Question

An investor purchased the following 5 bonds. Each bond had a par value of $1,000 and a 10% yield to maturity on the purchase day. Immediately after the investor purchased them, interest rates fell, and each then had a new YTM of 5%. What is the percentage change in price for each bond after the decline in interest rates? (Round all the answers below to two decimal places)

Price 10%	Price 5%	Percentage Change
10-year, 10% annual coupon
10-year zero
5-year zero
30-year zero
$100 perpetuity

Answer 1

Bond Valuation: The value of bond is the present value of the expected cashflows from the bond,discounted at Yield to Maturity(YTM).

10 year 10% annual coupon	Price @ 10% 1000.50	Price @ 5% 1386.20	Percentage Change 38.55%
10 year zero	385.55	613.87	59.22%
5 year zero	620.73	783.70	26.25%
30 year zero	57.31	231.37	303.72%
$100 perpetuity	1000.00	2000.00	100%

Working notes:

1.

Year	Cash flow	PVAF/PVF@10%	Present Value (Cashflow*PVAF/PVF)
1-10	100	6.145	614.50
10	1000	0.386	386.00

Bond Price = Cashflow*PVAF/PVF

= 614.50+386

= 1000.50

2. Price of a zero-coupon bond is the present value of future cashflows discounted at required rate of return. As far as a zero-coupon bond is concerned, it does not pay any coupon payments, it pays only a lump sum amount on its maturity.

Value of zero-coupon bond = Face vale / (1+required return)^{time to maturity}

Value of 10 year zero-coupon bond= 1000 / (1.10)^10

= 1000/2.5937

= 385.55

Value of 5 year zero-coupon bond= 1000 / (1.1)^5

= 1000/1.611

= 620.73

Value of 30 year zero-coupon bond= 1000 / (1.1)^30

= 1000/17.449

= 57.31

3. We have

Price of perpetual bond = Coupon per period / Discount rate

= 100/.1

= 1000

4.

Year	Cash flow	PVAF/PVF@5%	Present Value (Cashflow*PVAF/PVF)
1-10	100	7.722	772.20
10	1000	0.614	614.00

Bond Price = Cashflow*PVAF/PVF

= 772.20+614

= 1386.20

5.

Value of zero-coupon bond = Face vale / (1+required return)^{time to maturity}

Value of 10 year zero-coupon bond= 1000 / (1.05)^10

= 1000/1.629

= 613.87

Value of 5 year zero-coupon bond= 1000 / (1.05)^5

= 1000/1.276

= 783.70

Value of 30 year zero-coupon bond= 1000 / (1.05)^30

= 1000/4.322

= 231.37

Price of perpetual bond = Coupon per period / Discount rate

= 100/.05

= 2000