer science to solve problems and create new opportunities in the financial world.
Applications:

Product Development:

Financial engineers design new financial products like complex derivatives, structured products, and alternative investment vehicles.

Hedging:

The building blocks for financial engineering are forwards, futures, swaps, quanto swaps (multicurrency interest rate swap), options, and securitized products.
By using these tools a risk manager could hedge either a granular risk exposure or a basket of risk exposures.

Risk Management:

They develop and implement strategies to identify, measure, and manage financial risks. This includes tools like Value at Risk (VaR) and stress testing.

Trading Strategies:

They create quantitative trading models and algorithms to exploit market inefficiencies and generate profits.
This is often referred to as "algorithmic trading" or "high-frequency trading."

Investment Banking:

Financial engineers play a key role in investment banking activities like mergers and acquisitions, by valuing companies and structuring complex financial deals.

Case Study 1: Bankers Trust

Bankers Trust and Procter & Gamble (P&G) were involved in a very public legal battle in the mid-1990s concerning complex financial derivatives.
The P&G case is often referred to as one of the most prominent "derivatives debacles" in US financial history.
The case involved recordings of conversations between Bankers Trust brokers, where they allegedly admitted to misleading clients.
The fallout from the P&G lawsuit, along with other similar cases, contributed to increased scrutiny and regulatory reforms in the derivatives market.
The Parties: Bankers Trust, a major investment bank at the time and was known to be a skilled risk manager, and P&G, a consumer goods giant.
The Deal: In 1993-1994, P&G approached BT to help manage interest rate risk in both U.S. dollars and deutsche marks (German currency at the time). These swaps were supposed to help P&G manage interest rate fluctuations, but they turned sour.
Speculative Position: P&G made a speculative bet that interest rates would decline by entering into leverage swaps with BT. In these swaps, P&G would pay a fixed rate to BT and receive floating-rate payments in return. The leverage ratio of 20:1 indicates that P&G's exposure to the swaps was significantly amplified compared to its initial investment. At first, things went well. Maybe interest rates stayed steady or even went down a bit, and P&G made some money.

Leverage refers to using debt (borrowed funds) to amplify returns from an investment or project.
Leverage swaps are financial derivatives that allow investors to amplify their exposure to an underlying asset or index without directly owning it. These swaps involve an agreement between two parties to exchange cash flows based on the performance of the underlying asset or index, with the leverage factor determining the extent of the exposure.
Agreement: Two parties, often referred to as the "long" and "short" parties, enter into a leverage swap agreement. The long party agrees to pay a fixed rate of interest or return, while the short party agrees to pay a floating rate of interest or return.
Underlying Asset or Index: The swap's performance is tied to the movement of an underlying asset or index, such as a stock, bond, commodity, or market index. The parties agree on the specific asset or index that will serve as the basis for the swap.
Leverage Factor: The leverage factor determines the degree to which the swap amplifies exposure to the underlying asset or index. For example, a leverage factor of 2x means that for every 1% change in the underlying asset's value, the swap's value changes by 2%.
Cash Flows: Periodically, usually at predetermined intervals such as quarterly or annually, the parties exchange cash flows based on the performance of the underlying asset or index. The cash flows are calculated based on the difference between the actual performance of the asset or index and the agreed-upon fixed or floating rate.
Risk and Reward: Leverage swaps offer the potential for magnified returns compared to directly investing in the underlying asset or index. However, they also carry higher levels of risk, as losses are also amplified in proportion to the leverage factor. Therefore, investors must carefully consider the risk-reward trade-off before entering into leverage swap agreements.
Applications: Leverage swaps can be used for various purposes, including speculative trading, hedging, and portfolio management. For example, investors may use leverage swaps to gain leveraged exposure to a particular market or asset class without having to invest a large amount of capital upfront.
Overall, leverage swaps are complex financial instruments that offer investors the ability to amplify their exposure to underlying assets or indexes. While they can provide opportunities for enhanced returns, they also carry significant risks and should be used judiciously and with a thorough understanding of their mechanics and implications.

