Question

Question 1:

(a) A one-year zero coupon bond is currently priced at £96.154 and a two-year 10% coupon bond is currently priced at £107.515. Coupons are paid annually, the par value is £100 and all bonds are assumed to be issued by the UK government and are default risk-free. Calculate the one and two-year spot rates.

(b) Consider a three-year 10% annual coupon bond with a par value of £100. The term structure is flat at 6%

(i) Calculate the Macaulay duration and modified duration.

(ii) If the term structure shifts to 8% what is the actual change in the price of the bond? Approximate the change in the price of the bond using duration. How can we make the approximation more accurate?

(c) Bond A is a one-year zero coupon bond and is currently priced at £95.24. Bond B is a two-year 10% annual coupon bond and is currently priced at £107.42. Bond C is a two- year zero coupon bond. All bonds have a par value of £100 and are assumed to be issued by the UK government and are default risk-free. Calculate the the price of Bond C using the replicating portfolio method i.e. use Bond A and Bond B to replicate Bond C’s cash flows (do not calculate the price of Bond C using spot rates).

(d) The one-year spot rate is 3% and the two-year sport rate is 5%. A bond trader wants to invest £100 from t = 1 to t = 2 at the forward rate 1f1. How many units of a one-year zero coupon bond and a two-year zero coupon bond, par values £100, does the trader have to go long or short today, t = 0, to replicate a £100 investment from t = 1 to t = 2 that earns the forward rate 1f1? Show the resultant cash flows at t = 0, t = 1 and t = 2.

Question 2:

(a) Discuss the assumptions of the CAPM. Is a stock with a positive ↵ in relation to the secu- rity market line (SML) underpriced or overpriced? Explain.

(b) A stock is expected to pay its first dividend of £4 five years from today i.e. at t = 5. There- after, the dividend is expected to grow at an annual rate of 10% for the next four years and then grow at a constant rate of 2% per year forever. The appropriate discount rate for the dividends is 10% per year. What is the value of the stock today, t = 0?

(c) You are an investor and you want to form a portfolio that consists of two stocks, Stock A and Stock B, whose returns have the following characteristics:

Stock A Expected Return: 10%

Stock B Expected Return: 20%

Stock A Standard Deviation: 20%

Stock B Standard Deviation: 30%

Correlation Between A and B: 0.4

If you invest 50% of your wealth in Stock A and 50% of your wealth in Stock B what is your portfolio’s expected return and standard deviation? Without doing any calculations do you think your portfolio is the minimum variance portfolio (where the minimum variance portfolio is constructed using only Stock A and Stock B)? Explain.

(d) Now consider a third asset, the risk-free asset to combine with Stock A and Stock B. The risk-free rate has a return of 5%. If you invest 50% in the risk-free asset, 25% in Stock A and 25% in Stock B what is your portfolio’s expected return and standard deviation? Explain using your answer why a risk-averse investor would never want to hold Stock A on its own (i.e. a portfolio that has 100% invested in Stock A).

(e) Now consider only Stock A and Stock B but assume that the correlation between A and B is -1. If you want to construct a portfolio that has a standard deviation of 20% what is the maximum expected return possible? In this portfolio what weight would you have to hold in Stock A and Stock B?

Answer 1

Question 1:

(a) Given details

One-year zero coupon bond is currently priced at £ 96.154

Bonds are assumed to be issued by the UK government, so we first compute Risk Free rate

Risk free rate = (100 - 96.154)/96.154 = 0.399, ie 4%

Calculation of one and two-year spot rates of 10% coupon bond

two-year 10% coupon bond is currently priced at £107.515

One year spot rate = 107.515 * 1.04 = 111.8156

Two year spot rate = 107.515 * 1.04 * 1.04 =116.288

(b) Calculation of Macaulay duration and modified duration

given - three-year 10% annual coupon bond with a par value of £100, YTM = 6%

The Macaulay duration calculates the weighted average time before a bondholder would receive the bond's cash flows.

Modified duration measures the price sensitivity of a bond when there is a change in the yield to maturity.

The Macaulay duration is calculated by multiplying the time period by the periodic coupon payment and dividing the resulting value by 1 plus the periodic yield raised to the time to maturity. Next, the value is calculated for each period and added together. Then, the resulting value is added to the total number of periods multiplied by the par value, divided by 1, plus the periodic yield raised to the total number of periods. Then the value is divided by the current bond price.

Annual Coupon Payment = 10

Present Value of bond = 110.69

Modified Macaulay Duration = 2.5935

Computaional Notes

Years (T)	NCF	PV@6% NCF/(1+0.06)^T	Duration D (PV*T)
1	10	9.433962264	9.433962264
2	10	8.8999644	17.7999288
3	10+100 =110	92.358121134	277.074363402
		110.692047798	304.308254466

Present Value of Bond = £ 110.69

Macaulay Duration = (304.308254466) / 110.692047798 = 2.749

Modified Duration = 2.749 / 1.06 = 2.5935

If the term structure shifts to 8% what is the actual change in the price of the bond

Years (T)	NCF	PV@6% NCF/(1+0.08)^T
1	10	9.259259
2	10	8.573388
3	10+100 =110	87.32155
		105.1542

If the term structure shifts to 8% then the price of bond will be £ 105.1542.

That is, Bond Price reduces by £ 5.5358