Question

The following table summarizes current prices of various default-free bonds with different maturities. Interest is paid semi-annually.

Year to maturity	Counpon rate(%)	Spot price(￡)
0.5	0.0	96.15
1.0	0.0	92.19
1.5	8.5	99.45
2.0	9.0	99.64

3.1 Compute the yield to maturity (YTM) and the theoretical spot rate for each of the four bonds. (14%)

3.2 Based on your answer in 3.1, compute, under the pure expectations theory, the forward rate on the six-month default-free bond six months from now. (4%)

3.3 Suppose you wanted to lock in an interest rate for an investment that beginsin one year and matures in one year. Under the pure expectations theory, what rate would you obtain if there are no arbitrage opportunities? Show your calculations. (3%)

3.4 Explain each of the following theories for the term structure of interest rates and discuss how each of them could explain an upward slope of the yield curve. (1) Pure expectations (unbiased) (2) Liquidity preference (term premium) (3) Market segmentation （12.3%）

Answer 1

3.1 Spot rate for first six months, one year, 1.5 years, 2 years be Y1, Y2, Y3 and Y4

100/(1+Y1/2) = 96.15

Y1 = 8.01%

100/(1+Y2/2)^2 = 92.19

Y2 = 8.30%

For 1.5 years, coupon = 8.5%*100/2 = 4.25

4.25/(1+8.01%/2) + 4.25/(1+8.30%/2)^2 + 104.25/(1+Y3/2)^3 = 99.45

Y3 = 8.93%

For 2 years, coupon = 9%*100/2 = 4.5

4.5/(1+8.01%/2) + 4.5/(1+8.30%/2)^2+ 4.5/(1+8.93%/2)^3 + 104.5/(1+Y4/2)^3 = 99.64

Y4 = 9.25%

For zero coupon bond, YTM is equal to spot rate

For 0.5 years bond, YTM = 8.01%

For 1 years bond, YTM = 8.30%

For 1.5 years bond, YTM:

4.25/(1+YTM/2)+4.25/(1+YTM/2)^2+104.25/(1+YTM/2)^3 = 99.45

YTM = 8.90%

For 2 years bond, YTM:

4.5/(1+YTM/2)+4.5/(1+YTM/2)^2+4.5/(1+YTM/2)^3+104.5/(1+YTM/2)^4 = 99.64

YTM = 9.20%

3.2 Let the forward rate be F1

(1+8.01%/2)*(1+F1/2) = (1+8.30%/2)^2

F1 = 8.59%

3.3 Let the forward rate be F2

(1+8.01%/2)*(1+8.30%/2)* (1+F2/2)^2 = (1+9.25%/2)^4

F1 = 10.35%

3.4 Pure Expectations Theory: It states that long-term interest expectations are based on short-term rates and future expectations of short-term rates. An upward sloping curve is explained by expected future short rates being higher than the current short rate

Liquidity Preference: It states that investors demand a higher premium for longer maturity as it carries higher maturity risk. Yields on long-term bonds are greater than the expected return from rolling-over short-term bonds in order to compensate investors in long-term bonds for bearing interest rate risk

Market Segmentation: It states that demand and supply for various maturities of bonds are different. So investors have a higher demand for short-term bonds and higher supply for long-term bond, thus making the yield curve upward sloping.