Interest Rate Increase: However, the Federal Reserve unexpectedly raised interest rates by 250 basis points (2.5%) in 1994. This move was likely aimed at controlling inflation or addressing other macroeconomic concerns. The increase in interest rates had adverse effects on P&G's leverage swaps, as the floating-rate payments it had to make to BT increased significantly.
Substantial Losses: As a result of the interest rate hike, P&G incurred substantial losses on its leverage swaps. The losses were exacerbated by P&G's speculative position and the high level of leverage employed in the trades. Leverage can amplify both gains and losses, magnifying the impact of adverse market movements on an investor's portfolio.
Speculation vs. Hedging: The scenario highlights the distinction between speculation and hedging. While hedging involves using financial instruments to offset or mitigate specific risks, speculation entails taking directional bets on market movements in the hope of earning profits. In this case, P&G's decision to speculate on interest rate movements, rather than hedge its risk exposures, left it vulnerable to losses when interest rates moved against its position.
The Dispute: P&G alleged that Bankers Trust misled them about the complexity and risks of the swaps. They claimed the bank failed to properly explain the products and hid crucial details about their valuation. P&G sued Bankers Trust for substantial losses incurred on the transactions.
The Fallout: The legal battle dragged on for a couple of years. There were accusations of fraud, misrepresentation,and breach of fiduciary duty. It became a prominent case, shedding light on the potential dangers of complex financial derivatives and the importance of transparency in such transactions.
The Outcome: P&G won the judgement of $78million. Several other firms had a similar experience with BT and the resulting reputational damage was too much to overcome. BT was eventually acquired by Deutsche Bank.
Lessons Learned:

Understanding complex financial instruments: Companies like P&G entering into such agreements should thoroughly understand the risks involved.
Fiduciary duty of financial institutions: Banks and financial advisors have a responsibility to act in the best interests of their clients.
Regulation of derivatives: Proper oversight can help mitigate risks associated with complex financial instruments.

Case Study 2: Orange County

The Orange County crisis, also known as the Orange County bankruptcy, refers to a financial debacle that occurred in Orange County, California, in 1994.
Background:

Orange County was known for its affluent communities and was home to a large number of wealthy individuals and corporations.
The county's investment pool, managed by the Orange County Investment Pool (OCIP), held funds from various local government agencies, including school districts, cities, and special districts.
In the early 1990s, it had $7.7 billion in assets.

Aggressive Investment Strategy:

Under the leadership of Treasurer-Tax Collector Robert Citron, Orange County pursued an aggressive investment strategy that relied heavily on leveraging its investment portfolio through the use of repurchase agreements (repos) to borrow an additional 12.9 billion and other derivative securities.

Repos are short term borrowing mechanism where one party sells a security to a counterparty and agrees to repurchase it, at a later date, for a slightly higher price. The price increase is the "interest" on the transaction.

Citron subsequently invested all assets and all borrowed funds into complex inverse floating-rate notes. These securities have coupon rates that decline when interest rates rise.

Imagine a regular bond that pays a fixed interest rate (coupon) regardless of what happens to interest rates in the market.
Inverse floating-rate notes (IFRNs), also known as inverse floaters or inverse floaters notes, is the opposite. Its coupon rate goes down when interest rates go up, and vice versa.
By investing in IFRNs, Citron was essentially betting that interest rates would decrease and believed he could benefit from the inverse relationship between the coupon rates on the notes and prevailing interest rates.
However, this strategy was highly speculative and exposed Orange County to significant risks.
If interest rates were to continue rising or remain elevated, the value of the inverse floating-rate notes could decline sharply, leading to further losses for the county.
Additionally, investing borrowed funds into such high-risk securities exacerbated the county's leverage and financial vulnerability.
Ultimately, Citron's strategy of doubling down on complex derivative securities, including inverse

A Simpler Analogy: Imagine you have a bucket of water (your assets). You borrow another bucket (borrowed funds) and pour all the water from both buckets into a basket with holes (IFRNs) hoping it will rain (interest rates decrease) and fill the basket with more water (gains). But if the sun comes out instead (interest rates increase), the water leaks out quickly (losses) and you might be left with nothing.
Orange County was completely reliant on lenders being willing to roll forward the repo contracts when they matured i.e. a maturing repo was replaced with a new repo contract rather than paid for with cash.
Citron's strategy aimed to capitalize on interest rate differentials and generate higher returns for the county's investment pool.
Intially this strategy allowed Citron to earn 2% higher than his peers.

