Question

An investor purchased the following five bonds. Each bond had a par value of $1,000 and an 8% yield to maturity on the purchase day. Immediately after the investor purchased them, interest rates fell, and each then had a new YTM of 6%. What is the percentage change in price for each bond after the decline in interest rates? Fill in the following table. Enter all amounts as positive numbers. Do not round intermediate calculations. Round your monetary answers to the nearest cent and percentage answers to two decimal places. Price @ 8% Price @ 6% Percentage Change 10-year, 10% annual coupon $ $ % 10-year zero 5-year zero 30-year zero $100 perpetuity

Answer 1

Lets calculate the prices of each bond at 8% and 6%.

10 year, 10% annual coupon:

Price of the bond is calculated using the formula: P= C/(1+r)+C/(1+r)^2+....C/(1+r)^n+P/(1+r)^n; where C is the coupon payment per period, P is the face value of the bond, r is the yield to maturity and n is the number of years to maturity.

For the coupons part, we can use the formula of present value of annuity which is C*(1-(1+r)^-n)/r; where C is the annual cashflow, r is the discount rate and n is the number of years

For 8%, Price of the bond= (100*(1-1.08^-10)/8%)+(1000/1.08^10)= $1134.20

For 6%, Price of the bond= (100*(1-1.06^-10)/6%)+(1000/1.06^10)= $1294.40

10-year Zero:

For a zeo coupon bond, the price of the bond will be F/(1+r)^n where F is the Par value of the bond and r is the yield to maturity, and n is the number of years to maturity.

So, at 8%, Price of the bond= (1000/(1.08^10))= $463.19

at 6%, Price of the bond= (1000/(1.06^10))= $558.39

5-year Zero:

At 8%, Price of the bond= (1000/1.08^5)= $680.58

At 6%, Price of the bond= (1000/1.06^5)= $747.26

30-year Zero:

At 8%, Price of the bond= (1000/1.08^30)= $99.38

At 6%, Price of the bond= (1000/1.06^30)= $174.11

$100 Perpetuity:

Price of a perpetual bond is calculated as C/r; where C is the coupon payment and r is the YTM.

At 8%, Price of the bond= 1000/8%= $12500

At 6%, Price of the bond= 1000/6%= $16666.67

On Tabulating, we get,

	Price@8%	Price at 6%	% change in Price
10-year, 10% annual coupon	1134.20	1294.40	14.12%
10-year zero	463.19	558.39	20.55
5-year zero	680.58	747.26	9.80%
30-year zero	99.38	174.11	75.20
$100 perpetuity	12500	16666.67	33.33%