After completing this reading you should be able to:

- Distinguish between gross and net realized returns, and calculate the realized return for a bond over a holding period including reinvestments.
- Define and interpret the spread of a bond, and explain how a spread is derived from a bond price and a term structure of rates.
- Define, interpret, and apply a bond’s yield-to-maturity (YTM) to bond pricing.
- Compute a bond’s YTM given a bond structure and price.
- Calculate the price of an annuity and a perpetuity.
- Explain the relationship between spot rates and YTM.
- Define the coupon effect and explain the relationship between coupon rate, YTM, and bond prices.
- Explain the decomposition of P&L for a bond into separate factors including carry roll-down, rate change, and spread change effects.
- Identify the most common assumptions in carry roll-down scenarios, including realized forwards, unchanged term structure, and unchanged yields.

## Gross vs Net Realized Returns

The gross realized return on an investment has two components: Any increase in price of the asset plus income received while holding the investment. When dealing with bonds,

$$ { Gross\quad realized\quad return }_{ t-1,t }=\frac { { Ending\quad value }+{ Coupon }-{ Beginning\quad value }}{ Beginning\quad value } $$

**Example of a bond’s gross realized return over six months:**

What is the gross realized return for a bond that is currently selling for $1,060 if it was purchased exactly six-months ago for $1,000 and paid a $20 coupon today?

**Solution:**

$$ { Gross\quad realized\quad return }=\frac { { Ending\quad value }+{ Coupon }-{ Beginning\quad value }}{ Beginning\quad value } $$

$$ =\frac { 1,060+20-1000 }{ 1000 } =8\% $$

Now let’s assume the investor financed the purchase of the bond by borrowing cash to make the investment at a rate of **1% for 6 months**. The gross realized return less per period financing costs gives the **net realized return**.

$$ { Net\quad realized\quad return }=\frac { { Ending\quad value }+{ Coupon }-{ Beginning\quad value }}{ Beginning\quad value } – \frac { 1\% }{ 2 } $$

When calculating the gross/net realized return for multiple periods, it’s important to consider whether coupons received are reinvested. If reinvested, the coupons will grow at the reinvestment rate from the time they are received up to the end of the period. The risk that reinvested cash flows will grow by a rate that’s lower than the expected rate is known as **reinvestment risk**.

## Spread of a Bond

The spread of a bond is the difference between its market price and the price computed according to spot rates or forward rates – the term structure of interest rates.

As a relative measure, a bond’s spread helps us determine whether the bond is trading cheap or rich relative to the yield curve. This is how we incorporate spread in the bond price formula:

Recall that given a 2 year bond with a face value of \(P\), paying annual coupons each of amount \(C\), its price is given by:

$$ Market\quad bond\quad price=\frac { C }{ 1+f\left( 1.0 \right) } +\frac { C+P }{ \left( 1+f\left( 1.0 \right) \right) \times \left( 1+f\left( 2.0 \right) \right) } $$

To incorporate the spread \(s\), we assume that the bond is trading at a premium or discount to this computed price. We can find the bond’s spread using the following formula:

$$ =\frac { C }{ \left( 1+f\left( 1.0 \right)+s \right) } +\frac { C+P }{ \left( 1+f\left( 1.0 \right)+s \right) \times \left( 1+f\left( 2.0 \right) +s \right) } $$

## Yield to Maturity

Yield to maturity (YTM) of a fixed income security is the total return anticipated if the security is held until it matures. Yield to maturity is considered a long-term bond yield, but is expressed as an annual rate. In other words, it’s the security’s internal rate of return as long as the investor holds it up to maturity.

To compute a bond’s yield to maturity, we use the following formula:

$$ P=\frac { { C }_{ 1 } }{ { \left( 1+y \right) }^{ 1 } } +\frac { { C }_{ 2 } }{ { \left( 1+y \right) }^{ 2 } } +\frac { { C }_{ 3 } }{ { \left( 1+y \right) }^{ 3 } } +\cdots +\frac { { C }_{ N } }{ { \left( 1+y \right) }^{ N } } $$

Where:

\(P\)=price of the bond

\({ C }_{ t }\)=annual cash flow in year \(t\)

\(N\)=time to maturity in years

\(y\)=annual yield (YTM to maturity)

When cash flows are received multiple times every year, we can slightly modify the above formula such that:

$$ P=\frac { { C }_{ 1 } }{ { \left( 1+y \right) }^{ 1 } } +\frac { { C }_{ 2 } }{ { \left( 1+y \right) }^{ 2 } } +\frac { { C }_{ 3 } }{ { \left( 1+y \right) }^{ 3 } } +\cdots +\frac { { C }_{ n } }{ { \left( 1+y \right) }^{ n } } $$

Where:

\(P\)=price of the bond

\({ C }_{ t }\)=periodic cash flow in period \(t\)

\(n=N \times m\)=number of periods (\(=years \times number \quad of \quad periods \quad per \quad year\))

\(y\)=periodic yield

Provided all cash flows received are reinvested at the YTM, the yield to maturity is equal to the bond’s realized return.

For zero-coupon bonds that are not accompanied by recurring coupon payments, the yield to maturity is equal to the normal rate of return of the bond. The formula to determine YTM for zero-coupon bonds is given below:

$$ Yield\quad to\quad maturity={ \left( \frac { Face\quad value }{ Current\quad price\quad of\quad bond } \right) }^{ \frac { 1 }{ years\quad to\quad maturity } }-1 $$

**Exam tip:** The yield to maturity assumes cash flows will be reinvested at the YTM and assumes that the bond will be held until maturity.