Rising Interest Rates:

In 1994, interest rates began to rise unexpectedly, driven by actions taken by the Federal Reserve to combat inflationary pressures in the economy.
These interest rate increases had adverse effects on Orange County's leveraged investment portfolio, causing losses and triggering margin calls on its derivative positions.

Bankruptcy:

Unable to meet its financial obligations and facing mounting losses, Orange County filed for bankruptcy protection on December 6, 1994.
The bankruptcy filing was the largest municipal bankruptcy in U.S. history at the time, with the county's investment pool suffering losses of approximately $1.7 billion.

Lessons Learned:

Citron later admitted that he really did not understand the risk exposures of inverse floaters.
The lesson is to not invest in anything that you do not understand because the losses can be terminal.
Putting all your eggs in one basket (especially a risky one) is a dangerous strategy.

The Orange County crisis serves as a cautionary tale about the dangers of speculative investment strategies and excessive leverage, particularly when employed by government entities responsible for managing public funds. It underscored the importance of prudent risk management and oversight in financial decision-making, both in the public and private sectors.

Case Study 3: Sachsen Landesbank

In Germany there are a series of publicly owned banks known as Landesbanks. They specialize in lending to regional small and medium sized firms.
In mid-2000s, the Landesbanks saw the profit potential in the U.S. subprime lending market. They began to establish off-balance sheet entities in other countries that could hold securitized subprime mortgage loans. Sachsen Landesbank was one such bank.
The Sachsen Landesbank (SLB) crisis, which erupted in 2007, involved risky bets gone wrong and exposed the dangers of unregulated financial instruments.
The Culprit - Off-Balance Sheet Entities:

SLB, a German state-owned bank, had a subsidiary called Ormond Quay. This entity was an "off-balance sheet conduit," essentially a hidden account not reflected in SLB's main financial statements.
It created an off-balance sheet entity domiciled in Dublin, Ireland and bought a large amount of subprime assets.

The Purpose:

Through Ormond Quay, SLB could engage in riskier financial activities without directly impacting its reported financial health. This practice aimed to bypass regulations and present a rosier financial picture.

The Risky Bet - Subprime Mortgage Crisis

Ormond Quay heavily invested in complex financial instruments linked to subprime mortgages in the US housing market.
Subprime mortgages are loans issued to borrowers with poor credit history, who are more likely to default (fail to repay) on their loans.

The Underlying Bet:

SLB was essentially betting that the US housing market would continue to rise, making these subprime mortgage-linked investments profitable.

The Downfall:

In 2007, the US housing market bubble burst, leading to a subprime mortgage crisis.
As homeowners defaulted on their loans, the value of SLB's investments through Ormond Quay plummeted.

Hidden Losses Exposed:

With the collapse of the housing market, SLB could no longer hide the massive losses incurred by Ormond Quay.
The bank's true financial health was revealed, creating a major crisis.

The Aftermath:

SLB faced a liquidity crisis, meaning it struggled to access funds to meet its financial obligations.
The German government and other banks had to intervene with a bailout to prevent a complete collapse.

Regulatory Scrutiny:

The SLB crisis highlighted the dangers of off-balance sheet entities and unregulated financial instruments. It led to stricter regulations in Germany and other countries to prevent similar situations.

Lessons Learned:

The SLB crisis is a cautionary tale about the importance of transparency in financial institutions. Hiding risks through off-balance sheet entities can have disastrous consequences.
Regulation is crucial: The crisis showed the need for robust regulations to ensure financial institutions engage in responsible and transparent practices.
Risky bets can backfire: SLB's gamble on the housing market backfired spectacularly. It's a reminder that chasing high returns often comes with significant risks.

