Question

A government issues $4000000 of serial bonds paying interest at j₂ = 9%, of which $2000000 is redeemed in 10 years, $2000000 is redeemed in 15 years and $0 is redeemed in 20 years. Determine the purchase price on the day that the bonds were issued, in order for investors to receive a yield of j₂ = 4%. Please do not round intermediate answers.

Answer 1

Face Value of bonds issued = $4 million

Coupon rate = 9% (assuming that the bonds pay a semi-annual coupon at 9%/2 = 4.5% as is the convention for Government bonds)

Thus, semi-annual coupon as long as entire $4 million is outstanding = $4 million x 4.5% = $180,000

The semi-annual coupon once bonds worth $2 million are redeemed after 10 years and only $2 million is outstanding = $2 million x 4.5% = $90,000

This means, semi-annual coupons of $180,000 would be paid for the first 10 years (i.e. first 20 semi-annual periods) and at the end of this 10 year period $2 million would be redeemed (paid out to the investors). Thereafter, semi-annual coupons of $90,000 would be paid for the next 5 years (i.e. next 10 semi-annual periods) and at the end of the 15 year period, the remaining $2 million would be redeemed (paid out to the investors).

We need to find the Present Value of the bond which would lead to the payment stream tabulated below and result in a yield (semi-annual) of 4%. The way ahead is to discount this payment stream with a yield of 4% (semi-annual yield of 2%) and add all these discounted values of coupon payments and bond redemption. The total discounted value will give us the purchase price of the bond in order to receive a yield of j2=4%.

Semi-annual periods	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15	16	17	18	19	20	21	22	23	24	25	26	27	28	29	30
Coupon payments	180,000.00	180,000.00	180,000.00	180,000.00	180,000.00	180,000.00	180,000.00	180,000.00	180,000.00	180,000.00	180,000.00	180,000.00	180,000.00	180,000.00	180,000.00	180,000.00	180,000.00	180,000.00	180,000.00	180,000.00	90,000.00	90,000.00	90,000.00	90,000.00	90,000.00	90,000.00	90,000.00	90,000.00	90,000.00	90,000.00
Bond redemption																				2,000,000.00										2,000,000.00
Present value discounted at 4% yield (semi-annual yield of 2%)	176,470.59	173,010.38	169,618.02	166,292.18	163,031.55	159,834.85	156,700.83	153,628.27	150,615.95	147,662.69	144,767.35	141,928.77	139,145.85	136,417.50	133,742.65	131,120.25	128,549.26	126,028.69	123,557.54	1,467,077.51	59,379.82	58,215.51	57,074.03	55,954.93	54,857.78	53,782.14	52,727.58	51,693.71	50,680.11	1,153,828.16
Present value of all coupon and bond payments	5,937,394.44

As calculated in the above table, we get the purchase price on the issue date as $5,937,394.44

Note: Since the yield is supposed to be 4% and the coupon rate that the bonds were paying was 9%, naturally the purchase price had to be at a premium to the face value.