Interest Rate Risk and Return

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Topics

• Introduction
• Sources of Return from Investing in a Fixed-Rate Bond
• Investment Horizon and Interest Rate Risk
• Macaulay Duration

Table of Contents

Introduction

This learning module covers:

• Sources of return from investing in a fixed rate bond
• How investors are exposed to different interest rate risk if the same bond is held for different time periods
• Macaulay duration

Sources of Return from Investing in a Fixed-Rate Bond

The total return is the future value of reinvested coupon interest payments and the sale price (or redemption of principal if the bond is held to maturity). The horizon yield (or holding period rate of return) is the internal rate of return between the total return and the purchase price of the bond.

Total Return on a Bond = Reinvested Coupon Interest Payments + Sale/Redemption of Principal at Maturity

A bond investor has three sources of return:

• Receiving the full coupon and principal payments on the scheduled dates.
• Reinvesting the interest payments. This is also known as interest-on-interest.
• Potential capital gain or loss on sale of the bond, if the bond is sold before maturity date.

Now, we will look at a series of examples that demonstrate the effect on two investors’ realized rate of returns when one of these variable changes: time horizon, interest rate at which the coupons are reinvested, and market discount rates at the time of purchase and at the time of sale.

However, this holds good only if all of the following three conditions are true:

• The bond is held to maturity.
• The coupon and final principal payments are made on time (no default or delay).
• The coupon payments are reinvested at the same rate of interest.

To calculate the total return on the bond, we first need to calculate the interest earned when coupon payments are reinvested.

Coupon Reinvestment:

• The investor receives 5 coupon payments of 10 (per 100 of par value) for a total of 50, plus the redemption of principal (100) at maturity. The investor has the opportunity to reinvest the cash flows. If the coupon payments are reinvested at 12% (i.e., yield to maturity), the future value of the coupons on the bond’s maturity date is 63.53 per 100 of par value. $$\sum_{i=0}^4 \left(10 \times (1.12)^i\right) =63.53$$
• The first coupon payment of 10 is reinvested at 12% for 4 years until maturity, the second is reinvested for 3 years, and so on. The future value of the annuity is -63.53.
• The amount in excess of the coupons, 13.53 (= 63.53 – 50), is the interest-on-interest gain from compounding.
• The investor’s total return is 163.53, the sum of the reinvested coupons (63.53) and the redemption of principal at maturity (100).

The realized rate of return is 12%. $$92.79= \frac{163.53}{(1+r)^5} ;;;;;r=12%$$

This r is called the horizon yield, the internal rate of return between the total return and the purchase price of the bond.

Horizon yield is equal to the original yield to maturity if:

• Coupon payments are reinvested at the same yield to maturity calculated at the time of purchase of the bond.
• There are no capital gains or losses when the bond is sold. It is sold at a price on the constant-yield price trajectory. We arrive at the price 96.62 by taking 12% as the constant yield for the remaining two years.
  • If the yield is more than 12%, then losses occur.
  • If the yield is less than 12%, then capital gains occur.

Constant-Yield Price Trajectory for a 5-year, 10% Annual Payment Bond

The price of the bond at different time periods for a yield of 12% is plotted in the graph below:

There will be a capital gain if the bond is sold at a price above the trajectory at any point in time during the bond’s life. This will happen if the yield is below 12%.

Remember a bond’s price and interest rates are inversely related.

Similarly, there will be a capital loss if the bond is sold at a price below the trajectory at any point in time during the bond’s life. This will happen if the yield is above 12%. Any point on the trajectory represents the carrying value of the bond at that time.

Observation: Compared to Example 1 (12%), the realized return of this investor is higher because the coupons are reinvested at a higher rate. There is no capital gain or loss because the bond is held to maturity and the principal of 100 is redeemed.

Observation: Compared to Example 2 with a similar time horizon (12%), the realized return to this investor is lower at 10.91% because there is a capital loss. Even though the coupons are reinvested at a higher rate, the capital loss is greater than the gain from reinvesting coupons.

