Question

A financial institution’s liability looks like a zero coupon bond. The institution has to pay
$1,378,843 in six years. The current market value of the liability is $1 million. The institution
would like to immunize its exposure to the interest rate risk by matching the Macaulay
durations. Therefore, this institute plans to invest $1 million in one of the following two bonds
and hold it for six years. Both bonds pay coupon annually. Each coupon payment will be reinvested
in the same bond immediately after the coupon payment.

Bond	Term to maturity (years)	Coupon rate (% pa)	Yield to Maturity (% pa)	Par Value
A	9	5.5	5.5	$100
B	7	5.5	5.5	$100

Assume that the investor choose to invest in Bond B. Assume that interest rate
decreases by 1% immediately and stayed at that level. The investor hold the bond for
5 years and sold it immediately afterwards. Each coupon payment will be re-invested
in the same bond immediately after the coupon payment. What is the investor’s
income from coupon and coupon reinvestment? What is the investor’s income
from selling the bond?

A. $200,430 $1,000,000
B. $369,429 $1,009,569
C. $300,430 $1,500,000
D. $300,430 $1,200,000

Answer 1

Correct answer is second option i.e. option B. $369,429 $1,009,569

MV of liability x Duration of liability = MV of Bond B x Duration of Bond B

Please see below the duration calculation for bond B.

Hence, MV of liability x Duration of liability = MV of Bond B x Duration of Bond B

Or, 1,000,000 x 6 = 5.99553 (say 6) x MV of bond B

Hence, MV of bond B = 1,000,000 x 6 / 6 = $  1,000,000

Since the bond B has same coupon rate as the interest rate, the bonds must have been purchased at par. Hence, par value of the bond B = Market value = 1,000,000

Annual coupon = 5.5% x 1,000,000 =  55,000; Nper = 6; Rate = YTM - 1% = 5.5% - 1% = 4.5%

Hence, the investor’s income from coupon and coupon reinvestment = -FV (Rate, Nper, PMT, PV) = -FV (4.5%, 6, 55000, 0) = $ 369,429

And the sale price of the bond after 6 years = -PV (Rate, Nper, PMT, FV) = - PV(4.5%, 7-6, 55000, 1000784) = $1,009,569

Both the answers are visible in the second option i.e. option B. $369,429 $1,009,569