Question

Consider the three bonds quoted in the following table (settlement: 2/15/94). Calculate discount factors and spot rates at six-month intervals (d1, d2, d3 and y1, y2, y3), and implied six month forward rates (f1 and f2).

Coupon Rate	Maturity	Price
67/8	8/15/94	101:20
51/2	2/15/95	101:18
45/8	8/15/95	100:21

Answer 1

Coupon Rate	Coupon per period	Maturity	Price	Cash flow at t = 0.5	Cash flow at t = 1	Cash flow at t = 1.5
	(= Coupon rate / 2) x FV of $ 100
67/8 = 6.875%	3.4375	8/15/1994	101.20	103.44
51/2 = 5.5%	2.7500	2/15/1995	101.18	2.7500	102.75
45/8 = 4.625%	2.3125	8/15/1995	100.21	2.3125	2.3125	102.31

Price of the bond is present value of all the future payments.

Hence, 101.20 = 103.44 x d₁

Hence, d₁ = 101.20 / 103.44 =  0.9784

Also, d₁ = 1 / (1 + y₁ / 2). Hence, y₁ = 2 x (1 / d₁ - 1) = 4.42%

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For the second bond, 101.18 = 2.75 x d₁ + 102.75 x d₂ = 2.75 x 0.9784 + 102.75 x d₂

Hence, d₂ =  0.9585

Also, d₂ = (1 + y₂ / 2)^-2

Hence, y₂ = 2 x (d₂^-1/2 - 1) = 2 x (0.9585^-1/2 - 1) = 4.28%

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For the third bond, 100.21 = 2.3125 x d₁ + 2.3125 x d₂ + 102.31 x d₃ = 2.3125 x 0.9784 + 2.3125 x 0.9585 + 102.31 x d₃

Hence, d₃ =  0.9553

Also d₃ = (1 + y₃ / 2)^-3

Hence, y₃ = 2 x (d₃^-1/3 - 1) = 3.07%

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f₁ = 2 x (d₁ / d₂ - 1) = 4.14%

f₂ = 2 x (d₃ / d₂ - 1) = 0.68%