Question

Year	Zero-coupon bond yield (Sport rate)	Zero-coupon bond price	1-year implied forward rate	Par coupon rate
1	5%			a
2	b	c	d	6%

Please find out the values of a,b,c and d.

Answer 1

Par coupon rate = the coupon rate at which the price of a bond is equal to its par value

When coupon rate = yield, then the bond trades at par.

Hence, for 1 year maturity bond, Par coupon rate, a = zero-coupon bond yield = 5%

a = 5%

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Par coupon rate of 2 year coupon bond = 6%, Face value, FV = $ 1,000; Annual coupon, C = 6% x FV = 6% x $ 1,000 = $ 60

2 year Zero-coupon bond yield = Sport rate for year 2 = b

Hence, for a par coupon bond, Price = P = Face value = 1,000 = C / (1 + 5%) + (1,000 + C) / (1 + b)² = 60 / 1.05 + 1,060 / (1 + b)² = 57.14285714 + 1,060 / (1 + b)²

Hence, (1 + b)² = 1,060 / (1,000 - 57.14285714) = 1.124242424

Hence, b = 0.060302987 = 6.03%

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c = zero coupon bond price = 1 / (1 + b)² = 1 / (1 + 6.03%)² = 0.889487871 = 88.95%

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d = 1 year implied forward rate will be given by:

(1 + S_0,1) x ( 1 + d) = (1 + S_0,2)² Or, (1 + 5%) x (1 + d) = (1 + 6.03%)²

Hence, d = 1.0603² / 1.05 - 1 = 0.070707071 = 7.07%

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