###### Interest Rates

After completing this reading, you should be able to: Calculate and interpret the... **Read More**

**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 to derive a spread 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 perpetuity.
- Explain the relationship between spot rates and YTM.
- Define the coupon effect and explain the relationship between the 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.
- Explain the following four common assumptions in carry roll-down scenarios: realized forwards, unchanged term structure, consistent yields, and realized expectations of short-term rates; and calculate carry roll down under these assumptions.

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

$$ { \text{Gross realized return} }_{ \text{t}-1,\text{t} }=\cfrac { \text{Ending value}+ \text{Coupon – Beginning value} }{ \text{Beginning value} } $$

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**

$$ \begin{align*} { \text{Gross realized return} } & =\cfrac { \text{Ending value}+ \text{Coupon-Beginning value} }{ \text{Beginning value} } \\ & = \cfrac { 1,060+20-1000 }{ 1000 } =8\% \end{align*}$$

When calculating the gross realized return for multiple periods, it’s essential to consider whether coupons received are reinvested. If the coupons are reinvested, they will earn some interest at a given rate.

A bond purchased exactly six months ago for $1,000 paid a $20 coupon today. Suppose the coupon is reinvested at an annual rate of 4.4% for the next six months and that the bond is worth $1,080 after one year. What is the realized return on the bond over the one-year period?

**Solution**

$$ \begin{align*} { \text{Gross realized return} } & =\cfrac { \text{Ending value}+ \text{Coupon}+\text{Coupon Investment}-\text{Beginning value} }{ \text{Beginning value} } \\ & = \cfrac { 1,080+20+20\times1.022-1,000 }{ 1000 } =12.04\% \end{align*}$$

In addition to reinvestment income, we can also consider borrowing costs. If the investor buys the bond using borrowed funds, they will be expected to pay some interest at the end of the investment period. In these circumstances, the net realized return is calculated as follows:

$$\text{Net realised return}=\cfrac{\text{Ending value} + \text{Coupon}-\text{Beginning value} -\text{Financing costs}}{ \text{Beginning value} } $$

An investor purchased a bond exactly six months ago at $980 (per $1,000 nominal value). The purchase was entirely financed at an annual rate of 2%. Today, the bond is worth $995.

Given that the bond paid a coupon of $20 today, determine the net realized return

**Solution**

$ $\text{Net realised return}=\cfrac{\text{Ending value} + \text{Coupon}-\text{Beginning value} -\text{Financing costs}}{ \text{Beginning value} } $$

$$\text{Net realised return}=\cfrac{\text{995} + \text{20}-\text{980} -\text{9.8}}{ \text{980} }=2.57\% $$

where \(9.8 = \cfrac{2\%}{2}\times 980\)

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. We incorporate spread in the bond price formula as follows:

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:

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

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:

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

Yield to maturity (YTM) of fixed income security is the total return anticipated if we hold the security until it matures. Yield to maturity is considered a long-term bond yield, but we express it 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:

$$ \text{p}=\cfrac { { \text{C} }_{ 1 } }{ { \left( 1+\text{y} \right) }^{ 1 } } +\cfrac { { \text{C} }_{ 2 } }{ { \left( 1+\text{y} \right) }^{ 2 } } +\cfrac { { \text{C} }_{ 3 } }{ { \left( 1+\text{y} \right) }^{ 3 } } \dots +\cfrac { \text{F}+{\text{C} }_{ \text{N} } }{ { \left( 1+\text{y} \right) }^{ \text{N} } }$$

Where:

P = price of the bond

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

N = time to maturity in years

y = annual yield (YTM to maturity)

F = face value

Suppose a two-year bond with a coupon of 5% sells for USD 106. What is the yield to maturity expressed with semi-annual compounding?

We can find the yield \(y\) (expressed with semi-annual compounding) by solving the equation:

$$106=\frac{2.5}{1+y/2}+ \frac{2.5}{\left(1+y/2\right)^{2}}+ \frac{2.5}{\left(1+y/2\right)^{3}}+\frac{102.5}{\left(1+y/2\right)^{4}} $$

We can solve this by trial and error, to get y=1.93%

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

$$ \text{p}=\cfrac { { \text{C} }_{ 1 } }{ { \left( 1+\text{y} \right) }^{ 1 } } +\cfrac { { \text{C} }_{ 2 } }{ { \left( 1+\text{y} \right) }^{ 2 } } +\cfrac { { \text{C} }_{ 3 } }{ { \left( 1+\text{y} \right) }^{ 3 } } \dots +\cfrac { \text{F}+{ \text{C} }_{ \text{n} } }{ { \left( 1+\text{y} \right) }^{ \text{n} } } $$