Reputation Risk

A company's reputation is a public perception of its fairness, commitment to ethical behaviour, and treatment of stakeholders (i.e., customers, suppliers, counterparties).
One trending area with growing reputional influence is environmental, social, and governance (ESG) monitoring.
Reputation risk is the potential for negative operational outcomes due to a poor public perception.
The three big constituents to watch are customers, regulators and shareholders.
In the internet era, it is very easy for facts and rumors to be distributed to the general public.
Even a bad rumor can temporily destroy a firm's reputation in a matter of hours.
The most damaging situations occur when the rumors are true.
Case Study 1: Volkswagen

The Volkswagen emissions scandal, also known as Dieselgate, is a prime example of reputation risk in the auto industry.
The Deception:

In 2015, the US Environmental Protection Agency (EPA) discovered that Volkswagen had installed software, called a "defeat device," in millions of their diesel vehicles.
This software allowed the cars to cheat on emissions tests.
This meant that VW vehicles would appear clean during testing and passed all emissions screens with flying colours.
But the software stopped controlling emissions and emit significantly higher levels of pollutants in real-world driving conditions.
Between 2009 and 2015, this software management affected over 10 million cars worldwide.

The Fallout:

This revelation was a major blow to Volkswagen's reputation: The company was known for its reliable and supposedly clean diesel technology.
Share price drop: Its share price was cut by one-third as the scandal unfolded.
Environmental Damage: The excess pollution contributed to air quality issues.
Loss of Consumer Trust: Consumers felt deceived by Volkswagen's false claims about clean diesel technology.
Financial Penalties: Volkswagen faced billions of dollars in fines and settlements from regulatory bodies and lawsuits.
Management Shakeup: Top executives were fired or resigned due to the scandal.

Rebuilding Reputation:

Volkswagen has undertaken significant efforts to rebuild its reputation, including:

Public apologies
Recalls and repairs of affected vehicles
Investments in electric vehicle technology
Stronger focus on sustainability and environmental responsibility

Long-Term Impact:

The emissions scandal has severely tarnished Volkswagen's reputation.
It will likely take years for the company to fully regain consumer trust.
The scandal also serves as a warning to other automakers about the importance of ethical practices and environmental responsibility.
The scandal highlighted the increasing scrutiny on automakers regarding emissions standards and environmental impact.
It also raised questions about corporate culture and the potential for unethical behavior within large organizations.
At one point, the German government even expressed concern that the imprimatur (let it be printed) "Made in Germany" might suffer reputational damage in the aftermath of an epic lapse in judgement.

Overall, the Volkswagen emissions scandal is a cautionary tale about the significant costs associated with damaging a company's reputation. It emphasizes the importance of transparency, ethical conduct, and environmental responsibility in the auto industry.

Corporate Goverance Issues

Corporate goverance is a system of policies and procedures that direct how a firm is operated.
Governance checkpoints include adequate transperancy and accountability, supervision of senior lenders and risk management policies, deploying reasonable diversity, ensuring board member independence and representation of required skillsets for the company in question.
Case Study 1: Enron

In 1985, the highly leveraged merger of InterNorth and Houstan Natural Gas gave birth to Enron.
A subsequent wave of deregulation moved Enron into the role of being a gas broker.
The company would routinely purchase gas from various vendors and sell it to a network of customers at predertermined prices.
To cover its risk exposure to gas prices, Enron created a new market for energy derivatives. An innovative financial products tailored to the energy industry, allowing market participants to hedge their exposure to volatile energy prices or speculate on future price movements. These products included natural gas futures contracts, weather derivatives, and electricity swaps, among others.
As a result, Fortune magazine named Enron "America's Most Innovative Company" from 1995 through 2000.
At the end of the year 2000, Enron had 20,000 employees and nearly $101 billion in recorded revenue.
Reality caught up with Enron in December 2001, which is when it became the largest bankruptcy in U.S history.
Enron's corporate governance failure is a notorious example of ethical and managerial misconduct that led to one of the most significant corporate scandals in history.
Agency Risk:

Enron's senior management put self-interest above other stakeholders.
They pursued short-term profitability to maximize personal wealth and sacrificed the entire firm in the process.

Board of Directors:

Enron's board of directors failed to exercise effective oversight of the company's management and operations.
The board, which included prominent individuals from various industries, was criticized for lacking independence and failing to challenge management decisions.

