There are 3 sources of return in a fixed-income security:

- Receipt of the promised coupon and principal payments;
- Reinvestment of coupon payments; and/or
- Capital gains or losses on the sale of the bond prior to maturity.

A discount bond offers a “deficient” coupon rate, or a “below the market” discount rate. For a premium bond, the coupon rate exceeds the market discount rate. Through amortization, the carrying value of a bond reaches the par value at maturity.

A change in interest rates has a direct effect on an investor’s realized rate of return. Coupon payments are reinvested at the prevailing interest rate.

**Example**

An investor initially buys a 5-year, 8% annual coupon payment bond at the price of 85.00 per 100 of par value.

### Case 1: Holding the Bond to Maturity

First, note that the yield to maturity of the bond is calculated as follows.

$$85 = \frac { 8 }{ { (1+r) }^{ 1 } } +\frac { 8 }{ { (1+r) }^{ 2 } } +\frac { 8 }{ { (1+r) }^{ 3 } } +\frac { 8 }{ { (1+r) }^{ 4 } } +\frac { 108 }{ { (1+r) }^{ 5 } } ; \quad r = 12.18% $$

The bond’s yield-to-maturity is 12.18%. As seen previously, the easiest way to find the value of *r* is with the financial calculator:

n=5; PV=-85; PMT=8; FV=100; CPT => I/Y = 12.18

So the investor receives the series of 5 coupon payments of 8 (per 100 of par value) for a total of 40, plus the redemption of principal (100) at maturity. On top of collecting the coupon interest and the principal, there’s an opportunity to reinvest the cash flows.

If the coupon payments are reinvested at 12.18% immediately they are received, the future value of the coupons on maturity date will amount to 51.004 per 100 par value.

$$

\begin{array}{l|l|l|l|l}

\text{End of Year 1} & \text{End of Year 2} & \text{End of Year 3} & \text{End of Year 4} & \text{End of Year 5} \\

\hline

\$8\times { 1.1218 }^{ 4 } & \$8\times { 1.1218 }^{ 3 } & \$8\times { 1.1218 }^{ 2 } & \$8\times { 1.1218 }^{ 1 } & \$8\times { 1.1218 }^{ 0 } \\

\end{array}

$$

Total = $51.004

The 1^{st} coupon payment of $8 is reinvested at 12.18% for 4 years until the end of 5^{th} year, the 2^{nd} is invested for 3 years, and so forth. The amount in excess of the coupons, 11.004 (= 51.004 – 5 × 8), is called “interest-on-interest” gain from compounding.

The investor’s total return is 151.004, the sum of reinvested coupons (51.004) and the redemption of principal at maturity (100). The realized rate of return is 12.18%, as we have calculated previously.

$$85=\frac { 151.004 }{ { (1+r) }^{ 5 } } ; \quad r = 12.18\%$$

As case 1 demonstrates, the yield-to-maturity at the time of purchase equals the investor’s rate of return under three assumptions: (1) The investor holds the bond to maturity, (2) there is no default by the issuer, and (3) the coupon interest payments are reinvested at that same rate of interest.

### Case 2: Selling a Bond before Maturity

Let’s assume we have a second investor who buys the 5-year, 8% annual coupon payment bond but sells the bond after four years. Assuming that the coupon payments are reinvested at 12.18% for four years, the future value of the reinvested coupons is 38.3356 per 100 of par value.

$$

\begin{array}{l|l|l|l}

\text{End of Year 1} & \text{End of Year 2} & \text{End of Year 3} & \text{End of Year 4} \\

\hline

\$8\times { 1.1218 }^{ 4 } & \$8\times { 1.1218 }^{ 3 } & \$8\times { 1.1218 }^{ 2 } & \$8\times { 1.1218 }^{ 1 } \\

\end{array}

$$

Total = $38.3356

The 1^{st} coupon payment of 8 is reinvested at 12.18% for 3 years until the end of 4^{th} year maturity period, the 2^{nd} is invested for 2 years, and so forth.

The interest-on-interest gain from compounding is 6.3356 (= 38.3356 – 32).

At the time the bond is sold, it has one year remaining until maturity. If the yield-to-maturity remains 12.18%, the sale price of the bond (simply calculated as the PV of anticipated cash flows) is:

$$Price_{ t=4 }=\frac { 108 }{ 1.1218 } =96.2738$$

Therefore, the total return is 134.6094 (= 38.3356 + 96.2738) and the realized rate of return is 12.18%.

$$85=\frac { 134.6094 }{ { (1+r) }^{ 4 } } ; \quad r = 12.18\%$$

A horizon yield is the internal rate of return between the total return (the sum of reinvested coupon payments and the sale price or redemption amount) and the purchase price of the bond. In case 2, the horizon yield is 12.18% but for only 4 years as opposed to the 5 years of case 1.

Case 2 demonstrates that the realized horizon yield matches the original yield-to-maturity provided two conditions are met: (1) coupon payments are reinvested at the same interest rate as the original yield-to-maturity, and (2) the bond is sold at a price on the constant-yield price trajectory, i.e., the investor does not have any capital gains or losses when the bond is sold. The price trajectory is the time series of a bond’s prices from some date (usually the date that the bond is purchased) until the bond’s maturity

Capital gains are realized when a bond is sold at a price above its constant-yield price trajectory and capital losses may occur when a bond is sold at a price below its constant-yield price trajectory.

The investment horizon is critical in assessing interest rate risks and returns. The interest rate risk is composed of 2 offsetting risks, which are coupon reinvestment risk and market price risk.

**Reinvestment risk**

Reinvestment risk is the chance that an investor will not be able to * reinvest* cash flows from an investment at a rate equal to the investment’s current rate of return (yield to maturity). Two factors affect the degree of reinvestment risk:

**Maturity**: The longer the maturity of the bond, the higher the reinvestment risk because the higher the likelihood that interest rates will be lower than they were at the time the bond was purchased.**Coupon rate of the bond**: The higher the coupon rate, the bigger the payments that have to be reinvested, and, consequently, the reinvestment risk. In fact, a bond selling at a premium is more dependent on reinvestment income than another bond selling at par. The only fixed-income instruments that do not have reinvestment risk are zero coupon bonds since they have no interim coupon payments.

**Market (price) risk**

Bond market prices will decrease in value when the prevailing interest rates rise. In other words, if an investor wishes to sell the bond prior to maturity, the sale price will be lower if rates are higher.

As noted earlier, these two risks offset each other to an extent. The dominant risk depends in part on the investment horizon. The shorter the investment horizon, the smaller the reinvestment risk but the bigger the market risk.

QuestionWhich of the following sources of bond return is

most likelysubject to interest rate risk assuming the bond is held until maturity?A. Redemption of principal at maturity

B. Reinvestment of coupon payments

C. Original purchase price depending on the investment horizon

SolutionThe correct answer is B.

The interest rate risk results mainly from changes in coupon reinvestment rates. Higher interest rates mean higher income from the reinvestment of coupon payments.

*Reading 46 LOS 46a:*

*Calculate and interpret the sources of return from investing in a fixed-rate bond*