Primary and Secondary Fixed-income Mar ...
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There are 3 sources of return in a fixed-income security:
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. This is because coupon payments are reinvested at the prevailing interest rate.
An investor initially buys a 5-year, 8% annual coupon payment bond at the price of 85.00 per 100 of par value.
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 to use 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. Besides 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 after they are received, the coupon’s future value 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 1st coupon payment of $8 is reinvested at 12.18% for 4 years until the end of the 5th year, the 2nd 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). Therefore, 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 the following three assumptions:
Let’s assume we have a second investor who buys the 5-year, 8% annual coupon payment bond but sells it 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 first coupon payment of 8 is reinvested at 12.18% for 3 years until the end of the 5th year maturity period, the 2nd 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 (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 only for 4 years instead of 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:
Capital gains are realized when a bond is sold at a price above its constant-yield price trajectory. It is noteworthy that 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 comprises 2 offsetting risks:
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:
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 partially on the investment horizon. The shorter the investment horizon, the smaller the reinvestment risk, but the bigger the market risk.
Question
Which of the following sources of bond return is most likely subject to interest rate risk assuming the bond is held until maturity?
- Reinvestment of coupon payments.
- Redemption of principal at maturity.
- Original purchase price depending on the investment horizon.
Solution
The correct answer is A.
The interest rate risk results mainly from changes in coupon reinvestment rates. Higher interest rates mean higher income from the reinvestment of coupon payments.