Question

1.An investor buys a 9-year, 6.9% annual coupon bond at par ($100). After the purchase and before the first coupon is received, interest rates increase to 8.9% (assume a flat spot rate curve). The investor sells the bond after 7 years (right after receiving the 7th coupon payment). What is this investor's realized annual return in these 7 years?

Assume annual compounding, and that interest rates remain at 8.9% over the entire holding period.

2.An investor with an investment horizon of 1.0 year purchases a 8% 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

1)

No of periods = 9 years

Coupon per period = Coupon rate * Face value

Coupon per period = 6.9% * $100

Coupon per period = $6.9

Current Bond price

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

Bond Price = $6.9 / (1 + 8.9%)¹ + $6.9 / (1 + 8.9%)² + ...+ $6.9 / (1 + 8.9%)⁹ + $100 / (1 + 8.9%)⁹

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

Bond Price = $6.9 * ((1 - (1 + 8.9%)^-9) / 8.9%) + $100 / (1 + 8.9%)⁹

Bond Price = $41.5359 + $46.4247

Current Bond Price = $87.9606

Bond price after 7 years

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

Bond Price = $6.9 / (1 + 8.9%)¹ + $6.9 / (1 + 8.9%)² + $100 / (1 + 8.9%)²

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

Bond Price = $6.9 * ((1 - (1 + 8.9%)^-2) / 8.9%) + $100 / (1 + 8.9%)²

Bond Price = $12.1543 + $84.3226

Bond Price after 7 years = $96.4769

Let us calculate the coupon reinvestment amount  

Reinvested Coupon amount in year 7 = Coupon * (((1 + interest rate)^{no of periods} - 1) / interest rate)

Reinvested Coupon amount in year 7 = $6.9 * (((1 + 8.9%)⁷ - 1) / 8.9%

Reinvested Coupon amount in year 7 = $63.2886

Realized annual return = ((Bond Price after 7 years + Reinvested Coupon amount in year 7) / Current Bond Price)^{(1 / no of periods)} - 1

Realized annual return = (($96.4769 + $63.2886) / $87.9606)^(1/7) - 1

Realized annual return = 8.9%

2)

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 = (8% / 2) * $100

Coupon per period = $4

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 * $4

Present value of Cashflow = $3.9024

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

Weight = $3.9024 / $105.6430

Weight = 3.6940%

Weighted average of Time = Weight * Time period

Weighted average of Time = 3.6940% * 0.5

Weighted average of Time = 0.0185

Time period	Yield to Maturity	Discount Factor	Cashflow	Present value of Cashflow	Weight	Weighted average of Time
0.5	5%	0.9756	$4.00	$3.9024	3.6940%	0.0185
1	5%	0.9518	$4.00	$3.8073	3.6039%	0.0360
1.5	5%	0.9286	$4.00	$3.7144	3.5160%	0.0527
2	5%	0.9060	$104.00	$94.2189	89.1861%	1.7837
		Total	$116.00	$105.6430	100.00%	1.8910

Macaulay Duration = 1.8910 years

Duration gap = Macaulay Duration - Investment horizon

Duration gap = 1.8910 - 1.0

Duration gap = 0.8910 years