Question

Assume that the 1-year zero-coupon bond is sold at $89.78 and the yields to maturity for the coupon bonds selling at market prices equal to their face values are 11% and 13% for 1-year and 1.5-year issues respectively. Coupons are paid every 6 months and face values are $100 for all the bonds.

(a) Calculate the spot rate curve (s_0.5, s₁, s_1.5).

(Keep your answer in decimal format 4 decimal places, e.g. 0.1234. Do not give in percent format e.g. 12.34%.)

     s_0.5:     s₁:      s_1.5 :

(b) Compute the quasi-modified duration for each of these bonds. (Keep 2 decimal places, e.g. xx.12.)

     Zero-coupon bond:

11% coupon bond:  

    13% coupon bond:  

(c)  Determine the current price of an 14% coupon bond with face value $100 and 18 months to maturity. (Keep 2 decimal places, e.g. xx.12.)

Answer 1

Part (a)

Let S_0.5, S₁, S_1.5 be the spot rates. Then,

1-year zero-coupon bond is sold at $89.78

Hence, 89.78 = 100 / (1 + S₁ / 2)²

Hence, S₁ = [(100/89.78)^1/2 - 1] x 2 = 0.1108

The yields to maturity for the coupon bonds selling at market prices equal to their face values are 11% and 13% for 1-year and 1.5-year issues respectively. For bonds selling at par, coupon rate = yield

Hence, for the one year bond, coupon per period = 11% / 2 x 100 = 5.5% = 5.5

Price = 100 = 5.5 / (1 + S_0.5 / 2) + 105.5 / (1 + S₁ / 2)² = 5.5 / (1 + S_0.5 / 2) + 105.5 / (1 + 0.1108/ 2)²

Hence, S_0.5 =  0.0825

For the one and half year bond, coupon per period = 13% / 2 x 100 = 6.5

Price = 100 = 6.5 / (1 + S_0.5 / 2) + 6.5 / (1 + S₁ / 2)² + 10 / (1 + S_1.5 / 2)³ = 6.5 / (1 + 0.0825 / 2) + 6.5 / (1 + 0.1108 / 2)² + 106.5 / (1 + S_1.5 / 2)³

Hence, S_1.5 =  0.1320

Hence, spot rate curve is:

S_0.5 = 0.0825
S₁ = 0.1108
S_1.5 = 0.1320

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Part (b)

I have used Excel to get the quasi modified duration.

Please see the table below. The column highlighted in yellow contains your answer. Figures in parenthesis, if any, mean negative values. All financials are in $. Adjacent cells in blue contain the formula in excel I have used to get the final output.

Bond	Settlement	Maturity	Coupon	Yield	Frequency	Modified Duration
	A	B	C	D	E	F = MDURATION (A, B, C, D, E)
Zero Coupon	1/1/2019	1/1/2020	0.00%	8.25%	2.0	0.9604
1 year, 11% coupon bond	1/1/2019	1/1/2020	11.00%	11.00%	2.0	0.9232
1.5 year, 13% coupon bond	1/1/2019	1/7/2020	13.00%	13.00%	2.0	0.8704

Part (c)

Coupon per period = 14% / 2 x 100 = 7

Hence, Price = 7 / (1 + S_0.5 / 2) + 7 / (1 + S₁ / 2)² + 107 / (1 + S_1.5 / 2)³ = 7 / (1 + 0.0825 / 2) + 7 / (1 + 0.1108 / 2)² + 107 / (1 + 0.1320 / 2)³ = $ 101.34