Conflicts of Interest:

Several members of Enron's board had financial ties to the company or its executives, creating conflicts of interest that compromised their ability to act in the best interests of shareholders.
Ken Law was both chairman of the board and the CEO, he was effectively supervising himself.
The board also allowed Enron's CFO to operate his own private equity firm as a side business. This was a business in name only and served the function of funding even more money to the CFO's personal control.
These conflicts hindered the board's ability to provide independent oversight and hold management accountable.

Executive Compensation:

Enron's executive compensation structure incentivized risky behavior and short-term financial gains at the expense of long-term sustainability.
The company would routinely construct a physical asset and reconize it as revenue as soon as production was complete, regardless whether an actual sale occurred. Any losses when actual sale happended were burried in an off-balance sheet SPV.
Executives were rewarded with generous bonuses and stock options based on the company's reported earnings and stock price performance, leading to aggressive accounting practices and financial manipulation to inflate profits.

Accounting Irregularities:

Enron engaged in complex accounting schemes and off-balance-sheet transactions to conceal debt, inflate revenues, and manipulate financial statements.
These accounting irregularities, including the use of special-purpose entities (SPEs) and mark-to-market accounting, allowed Enron to misrepresent its financial position and performance to investors and regulators.

Special-Purpose Entity (SPE): An SPE is a legal entity created for a specific purpose, often to isolate financial risk or facilitate complex financial transactions. SPEs can take various legal forms, such as corporations, partnerships, or trusts, depending on the jurisdiction and the nature of the transaction. These entities are typically used to hold specific assets or liabilities off the balance sheet of the sponsoring company.

Enron used special-purpose vehicles (SPVs) and other creative accounting tricks to commit outright fraud. It would create fake sales and then move any resulting losses to an SPV to hide them from public scrutiny.

Special-Purpose Vehicle (SPV): An SPV is a type of SPE specifically designed to carry out a particular financial transaction or project. SPVs are commonly used in structured finance to pool assets, issue securities, or facilitate off-balance-sheet financing. They are often bankruptcy-remote entities, meaning that their assets and liabilities are separate from those of the sponsoring company, thereby protecting investors from the risks associated with the sponsor's financial condition.

Lack of Transparency:

Enron's management actively obscured the company's true financial condition through opaque financial reporting and disclosures.
Investors and analysts were unable to assess the risks associated with Enron's business activities, leading to misplaced confidence in the company's prospects and financial health.

Weak Internal Controls:

Enron's internal controls and risk management systems were inadequate to detect or prevent fraudulent activities.
There was a lack of segregation of duties, inadequate oversight of financial transactions, and insufficient whistleblower protections, allowing misconduct to go undetected for an extended period.

Auditor Failure:

Arthur Anderson was one of the big five accounting firms and it was the sole auditor of Enron.
Congressional inquiry later discovered that Arthur Anderson was implicit in this fraud and revoked its ability to practice accounting.
Enron's fraud caused both firms to fail.

Regulatory Capture:

Enron had significant influence over regulatory agencies and accounting firms, which further enabled its misconduct.
Regulatory agencies, such as the Securities and Exchange Commission (SEC), and accounting firms, including Arthur Andersen, failed to adequately scrutinize Enron's financial practices or challenge its accounting methods.

Lessons Learned:

The collapse of Enron in 2001 exposed the failures of its corporate governance system and highlighted the need for stronger regulatory oversight, enhanced transparency, and ethical leadership within corporations. Enron's failure was the fuel needed to bring Sarbanes-Oxley (SOX) to life in 2002. This law requires accountability for key corporate officers relative to the reliablity of reported financial statements.
The Enron scandal led to reforms in corporate governance practices, accounting standards, and securities regulations aimed at restoring investor confidence and preventing similar corporate failures in the future. Created Public Company Accounting Oversight Board (PCAOB), which holistically promotes a high standard for corporate governance.

