Question

Suppose that you have some obligation to pay $1,601.81 in 4 years time. You have elected to fund this liability by purchasing a 5 year bond with a 12.50% annual pay bond (the annual coupons are paid at the end of each year) with a face value of $1,000 at a purchase price of par ($1,000). Demonstrate that your liability will be fully funded at its due date in 4 years by this bond, irrespective of any interest rate changes. (Assume that any interest rate change occurs an instant after purchasing the bond.) Complete the following table showing that the combined value of the sales price of your bond at the end of 4 years, and the accumulated interest earned over the 4 year period is at least equal to your liability payment. New Interest Rate Accumulated Interest Selling Price Total Terminal Value 8% 10% 14% 16%?

Answer 1

The obligation to be paid after 4 years = $1601.81

The details of the exercise is as given

Head	Value
Annual Coupon (PMT)	125
Time (NPER)	4
Current Price of Bond (PV)	1000
Type (Coupon received at the end)	0
Time for Maturity of Bond (NPER)	5
Maturity Value of Bond (FV)	1000

As any point of time, if the interest rate changes, it will have a 3 fold effect

1. It will cause a change in the current price of Bond (An increase if rates go down, a decrease if rates go up)

2. The Future value of the bond after 4 years will change, not just because of rate, but because the current price of bond (assumed purchase price) also changes as mentioned above.

3. A change in interest rate will cause a change in accumulated future value of Coupons

As can be seen in below table, the total of FV of Bond after 4 years plus accumulated interest after 4 years is always greater than $1601.80  

Rate	A = Current Price of Bond Using PV	B = Price of Bond after 4 years using FV	C =Accumulated Value of Coupons received	Total D = B +C
0.00%	($1,625.00)	$1,125.00	$500.00	$1,625.00
4.00%	($1,378.40)	$1,081.73	$530.81	$1,612.54
8.00%	($1,179.67)	$1,041.67	$563.26	$1,604.93
10.00%	($1,094.77)	$1,022.73	$580.13	$1,602.85
12.00%	($1,018.02)	$1,004.46	$597.42	$1,601.88
12.50%	($1,000.00)	$1,000.00	$601.81	$1,601.81
14.00%	($948.50)	$986.84	$615.14	$1,601.99
16.00%	($885.40)	$969.83	$633.31	$1,603.14
18.00%	($828.01)	$953.39	$651.93	$1,605.32
20.00%	($775.70)	$937.50	$671.00	$1,608.50

Calculations**

Purchase price of bond at a given rate, is calculated using PV function in excel

Future Value of bond after 4 years at a given rate, is calculated using FV function in excel

Value of Coupons with accumulated interest is calculated using FV function

For example: Let's calculate all values at a interest rate of 8%

1. Purchase price of bond = PV(8%,5,125,1000,0)

= - $1179.67 (The term here is 5 years, the negative sign indicates cash outflow for purchasing the bond)

2. Future value of bond after 4 years = FV(8%,4,125,-1179.67,0)

= $1041.67 (here term is 4years, and PV is current PV found above = -1179.67)

3. Value of coupons with accumulated interest = FV(8%,4,-125,0,0)

= $563.26 (Coupons are invested, so a cash flow, therefore with a negative sign)

The sum of FV of bond and FV of coupons at 8% = 1041.67 + 563.26 = $1604.93

Similarly other values are calculated