Increase in the value of reinvested coupons = 0.99
Capital loss = 91.87 – 96.62 = -4.75

Capital gain or loss is always calculated relative to the carrying value at that point in time.

Observation: The realized return is lower than that in Example 1 (12%) because the coupons are reinvested at a lower rate of return. Since the bond is held to maturity, there is no capital gain or loss.

Decrease in the value of reinvested coupons = 58.67 – 63.53 = – 4.86

Observation: The realized return is greater than that of the investors in Examples 2 and 4 with a similar time horizon. It is primarily due to the capital gains.

Capital gain = 103.57 – 96.62 = 6.95
Decrease in the value of reinvested coupons = 32.46 – 33.74 = -1.28

As you can see, the capital gain is far greater than the decrease in the value of reinvested coupons.

Interest rate risk affects the realized rate of return for any bond investor in two ways: coupon reinvestment risk and market price risk. But what is interesting is that these are offsetting types of risk. Two investors with different time horizons will have different exposures to interest rate risk.

From the examples above, let us sum up what happens when interest rates go up or down:

When interest rates go up or down:

• Reinvestment income is directly proportional to interest rate movements. The value of reinvested coupons increases when the interest rate goes up.
• Bond price is inversely proportional to interest rate movements. Bond price decreases when the interest rate goes up.

When does coupon reinvestment risk matter?

• Coupon reinvestment risk matters when an investor has a long-term horizon. If the investor buys a bond and sells it before the first coupon payment, then this risk is irrelevant. But a buy-and-hold investor as in Examples 1, 3, and 5 has only coupon reinvestment risk.

When does market price risk matter?

• Market price risk matters when an investor has a short-term horizon relative to the time to maturity. If the investor buys a bond and sells it before the first coupon payment, then he is exposed to market price risk only. But a buy-and-hold investor such as those in Examples 1, 3, and 5 has only coupon reinvestment risk, and no market price risk.

Investment Horizon and Interest Rate Risk

The impact of a sudden change in yield on the price of a bond is of particular concern to short-term investors (price risk). Long-term investors will also be concerned about the impact of a change in yield on the reinvestment income (reinvestment risk). An investor who plans to hold the bond to maturity will only be concerned about reinvestment risk.

• If the investment horizon is 4 years, then the major concern is price risk. In this case, the price effect dominates relative to the gain/loss from reinvestment of coupons.
• If the investment horizon is 7 years, the reinvestment risk and price risk offset each other. For this particular bond the Macaulay duration is 7 years.

Macaulay duration (explained in the next section) is the investment horizon for which coupon reinvestment risk and market price risk offset each other.

• The assumption is a one-time parallel shift in the yield curve.
• Investment horizon is different from the bond’s maturity. In this case, the maturity is 10 years while the horizon is 7 years.

The duration gap of a bond is defined as the Macaulay duration – investment horizon.

Duration Gap = Macaulay Duration – Investment Horizon

• If Macaulay duration < investment horizon, the duration gap is negative: coupon reinvestment risk dominates.
• If Macaulay duration = investment horizon, the duration gap is zero: coupon reinvestment risk offsets market price risk.
• If Macaulay duration > investment horizon, the duration gap is positive: market price risk dominates.

Macaulay Duration

Macaulay duration is a weighted average of the time to receipt of the bond’s promised payments, where the weights are the shares of the full price that correspond to each of the bond’s promised future payments.

We can also use the following formula to calculate Macaulay Duration: $$\text{MacDur}=\left(\frac{1+r}{r}-\frac{1+r+(N(c-r))}{c((1+r)^N-1)+r} \right) - \frac{t}{T} $$
where:

• r = Yield to maturity
• t = # of days from the last coupon payment date to the settlement date
• T = # of days in the coupon period
• c = Coupon rate per period
• N = # of periods to maturity