Cyber Risk

Cyber risk is the risk of financial or reputational loss resulting from a breach in internal technology infrastructures.
This risk revolves around a hacker accessing system that result in thefy of money, information, or identity data (i.e., social security numbers, emails, passwords) of customers and employees.
Corporations spend billions of dollars every year to ensure the integrity of the technology systems, and some even purchase cyber insurance to outsource the risk of loss to the third party.
Case Study 1: The SWIFT Case

The SWIFT cyberattack case refers to a series of high-profile security breaches involving the SWIFT messaging system, which facilitates international financial transactions among banks and other financial institutions.
Background on SWIFT:

The Society for Worldwide Interbank Financial Telecommunication (SWIFT) is the global leader in electronically transferring funds between financial institutions.
SWIFT is a cooperative organization headquartered in Belgium that operates a secure messaging network used by banks and financial institutions worldwide to send and receive information about financial transactions.
It is supervised by a consortium of central bankers from Belgium, U.S, England, the European Central Bank, Japan and others.
If a business needs to move a sum of money from a bank in Tokyo to a bank in Berlin, the SWIFT will be the vehicle used to transfer the money. It conducts between 25 to 30 million transactions every day.
The SWIFT network facilitates the transfer of funds, securities, and other financial messages between banks across the globe.

Cyberattacks on Banks:

Beginning in 2015, a series of cyberattacks targeted banks and financial institutions connected to the SWIFT network.
The attackers used sophisticated malware and social engineering techniques to gain unauthorized access to banks' internal systems and manipulate SWIFT messages to facilitate fraudulent transfers of funds.

Modus Operandi:

The attackers typically compromised the security of banks' internal networks through various means, such as phishing emails, malware infections, or exploiting vulnerabilities in software or systems.
Once inside the network, they used specialized malware to manipulate or alter SWIFT messages to authorize fraudulent transactions or hide their tracks.

Notable Incidents:

One of the most prominent incidents occurred in February 2016 when hackers accessed the SWIFT system and stole $81 million from the Bangladesh Central Bank's account at the Federal Reserve Bank of New York.
The access was through the use of employee credentials (how these employee credentials were obtained was never fully resolved) and series of requests to transfer funds to various locations throughout Asia.
The goal was to steal $1 billion, but the Bank of New York stopped all the transactions after discovering the typo ("fandations" instead of "foundations") in one of the otherwise legimate looking requests.
After the $81 million was transferred to accounts in the Philippines, the hackers used malware to delete the record of the transfer and disable transaction confirmation notifications.
When the Bangladesh Bank realized that it was not receiving any transfer notifications, it rebooted their system and immediately received a flood of transfer notices, which brought the cyber theft into the light.
The money was never recovered because it was transferred from the bank of Philliphines to a series of casinos and promptly withdrawn.

Impact and Response:

The SWIFT cyberattacks raised concerns about the security of the global financial system and prompted SWIFT to enhance its security measures and protocols.
SWIFT introduced measures such as mandatory security controls, increased information sharing and collaboration among members, and improved fraud detection capabilities to mitigate the risk of similar attacks in the future.

Attribution and Investigation:

While the identity of the attackers behind the SWIFT cyberattacks remains unclear, cybersecurity experts have linked the attacks to sophisticated cybercriminal groups or state-sponsored actors with advanced capabilities and significant resources.
Investigations into the attacks have involved law enforcement agencies, cybersecurity firms, and international organizations collaborating to identify and disrupt cyber threats targeting the financial sector.

Overall, the SWIFT cyberattack case underscores the growing cybersecurity risks facing the financial industry and the need for robust security measures, collaboration, and vigilance to protect critical financial infrastructure from cyber threats.

References and Credits

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FRM 2023 Notes
Reading 9
https://chatgpt.com/
https://gemini.google.com/

HelloWorldEngineer

Thursday, 11 July 2024

Bond Yields and Return Calculations

Realized Returns

Bond Spreads

Yield to Maturity

Annuity and Perpetuity

Japanese Yields

The Relationships

Spot Rates and YTM

YTM, Coupon Rate and Price

Bond Return Decomposition

Carry Roll Down Scenarios

Credits and References

Thursday, 27 June 2024

Lessons from Financial Disasters - Part II

Introduction

Operational Risk

Financial Engineering Issues

Reputation Risk

Corporate Goverance Issues

Cyber Risk

References and Credits

Scarcity Brings Efficiency: Python RAM Optimization