Question

You need a payout of $550,000 in 5 years. You find this semi-annual bond available for
purchase:

PV $941.34
FV $1000.00
YTM 8.00%
Coupon 6.75%
Years to maturity 6
Duration 4.9936

a) How many bonds do you need to purchase today to achieve the $550,000 payout in
5 years? (Show in dollars and bonds. Round to the nearest bond.) (2 make-up points

b) Show that if rates rise to 9% immediately after you purchase the bond that you will
still achieve your necessary payout of $550,000. That is, show what you will receive
in year 5 from selling and bond and from reinvested coupons.

Answer 1

a) Payout at the end of 5 years is a combination of 2 things:

1. Reinvested coupons
2. Price of the bond

First, we shall find the accumulated value of reinvested coupons over 5 years:

Time in semi-annual periods	1	2	3	4	5	6	7	8	9	10
Coupon payments	33.75	33.75	33.75	33.75	33.75	33.75	33.75	33.75	33.75	33.75
Compounding factor @4%	1.423311812	1.36856905	1.31593178	1.26531902	1.2166529	1.16985856	1.124864	1.0816	1.04	1
Future Value at the end of 5 years	48.03677367	46.18920545	44.4126975	42.7045169	41.0620355	39.4827264	37.96416	36.504	35.1	33.75
Total future value at the end of 5 years	405.2061154

It comes to $405.2061154 per bond

Now, we shall find the price per bond at the end of 5 years

The price shall be the present value of the future cash flows from the bond. The formula of the price of a bond:

,
where MP = Market price
C = coupon = 67.5/2 = $33.75
r = rate of interest = YTM = (8/2)% = 4%
n = time period = 1,2 for coupons and 2 for FV
FV = future value of bond = $1000

Price of the bond after 5 years:

Time Period after 5 years	1	2
Coupon	33.75	33.75
Nominal Value		1000
Total	33.75	1033.75
Discounting factor @4%	0.961538462	0.924556213
Present Value	32.45192308	955.7599852
Price	988.2119083

Payout needed =$550,000
Accumulated Coupon per bond = $405.2061154
Price per bond = $988.2119083
Total payout per bond = $1393.418024

Bonds needed = payout needed / total payout per bond = 550,00/1393.418024

= 394.71285 bonds

Rounding off, we have to purchase 395 bonds (As the payout of 550,000 will not be achieved with 394 bonds)

$ value of bonds purchased = 395*941.34 = $371,829.3

b) If interest rates rise to 9%, both the reinvested value of coupons at the end of five years and the price of the bond shall be the same. The reinvestment effect and the price effect shall cancel each other and the payout shall be more or less the same.
Let us see how.
We shall find the reinvested value of coupons and the price per bond using the same formulae, except that a rate of (9/2)% per semi-annual period shall be used instead of 4% used earlier.

Accumulated coupon Reinvestment amount:

Time in semi-annual periods	1	2	3	4	5	6	7	8	9	10
Coupon payments	33.75	33.75	33.75	33.75	33.75	33.75	33.75	33.75	33.75	33.75
Compounding factor @4%	1.48609514	1.422100613	1.36086183	1.30226012	1.24618194	1.1925186	1.14116613	1.092025	1.045	1
Future Value at the end of 5 years	50.15571099	47.99589568	45.9290868	43.9512792	42.0586404	40.2475028	38.5143567	36.8558438	35.26875	33.75
Total future value at the end of 5 years	414.7270663

Note that the accumulated amount has slightly increased per bond.

Price per bond:

Time Period after 5 years	1	2
Coupon	33.75	33.75
Nominal Value		1000
Total	33.75	1033.75
Discounting factor @4%	0.956937799	0.915729951
Present Value	32.29665072	946.6358371
Price	978.9324878

Notice that the price has slightly reduced.

The two effects shall balance each other and the total payout per bond shall be 414.7270663+978.9324878 = $1393.659554. This is almost the same payout per bond as was earlier.

Total payout = 350*1393.659554 =

550495.5239

Hence, the payout target is fulfilled.