Question

The YTM on a bond is the interest rate you earn on your investment if interest rates don’t change. If you actually sell the bond before it matures, your realized return is known as the holding period yield (HPY).

a.	Suppose that today you buy an annual coupon bond with a coupon rate of 8.3 percent for $835. The bond has 9 years to maturity and a par value of $1,000. What rate of return do you expect to earn on your investment? (Do not round intermediate calculations and enter your answer as a percent rounded to 2 decimal places, e.g., 32.16.)
b-1.	Two years from now, the YTM on your bond has declined by 1 percent, and you decide to sell. What price will your bond sell for? (Do not round intermediate calculations and round your answer to 2 decimal places, e.g., 32.16.)
b-2.	What is the HPY on your investment? (Do not round intermediate calculations and enter your answer as a percent rounded to 2 decimal places, e.g., 32.16.)

Answer 1

Part a)

Approximate YTM = {Interest+(Maturity value-Current price)/n}/{(Maturity value+Current price)/2}

where, interest = par value*coupon rate = $1,000*8.3% = $83; Current price = $835; Maturity value = par value = $1,000; n = 9years

Approximate YTM = {83+(1000-835)/9}/{(1000+835)/2} = {83+(165/9)}/{1835/2} = (83+18.33)/917.5 = 101.33/917.5 = 11.04%

Computation of YTM based on discounted cashflow method:

Year	Type	Cashflow	PVF @ 11%	Discounted cashflow (Cashflow * PVF@11%)	PVF @ 11.5%	Discounted cashflow (Cashflow * [email protected]%)
1	Coupon	83	1/(1+discount rate) = 1/(1.11) = 0.9009	74.77	1/(1+discount rate) = 1/(1.115) = 0.8969	74.44
2	Coupon	83	1/[(1+discount rate)^2] = 1/[(1.11)^2] = 0.8116	67.36	1/[(1+discount rate)^2] = 1/[(1.115)^2] = 0.8044	66.77
3	Coupon	83	1/[(1+discount rate)^3] = 1/[(1.11)^3] = 0.7312	60.69	1/[(1+discount rate)^3] = 1/[(1.115)^3] = 0.7214	59.88
4	Coupon	83	1/[(1+discount rate)^4] = 1/[(1.11)^4] = 0.6587	54.67	1/[(1+discount rate)^4] = 1/[(1.115)^4] = 0.647	53.70
5	Coupon	83	1/[(1+discount rate)^5] = 1/[(1.11)^5] = 0.5934	49.25	1/[(1+discount rate)^5] = 1/[(1.115)^5] = 0.5803	48.16
6	Coupon	83	1/[(1+discount rate)^6] = 1/[(1.11)^6] = 0.5346	44.37	1/[(1+discount rate)^6] = 1/[(1.115)^6] = 0.5204	43.19
7	Coupon	83	1/[(1+discount rate)^7] = 1/[(1.11)^7] = 0.4816	39.97	1/[(1+discount rate)^7] = 1/[(1.115)^7] = 0.4667	38.74
8	Coupon	83	1/[(1+discount rate)^8] = 1/[(1.11)^8] = 0.4339	36.01	1/[(1+discount rate)^8] = 1/[(1.115)^8] = 0.4186	34.74
9	Coupon	83	1/[(1+discount rate)^9] = 1/[(1.11)^9] = 0.3909	32.44	1/[(1+discount rate)^9] = 1/[(1.115)^9] = 0.3754	31.16
9	Maturity value	1000	1/[(1+discount rate)^9] = 1/[(1.11)^9] = 0.3909	390.90	1/[(1+discount rate)^9] = 1/[(1.115)^9] = 0.3754	375.40
				850.45		826.18

YTM = Base rate +  (Discounted cashflow @ 11%-Current price)*(difference in discount rate)/(Discounted cashflow @ 11%-Discounted cashflow @ 11.5%)

= 11% + (850.45-835)*0.5%/(850.45-826.18)

= 11% + 15.45*0.5%/24.27

= 11% + 0.32% = 11.32%

Part b-1)

Current YTM = YTM (refer part a) - 1% = 11.32%-1% = 10.32%

Year	Type	Cashflow	PVF @ 10.32%	Discounted cashflow (Cashflow*[email protected]%)
3	Coupon	83	1/(1+discount rate) = 1/(1.1032) = 0.9065	75.24
4	Coupon	83	1/[(1+discount rate)^2] = 1/[(1.1032)^2] = 0.8217	68.20
5	Coupon	83	1/[(1+discount rate)^3] = 1/[(1.1032)^3] = 0.7448	61.82
6	Coupon	83	1/[(1+discount rate)^4] = 1/[(1.1032)^4] = 0.6751	56.03
7	Coupon	83	1/[(1+discount rate)^5] = 1/[(1.1032)^5] = 0.6119	50.79
8	Coupon	83	1/[(1+discount rate)^6] = 1/[(1.1032)^6] = 0.5547	46.04
9	Coupon	83	1/[(1+discount rate)^7] = 1/[(1.1032)^7] = 0.5028	41.73
9	Maturity value	1000	1/[(1+discount rate)^7] = 1/[(1.1032)^7] = 0.5028	502.80
			Selling price at the end of year 2	902.65

The purchaser wants 10.32% yield hence they will buy at $902.65 per bond

Part b-2)

Approximate HPY = (Sale value-purchase price+interest received for 2 years)/(purchase price*holding period) = ($902.65-$835+$166)/)($835*2) = 233.65/1670 = 13.99%

Computation of HPY based on discounted cashflow:

Year	Type	Cashflow	PVF @ 14%	Discounted cashflow (Cashflow *PVF@14%)	PVF @ 13.5%	Discounted cashflow (Cashflow *[email protected]%)
1	Coupon	83	1/(1+discount rate) = 1/(1.14) = 0.8772	72.81	1/(1+discount rate) = 1/(1.135) = 0.8811	73.13
2	Coupon	83	1/(1+discount rate) = 1/[(1.14)^2] = 0.7695	63.87	1/(1+discount rate) = 1/[(1.135)^2] = 0.7763	64.43
2	Sale value	902.65	1/(1+discount rate) = 1/[(1.14)^2] = 0.7695	694.59	1/(1+discount rate) = 1/[(1.135)^2] = 0.7763	700.73
				831.27		838.29

HPY = Base rate +  (Discounted cashflow @ 13.5%-purchase price)*(difference in discount rate)/(Discounted cashflow @ 13.5%-Discounted cashflow @ 14%)

= 13.5% + (838.29-835)*0.5%/(838.29-831.27)

= 13.5% + 3.29*0.5%/7.02

= 13.5% + 0.23% = 13.73%