Question

Pete plans to buy an 8 percent, $1,000 par bond that matures in three years and the interest is paid semiannually, and the bond’s YTM is 10 percent.

1. Calculate the bond’s duration.   

(b) Calculate the bond’s modified duration.   

(c) Assuming the bond’s YTM declines from 10 percent to 9.5 percent, calculate the bond’s price change. Explain your answer.

(d) Explain how changes in YTM affects the bond’s market price risk and reinvestment risk.

Answer 1

(a) Calculate the bond’s duration.  

Year (t)	Payments (n)	Cash Flow from coupon payments (8%/2 of $1000)	Cash Flow from maturity amount	Total Cash Flow from coupon payments and maturity amount (CF)	Present value (PV) discounted at 10%/2 =5% semiannual yield to maturity	PV *t
0.5	1.0	$40.0		$40.0	$38.10	$19.05
1.0	2.0	$40.0		$40.0	$36.28	$36.28
1.5	3.0	$40.0		$40.0	$34.55	$51.83
2.0	4.0	$40.0		$40.0	$32.91	$65.82
2.5	5.0	$40.0		$40.0	$31.34	$78.35
3.0	6.0	$40.0	$1,000.0	$1,040.0	$776.06	$2,328.19
sum					$949.24	$2,579.52
					Bond's Price↑
				Duration = sum of (PV*t)/sum of PVs =	$2,579.52/$949.24	2.72 year

(b) Calculate the bond’s modified duration.  

Modified Duration = Duration / (1 + YTM/n)

Where,

Duration = 2.72 years

Yield to maturity, YTM = 10% per year

Number of discounting periods in year, n = 2 (for semiannual coupon payments)

Therefore,

Modified Duration = 2.72/ (1+ 10%/2)

= 2.59 years

(c) Assuming the bond’s YTM declines from 10 percent to 9.5 percent, calculate the bond’s price change. Explain your answer.

Predicted price change = – Duration * (change in y)/ (1+y)* P0

= - 2.72 *(-0.50%)/ (1+10%) * $949.24

= $11.73

Therefore, predicted new price = $11.73 + $949.24 = $960.97

(d) Explain how changes in YTM affect the bond’s market price risk and reinvestment risk.

The changes in YTM affect the bond’s market price in reverse direction. The cash flows from the coupons of the bond are exposed to reinvestment risk because cash flows (coupon payments) are invested at rates that can differ from the initial YTM due the changes in YTM.

Formulas used in excel calculation: