INTEREST RATE TREES FOR CALLABLE BONDS

Question

The following yield curve is given in annual effective yields.

maturity => zero coupon spot rate

1year => 3%

2years => 3.5%

3years => 3.75%

4 years => 3.95%

a) 1 year from now you plan to take out a 2 year loan. What interest rate should you expect to pay on your loan?

HINT: This is effectively a zero coupon loan in which you borrow money at time 1 and pay back principle and interest at the end of year 3

b) You plan to invest $10,000 to invest for a period of 1 year. You can choose to buy a zero-coupon bond maturing in 1, 2, 3, or 4 years. Remember, you will be selling it in 1 year! If you assume the yield curve will not change over the next year, which investment choice has the highest rate of return? JUSTIFY your choice using the appropriate math.

Answer 1

a) To calculate the interest rate on the 2-year loan, we need to find the implied forward rate between years 2 and 3. Using the formula for calculating forward rates:

(1 + r3)^3 = (1 + r2)^2 * (1 + f(2,3))

where r3 is the 3-year spot rate, r2 is the 2-year spot rate, and f(2,3) is the implied forward rate between years 2 and 3.

Plugging in the values given:

(1 + 0.0375)^3 = (1 + 0.035)^2 * (1 + f(2,3))

Solving for f(2,3):

f(2,3) = 0.0437 or 4.37%

Therefore, the interest rate on the 2-year loan should be approximately 4.37%.

b) To determine the investment choice with the highest rate of return, we need to calculate the price of each zero-coupon bond and choose the bond with the lowest price, as that bond will give us the highest rate of return. The price of a zero-coupon bond is calculated as:

Price = Face Value / (1 + Yield)^(Maturity)

where Face Value is the amount received at maturity, Yield is the annual effective yield, and Maturity is the time to maturity in years.

For a zero-coupon bond with 1-year maturity, the price is:

Price(1 year) = $10,000 / (1 + 0.03)^1 = $9,708.74

For a zero-coupon bond with 2-year maturity, the price is:

Price(2 years) = $10,000 / (1 + 0.035)^2 = $9,558.11

For a zero-coupon bond with 3-year maturity, the price is:

Price(3 years) = $10,000 / (1 + 0.0375)^3 = $9,452.15

For a zero-coupon bond with 4-year maturity, the price is:

Price(4 years) = $10,000 / (1 + 0.0395)^4 = $9,380.45

Therefore, buying the 3-year zero-coupon bond will give us the highest rate of return, as it has the lowest price and hence the highest yield. The rate of return can be calculated as:

Rate of Return = (Price at Maturity / Price at Purchase)^(1 / Time) - 1

where Time is the investment horizon in years.

In this case, the rate of return for the 3-year zero-coupon bond is:

Rate of Return = ($10,000 / $9,452.15)^(1/1) - 1 = 0.0571 or 5.71%