Spot Rates and Bootstrapping

EXERCISE 01

P_2=\frac{100}{1.055^2}\approx89.85

Two-year price = 89.85.

P_4=\frac{100}{1.0595^4}\approx79.36

Four-year price = 79.36.

No. A five-year price or an explicit extrapolation assumption is required; the four supplied spot rates do not uniquely determine it.

EXERCISE 02

Two default-free bonds have the following annual cash flows. Assume frictionless trading, with borrowing and short selling permitted.

Bond	Price today	Year 1	Year 2
A	100.98	4	104
B	102.89	5	105

A.Recover the discount factors and effective annual spot rates, then the one-year forward rate from year 1 to 2.

P=\sum_k CF_kd_k,\qquad z_t=d_t^{-1/t}-1

100.98=4d_1+104d_2

102.89=5d_1+105d_2

d_1+d_2=102.89-100.98=1.91

100.98=4(1.91-d_2)+104d_2

d_2=0.9334

d_1=1.91-0.9334=0.9766

z_1=\frac1{0.9766}-1

2.396068%.

z_2=0.9334^{-1/2}-1

3.506137%.

f_{s,t}=\left(\frac{d_s}{d_t}\right)^{1/(t-s)}-1

f_{1,2}=\frac{0.9766}{0.9334}-1

4.628241%.

EXERCISE 03

Three zero-coupon bonds trade at the prices below. Use a 360-day year and effective annual compounding for all quoted spot and forward rates.

Bond	Price	Payment day	Redemption
A	93	90	100
B	168	180	200
C	240	360	300

P=\sum_k CF_kd_k,\qquad z_t=d_t^{-1/t}-1

d_{0.25}=93/100=0.93

d_{0.5}=168/200=0.84

d_1=240/300=0.80

z_{0.25}=0.93^{-4}-1,\quad z_{0.5}=0.84^{-2}-1,\quad z_1=0.80^{-1}-1

f_{s,t}=\left(\frac{d_s}{d_t}\right)^{1/(t-s)}-1

f_{0.25,0.5}=(0.93/0.84)^4-1

f_{0.25,1}=(0.93/0.80)^{4/3}-1

f_{0.5,1}=(0.84/0.80)^2-1

P_D=50(0.93)+150(0.84)+200(0.80)

332.50