Question

(just need to answer part b)

a) You are a financial consultant and an expert on contingent immunization strategies. A client has asked you to make sure that her portfolio has a value of at least $500,000 at the end of six years. The current value of her portfolio is $390,000. The bonds that you have at your disposal currently yield an effective annual rate of 5%.

1. Given the current interest rate, what amount would need to be invested today to achieve the requested goal? [1 mark]

2. Suppose that four years have passed, and the interest rate is 9%. What is the trigger point of your client’s portfolio currently? That is, how low can the value of the portfolio get before you are forced to immunize your strategy in order to achieve the requested goal? [2 mark]

3. If the portfolio's value after four years is $405,811 (i.e., there are two years left to meet the original goal) what should you do? What effective annual rate is required to meet the original goal? [2 marks]

b)You are about to purchase a 10-year par bond with a 5% coupon rate paid annually.

1.What are the duration and the convexity of this bond? [4 marks] Using Macaulay's duration formula

2. Assume that right after you purchase the bond an economic announcement drives the YTM to 7%. What is the new price of the bond? [1 mark]

3. What price would be predicted by the duration rule after the YTM increases to 7%? Is this answer the same as the one reported in part (2)? Why? [2 marks]

4. What price would be predicted by the duration-with-convexity rule after the YTM increases to 7%? Is this answer the same as that reported in part (2)? Why? [2 marks]

Answer 1

Since it is a par bond, assume that par value is 100 so current price is also 100 and annual yield (YTM) is equal to the coupon rate of 5%. Annual coupon will be 5%*100 = 5.

Macaulay duration = 8.11 years (Calculations in the table below)

Modified duration = macaulay duration/(1+YTM) = 8.11/(1+5%) = 7.72 years

Using the total from Col.[6] in the table above, convexity = 8,268.49/(price*(1+YTM)^2) = 8,268.49/(100*(1+5%)^2) = 75.00

a). If new YTM = 7% then price of the bond is: FV = 100; PMT = 5; N = 10; rate = 7%, solve for PV. New price = 85.95

b). New price using macaulay duration:

Change in price = -current price*change in YTM*duration/(1+current YTM) = -100*0.02*8.11/(1+5%) = -15.44

New price = old price + change in price = 100 -15.44 = 84.56

This answer is slightly different from the one calculated in part (a) as this is an approximation, not the exact formula.

c). New price using duration plus convexity:

Change in price = [(-modified duration*change in YTM) + (convexity/2*change in YTM^2)]*current price

= [(-7.72*0.02) + (75/2*0.02^2)]*100 = -13.94

New price = 100-13.94 = 86.06

This answer is also slightly different from the one calculated in part (a) as this is also an approximation but it is a better approximation than using only duration as done in part (b).