Forward Rates

EXERCISE 01

Bond A costs 98 and pays 121 at year 2. Bond B costs 98 and pays 10 at year 1 and 110 at year 2. Assume default-free cash flows, frictionless borrowing and short selling.

A.Check for arbitrage, recover spot and forward rates, and price bond C paying 5 at year 1 and 115 at year 2.

A.

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

98=121d_2,\qquad98=10d_1+110d_2

d_2=\frac{98}{121}

d_1=\frac{98-110d_2}{10}

Discount factors are 0.890909091 and 0.809917355. Both are positive, so both prices admit a common positive linear pricing rule: there is no arbitrage among these traded cash flows. They also decrease with maturity, consistent here with positive rates.

z_1=d_1^{-1}-1

12.244898%.

z_2=d_2^{-1/2}-1

11.116780%.

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

f_{1,2}=d_1/d_2-1

10.000000%.

P_C=5d_1+115d_2

97.595041

A replicating portfolio is 0.5 units of B plus (60/121) units of A; its cash flows are exactly 5 and 115.

P_C=0.5(98)+\frac{60}{121}(98)

Forward Rates

EXERCISE 02

Effective annual spot rates are 5% for one year, 7% for two years and 6% for three years. Bond A pays 5, 5 and 105 at years 1, 2 and 3. Bond B pays 0, 3 and 121. Default-free zero-coupon instruments at the given spot rates can be traded and shorted without costs.

A.Find the forward rates for years 1–2, 1–3 and 2–3.
B.Find arbitrage-free prices and trading directions if A is quoted at 98 and B at 104.

A.

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

d_1=1.05^{-1},\quad d_2=1.07^{-2},\quad d_3=1.06^{-3}

f_{1,2}=\frac{1.07^2}{1.05}-1

9.038095%.

f_{1,3}=\left(\frac{1.06^3}{1.05}\right)^{1/2}-1

6.503566% annually over the two-year interval.

f_{2,3}=\frac{1.06^3}{1.07^2}-1

4.027950%.

B.

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

P_A=\frac5{1.05}+\frac5{1.07^2}+\frac{105}{1.06^3}

97.289123

P_B=\frac3{1.07^2}+\frac{121}{1.06^3}

104.214249

A is overpriced: short A for 98 and buy its zero-coupon replicating cash flows for 97.289123, locking in 0.710877. B is underpriced: buy B for 104 and short its replicating cash flows for 104.214249, locking in 0.214249. Future cash flows cancel in each strategy.