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

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

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