Question

An investor with an investment horizon of 1.6 year purchases a 5% coupon bond with 2 years to maturity and a face value of $100? The bond is trading at a yield of 5%. Coupons are paid semi-annually. What is this investor's duration gap?

Assume semi-annual compounding. Round your answer to 4 decimal places.

Answer 1

No of periods = 2 years * 2 = 4 semi-annual periods

Coupon per period = (Coupon rate / No of coupon payments per year) * Face value

Coupon per period = (5% / 2) * $100

Coupon per period = $2.5

Illustrating for Time period 0.5

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

Discount factor = 1 / (1 + 5% / 2)^{(0.5 * 2)}

Discount factor = 0.9756

Present value of Cashflow = Discount factor * Cashflow

Present value of Cashflow = 0.9756 * $2.5

Present value of Cashflow = $2.44

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

Weight = $2.44 / $100

Weight = 2.44%

Weighted average of Time = Weight * Time period

Weighted average of Time = 2.44% * 0.5

Weighted average of Time = 0.0122

Time period	Yield to Maturity	Discount Factor	Cashflow	Present value of Cashflow	Weight	Weighted average of Time
0.5	5%	0.9756	$2.50	$2.44	2.44%	0.0122
1	5%	0.9518	$2.50	$2.38	2.38%	0.0238
1.5	5%	0.9286	$2.50	$2.32	2.32%	0.0348
2	5%	0.9060	$102.50	$92.86	92.86%	1.8572
		Total	$110.00	$100.00	100.00%	1.9280

Macaulay Duration = 1.9280 years

Duration Gap = Macaulay Duration - Investment horizon

Duration Gap = 1.9280 - 1.6

Duration Gap = 0.3280 years