## Prices of Annuities and Perpetuities

An annuity is a series of annual payments of *PMT* until final time *T*. The value of an ordinary annuity is given by:

$$ { PV }_{ annuity }=PMT \frac { 1-{ (1+r) }^{ -T }}{ r } $$

Where:

\(r\)=discount rate

A perpetuity is a type of annuity whose cash flows continue for an **infinite amount of time**. The present value of a perpetuity is given by:

$$ { PV }_{ perpetuity }=\frac { PMT }{ r } $$

## The Relationship between Spot Rates and YTM

The spot rate and the yield to maturity can both be used to determine the fair market price of a bond. However, while the yield to maturity is constant, the spot rate varies from one period to the next to reflect interest rate expectations as time goes.

The spot rate is a truer measure of the fair market price when interest rates are believed to rise and fall over the coming years.

Given a bond’s cash flows and the relevant spot rates, you can easily calculate the price of a bond. You can then determine the bond’s YTM by equating the price to the present values of cash flows discounted at the YTM.

## The Coupon Effect

The coupon effect describes the fact that fairly priced bonds of the same maturity but different coupons have different yields to maturity. This implies that yield is not a reliable measure of relative value. Even if fixed-income security A has a higher yield than fixed security B, A is necessarily not a better investment.

It also follows that if two bonds have identical features save for the coupon, the bond with the smaller coupon is more sensitive to interest rate changes. In other words, given a change in yield, the smaller coupon bond will experience a higher percentage change in price compared to the bond with larger coupons. The most sensitive bonds are obviously zero-coupon bonds, which do not make any coupon payments.

**Exam tips:**

- The lower the coupon rate, the higher the interest-rate risk. The greater the coupon rate, the lower the interest rate risk.
- If coupon rate > YTM, the bond will sell for more than par value, or at a premium.
- If coupon rate < YTM, the bond will sell for less than par value, or at a discount.
- If coupon rate = YTM, the bond will sell for par value.

Over time, the price of premium bonds will gradually fall until they trade at par value at maturity. Similarly, the price of discount bonds will gradually rise to par value as maturity gets closer. This phenomenon is known as “pulling to par”.

## Components of a Bond’s P&L

A bond’s profitability or loss is generated through price appreciation and explicit cash flows. The total price appreciate is as follows:

$$𝑃_{𝑡+1} (𝑅_{𝑡+1} 𝑠_{𝑡+1} )− 𝑃_{𝑡} (𝑅_{𝑡} 𝑠_{𝑡} )$$

There are three components of price appreciation:

**Carry-roll-down component**: The carry-roll-down component comprises of price changes that emanate from a deviation of term structure from the original structure to an expected term structure, denoted as \({ R }^{ e }\), as maturity approaches. It does not account for spread changes$$ Carry\quad roll\quad down\quad component=𝑃_{𝑡+1} (𝑅^{e}_{𝑡+1} 𝑠_{𝑡} )− 𝑃_{𝑡} (𝑅_{𝑡} 𝑠_{𝑡} ) $$

**Rate changes component**: The rate changes component accounts for price changes due to interest rate movements from an expected term structure to the term structure that exists at time \(t+1\).$$ Rate\quad changes\quad component=𝑃_{𝑡+1} (𝑅_{𝑡+1} 𝑠_{𝑡} )− 𝑃_{𝑡} (𝑅^{e}_{𝑡+1} 𝑠_{𝑡} ) $$

This component also doesn’t account for spread changes.

**Spread change component**: As the words suggest, the spread change component accounts for price changes emanating from changes in the bond’s spread from time \(t\) to \(t+1\).$$ Spread\quad change\quad component=𝑃_{𝑡+1} (𝑅_{𝑡+1} 𝑠_{𝑡+1} )− 𝑃_{𝑡+1} (𝑅_{𝑡+1} 𝑠_{𝑡} )$$

**Assumptions ****in ****Carry-Roll-Down Scenarios**

**Realized forwards:**The return to a bond held to maturity is the same as rolling the investment one period at a times at the forwards rates. However, in reality, some forwards are realized above or below the initial forwards.**Unchanged term structure****Unchanged yields**

## Questions

### Question 1

A bond currently selling for $1,060 was purchased exactly 12 months ago for $1,000 and paid a $20 coupon six months ago. Today, the bond paid a $20 coupon. The coupon received 6 months ago was reinvested at an annual rate of 2%. Given that the purchase price was entirely financed at an annual rate of 1%, the net realized return of the bond is closest to:

- 11.0%
- 9.0%
- 10.0%
- 9.52%

The correct answer is **D**.

Solution:

$$ { Net\quad realized\quad return }_{ t-1,t }=\frac { { BV }_{ t }+{ C }_{ t }-{ BV }_{ t-1 } }{ { BV }_{ t-1 } } -financing\quad costs $$

$$ =\frac { 1,060+20+20\left( 1+\cfrac { 2\% }{ 2 } \right) -1000 }{ 1000 } -\cfrac { 1.0\% }{ 2 } $$

$$ =10.02\%-0.5\%=9.52\% $$

### Question 2

On Jan 1 2017, Commercial Bank of India issued a six-year bond paying an annual coupon of 6% at a price reflecting a yield to maturity of 4%. As of 31 Dec 2017, interest rates remain unchanged. Holding all other factors constant, and assuming a flat term structure of interest rates, how was the bond’s price affected? The price:

- Remained constant
- Decreased
- Increased
- Increased, but only in the second half of the year

The correct answer is **B**.

From the data given, it’s clear that the bond’s coupon is greater than the yield. As such, the bond must have traded at a premium – implying the price must have been greater than the face value. Provided the yield doesn’t change, a bond’s price will always converge to its face value. Since the price starts higher, it must decrease. This phenomenon is called ‘pulling to par.”