Question

A five-year bond with a yield of 7% (continuously compounded) pays a 5.5% coupon at the end of each year.

1. What is the bond’s price?
2. What is the bond’s duration?
3. Use the duration to calculate the effect on the bond’s price of a 0.3% decrease in its yield.
4. Recalculate the bond’s price on the basis of a 6.7% per annum yield and verify that the result is in agreement with your answer to (c).

Answer 1

a.) Price of the bond is sum of all present value (discounted value) cash flow generated by that bond.Now suppose that the face value of bond is $ 1,000. And coupon rate is given 5.5% payable at the end of the each year and Yield is 7%. So price of the bond will calculated by Present value application.

Present value of the future cash flow is calculated:

PV (present value) = FV(future value) / (1+r)^n

r = discounting rate

n = time period

Here the discounting rate is 7% (yield)

So cash flow for the above bond is given below:

Year	Cash flow from the bond
0	($ 1,000)
1	$ 55
2	$ 55
3	$ 55
4	$ 55
5	$ 1055

At 0 year you purchase the bond. So there is out flow of cash of $ 1,000 and after that you are receiving cash and the end of 5 year bond is redeem and you get $ 1055 (principal + coupon payment).

So Price of Bond = C1 / (1+r)^1 +C2 / (1+r)^2 + C3 / (1+r)^3 + .......... +(Cn + FV) / (1+r)^n

PB = price of bond to present value

Ct = coupon payment

r= yield rate

n = number of periods to maturity

FV = Face value

So present value of the all cash flow is calculated below:

Year	Cash flow from the bond	Present value
1	$ 55	$ 51.40
2	$ 55	$ 48.04
3	$ 55	$ 44.90
4	$ 55	$ 41.96
5	$ 1055	$ 752.20
Total		$ 938.5

Note for Calculation: r= 7%, coupon rate = 5.5%

( $ 55) / (1+7%)^1 + ( $ 55) / (1+7%)^2 + ...... + ($ 752.20) / (1+7%)^5 = $ 938.5

Note all figure in brackets in table is in negative.

So the price of the bond is $ 938.5

b.) Duration of bond: Duration of bond is to measure price sensitivity of the bond by changing in interest rate. And for finding duration of the bond Macaulay Duration formula is used. Actually duraiton measure how much time required in years by investor to repay bond price by bond's cash flow.

To find Macaulay duration:

Step 1: Find cash flow from bond (se first table above)

Step 2: Find present value of all cash flow and then do sum of it. (see second table above)

Step 3: Divide each present value of each cash flow by Total of them. and multiply by year

Step 4: Do total of step 3 (Look below table)

Year	Cash flow from the bond	Present value	(presenr value / Total) * (year)
1	$ 55	$ 51.40	0.05477
2	$ 55	$ 48.04	0.1024
3	$ 55	$ 44.90	0.1435
4	$ 55	$ 41.96	0.1788
5	$ 1055	$ 752.20	4.0075
Total		$ 938.5	4.4870

The Maculay duration of the bond is 4.487 year.

c.) Effect of 0.3% decrease in yield

For this we need to find Modified duration. It's show how the price of bond will change due to increase or decrease in yield.

Formula to find modified duration = Maculay duration / (1+ Yield rate)

modified duration = 4.487 / (1+7%) = 4.19%

This show that if Yield rate increase or decreasse by 1% then price of bond will increase or decrease by 4.19%. Remember one thing that bond price has inverse relationship with it's yield.

So if there is decrease of 0.3% in yield of the bond then the price of bond will increase by 4.19 * 0.3 = 1.26%. So price of bond will be $ 938.50 * (1+1.26%) = $ 950.33

d.) Verify:

Calculation of bond price will remain same only there will change in yield rate (i.e 6.7%)

Year	Cash flow from the bond	Present value
1	$ 55	$ 51.55
2	$ 55	$ 48.31
3	$ 55	$ 45.28
4	$ 55	$ 42.43
5	$ 1055	$ 762.83
Total		$ 950.4

So here the bond price is same as in case of 'C'.