Where:

P = price of the bond

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

n = N × m = number of periods (= years × number of periods per year)

F = face value

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. We use the formula below to determine YTM for zero-coupon bond:

$$ \text{Yield to maturity}={ \left( \cfrac { \text{Face value} }{ \text{Current price of bond} } \right) }^{ \cfrac { 1 }{ \text{year to maturity} } }-1 $$

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

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

$$ { \text{PV} }_{ \text{annuity} }=\text{PMT}\cfrac { 1-{ \left( 1+\text{r} \right) }^{ -\text{T} } }{ \text{r} } $$

Where:

r=discount rate

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:

$${ \text{PV} }_{ \text{perpetuity} }=\cfrac { \text{PMT} }{ \text{r} }$$

Suppose we receive a semi-annual coupon at the rate of USD 3 per annum forever. Suppose further that the yield to maturity is 6%. Then, the present value of the perpetuity is

$$\frac{3}{0.06}=\text{USD 50}$$

We can use both the spot rate and the yield to maturity 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 more accurate 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 applicable 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 describes the fact that reasonably priced bonds of the same maturity but different coupons have different yields to maturity, which 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 lower coupon bond will experience a higher percentage change in price compared to the bond with larger coupons. The most sensitive bonds are zero-coupon bonds, which do not make any coupon payments.

- 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 the 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.”

We generate the bond’s profitability or loss through price appreciation and explicit cash flows. The total price appreciates as follows:

$$ { \text{P} }_{ \text{t}+1 }\left( { \text{R} }_{ \text{t}+1 }{ \text{s} }_{ \text{t}+1 } \right) -{ \text{P} }_{ \text{t} }\left( { { \text{R} } }_{ \text{t} }{ \text{s} }_{ \text{t} } \right) $$

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 \(\text{R}^{e}\), as maturity approaches. It does not account for spread changes$$ \text{Carry roll down component}={ \text{P} }_{ \text{t}+1 }\left( { \text{R} }_{ \text{t}+1 }^{ \text{e} }{ \text{s} }_{ \text{t} } \right) -{ \text{P} }_{ \text{t} }\left( { \text{R} }_{ \text{t} }{ \text{s} }_{ \text{t} } \right) $$**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.$$ \text{Rate changes component}={ \text{P} }_{ \text{t}+1 }\left( {\text{R} }_{ \text{t}+1 }{ \text{s} }_{ \text{t} } \right) -{ \text{P} }_{ \text{t} }\left( { { \text{R} } }_{ \text{t}+1 }^{ \text{e} }{ \text{s} }_{ \text{t} } \right) $$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.

$$ \text{Spread change component} = { \text{P} }_{ \text{t}+1 }\left( { \text{R} }_{ \text{t}+1 }{ \text{s} }_{ \text{t}+1 } \right) -{ \text{P} }_{ \text{t}+1 }\left( { { \text{R} } }_{ \text{t}+1 }{ \text{s} }_{ \text{t} } \right) $$

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

## Practice 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 six months ago was reinvested at an annual rate of 2%. Given that the purchase price was entirely financed at a yearly rate of 1%, the net realized return of the bond is closest to:

A. 9%

B. 8%

C. 10%

D. 11%

The correct answer is

A.$$ \begin{align*} { \text{Netrealised return} }_{ \text{t}-1,\text{t} } & =\cfrac { { \text{BV} }_{ \text{t} }+{ \text{C} }_{ \text{t} }-{ \text{BV} }_{ t-1 } }{ { \text{BV} }_{ \text{t}-1 } } -\text{\% Financing Costs} \\ & =\cfrac { 1,060+20+20\left( 1+\cfrac { 2\% }{ 2 } \right) -1000 }{ 1000 } -1.0\% \\ & =10.02\%-1\%=9.02\% \approx 9\% \end{align*} $$

Alternatively, \(\text{Financial Costs}=0.01\times 1000=10\)

So that,

$$ \begin{align*} { \text{Netrealised return} }_{ \text{t}-1,\text{t} } & =\cfrac { { \text{BV} }_{ \text{t} }+{ \text{C} }_{ \text{t} }-{ \text{BV} }_{ t-1 }-\text{Financing Costs} }{ { \text{BV} }_{ \text{t}-1 } } \\ & =\cfrac { 1,060+20+20\left( 1+\cfrac { 2\% }{ 2 } \right) -1000 -10}{ 1000 } \\ & =9.02\% \approx 9\% \end{align*} $$

## 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 Dec 31, 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 higher than the yield. As such, the bond must have traded at a premium – implying the price must have been higher 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.”