Question

A newly issued bond has a maturity of 10 years and pays a 5.5% coupon rate (with coupon payments coming once annually). The bond sells at par value.

a. What are the convexity and the duration of the bond? Use the formula for convexity in footnote 7.
b. Find the actual price of the bond assuming that its yield to maturity immediately increases from 5.5% to 6.5% (with maturity still 10 years). Assume a par value of 100.
c. What price would be predicted by the modified duration rule ΔPP=−D*Δy?ΔPP=−D*Δy? What is the percentage error of that rule?
d. What price would be predicted by the modified duration-with-convexity rule ΔPP=−D*Δy+12×Convexity×(Δy)2?ΔPP=−D*Δy⁢+12×Convexity×(Δy)2? What is the percentage error of that rule?

Answer 1

Term to Maturity = 10 years

Coupon rate = 5.5%

Assuming par value = $1000

The bond sells at Par so the Yield to Maturity(YTM) = Coupon rate = 5.5%

a) For time period 1:-

Discount factor = 1 / (1+ YTM)^{Time period}

Discount factor = 1 / (1+ 5.5%)¹

Discount factor = 0.9479

Present Value of Cashflow = Discount factor * Cashflow

Present Value of Cashflow = 0.9479 * $55

Present Value of Cashflow = $52.13

Weight = (Present Value of Cashflow / Total (Present Value of Cashflow )

Weight = ($52.13 / $1000)

Weight = 5.21%

Weighted average of Time = Weight * Time period

Weighted average of Time = 5.21% * 1

Weighted average of Time = 0.0521

Time period	Yield to Maturity	Discount Factor	Cashflow	Present value of Cashflow	Weight	Weighted average of Time
1	5.50%	0.9479	$55	$52.13	5.21%	0.0521
2	5.50%	0.8985	$55	$49.41	4.94%	0.0988
3	5.50%	0.8516	$55	$46.84	4.68%	0.1405
4	5.50%	0.8072	$55	$44.40	4.44%	0.1776
5	5.50%	0.7651	$55	$42.08	4.21%	0.2104
6	5.50%	0.7252	$55	$39.89	3.99%	0.2393
7	5.50%	0.6874	$55	$37.81	3.78%	0.2647
8	5.50%	0.6516	$55	$35.84	3.58%	0.2867
9	5.50%	0.6176	$55	$33.97	3.40%	0.3057
10	5.50%	0.5854	$1,055	$617.63	61.76%	6.1763
		Total	$1,550	$1,000.00	100.00%	7.9522

Macaulay Duration = 7.9522 7.95 years

Modified Duration = Macaulay Duration / (1 + YTM)

Modified Duration = 7.95 / (1 + 5.5%)

Macaulay Duration = 7.5355%

For time period 1

Weighted average of Time = Weight * Time period * (1 + Time period)

Weighted average of Time = 5.21% * 1 * (1 + 1)

Weighted average of Time = 0.1043

Time period	Yield to Maturity	Discount Factor	Cashflow	Present value of Cashflow	Weight	Weighted average of Time
1	5.50%	0.9479	$55	$52.13	5.21%	0.1043
2	5.50%	0.8985	$55	$49.41	4.94%	0.2965
3	5.50%	0.8516	$55	$46.84	4.68%	0.5621
4	5.50%	0.8072	$55	$44.40	4.44%	0.8879
5	5.50%	0.7651	$55	$42.08	4.21%	1.2625
6	5.50%	0.7252	$55	$39.89	3.99%	1.6753
7	5.50%	0.6874	$55	$37.81	3.78%	2.1173
8	5.50%	0.6516	$55	$35.84	3.58%	2.5803
9	5.50%	0.6176	$55	$33.97	3.40%	3.0573
10	5.50%	0.5854	$1,055	$617.63	61.76%	67.9392
		Total	$1,550	$1,000.00	100.00%	80.4827

Convexity = 80.4827

b)Bond price at YTM = 6.5%

Bond price = Coupon / (1 + YTM)^period + Coupon / (1 + YTM)^period + ...+Coupon / (1 + YTM)^period + Face value / (1 + YTM)^period

Bond price = $55 / (1 + 6.5%)¹ + $55 / (1 + 6.5%)² +.. +$55 / (1 + 6.5%)¹⁰ + $1000 / (1 + 6.5%)¹⁰

Using PVIFA = (1 - (1 + interest rate)^{-no of periods}) / interest rate to value coupons

Bond price = ((1 - (1 + 6.5%)^-10) / 6.5%) * $55 + $1000 / (1 + 6.5%)¹⁰

Bond price = $395.38 + $532.73

Bond price = $928.11

c)

Price change predicted by Modified duration

ΔPrice change predicted =−D*Δy

ΔPrice change predicted = -7.5355 * 1%

ΔPrice change predicted = -7.5355%

Actual price change = (Bond price at YTM(6.5%) - Bond price at YTM(5.5%)) / Bond price at YTM(5.5%)

Actual price change = ($928.11 - $1000) / $1000

Actual price change = -7.189%

Error = ΔPrice change predicted - Actual price change

Error = -7.5355% - (-7.189%)

Error = -0.3465%

d)

Price change predicted by Modified duration & Convexity

ΔPrice change predicted =−D*Δy + (1 / 2) * Convexity * (Δy)²

ΔPrice change predicted = -7.5355 * 1% + (1 / 2) * 80.4827 * (1%)²

ΔPrice change predicted = -7.1331%

Error = ΔPrice change predicted - Actual price change

Error = -7.1331% - (-7.189%)

Error = 0.0559%