Question

A 10-Year maturity bond making annual coupon payments with a coupon rate of 5% and currently selling at a yield to maturity of 4% has a convexity of 145.4.

Compute the modified duration of the bond.

Based on the information above, compute the approximated new price using the Duration & Convexity adjustment if the yield to maturity increases by 75 basis points.

What is the percentage error?

Answer 1

No of periods = 10 years

Coupon per period = Coupon rate * Face value

Coupon per period = 5% * $1000

Coupon per period = $50

Bond price at YTM = 4%

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

Bond Price = $50 / (1 + 4%)¹ + $50 / (1 + 4%)² + ...+ $50 / (1 + 4%)¹⁰ + $1000 / (1 + 4%)¹⁰

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

Bond Price = $50 * ((1 - (1 + 4%)^-10) / 4%) + $1,000 / (1 + 4%)¹⁰

Bond Price = $405.54 + $675.56

Bond Price = $1081.11

Bond price at 75 bps increase in Yield to Maturity i.e. YTM = 4.75%

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

Bond Price = $50 / (1 + 4.75%)¹ + $50 / (1 + 4.75%)² + ...+ $50 / (1 + 4.75%)¹⁰ + $1000 / (1 + 4.75%)¹⁰

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

Bond Price = $50 * ((1 - (1 + 4.75%)^-10) / 4.75%) + $1,000 / (1 + 4.75%)¹⁰

Bond Price = $390.82 + $628.72

Bond Price = $1019.54

Change in Bond price = (Bond Price at 4.75% yield - Bond Price at 4% yield) / Bond Price at 4% yield

Change in Bond price = ($1019.54 - $1081.11) / $1081.11

Change in Bond price explained by YTM = -5.6949%

Computing Modified Duration

Illustrating for Time period 0.5

Discount factor = 1 / (1 + YTM / 2)^{(Time period * 2)}

Discount factor = 1 / (1 + 4% / 2)^{(0.5 * 2)}

Discount factor = 0.9615

Present value of Cashflow = Discount factor * Cashflow

Present value of Cashflow = 0.9615 * $50

Present value of Cashflow = $48.08

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

Weight = $48.08 / $1081.11

Weight = 4.45%

Weighted average of Time = Weight * Time period

Weighted average of Time = 4.45% * 1

Weighted average of Time = 0.0455

Time period	Yield to Maturity	Discount Factor	Cashflow	Present value of Cashflow	Weight	Weighted average of Time
1	4.00%	0.9615	$50	$48.08	4.45%	0.0445
2	4.00%	0.9246	$50	$46.23	4.28%	0.0855
3	4.00%	0.8890	$50	$44.45	4.11%	0.1233
4	4.00%	0.8548	$50	$42.74	3.95%	0.1581
5	4.00%	0.8219	$50	$41.10	3.80%	0.1901
6	4.00%	0.7903	$50	$39.52	3.66%	0.2193
7	4.00%	0.7599	$50	$38.00	3.51%	0.2460
8	4.00%	0.7307	$50	$36.53	3.38%	0.2703
9	4.00%	0.7026	$50	$35.13	3.25%	0.2924
10	4.00%	0.6756	$1,050	$709.34	65.61%	6.5612
		Total	$1,500	$1,081.11	100.00%	8.1909

Macaulay Duration = 8.1909

Modified Duration = Macaulay Duration / (1 + YTM)

Modified Duration = 8.1909 / (1 + 4%)

Modified Duration = 7.8759

Change in Bond price expalined by Modified duration & Convexity

Change in Bond price = - Modified Duration * Change in yield + 0.5 * Convexity * (Change in yield)²

Change in Bond price = - 7.8759 * (0.75%) + 0.5 * 145.4 * (0.75%)²

Change in Bond price explained by Duartion & Convexity = -5.4979%

%Error = Change in Bond price explained by YTM - Change in Bond price explained by Duartion & Convexity

%Error = - 5.6949% - (- 5.4980%)

%Error = -0.1969%