Suppose that a bond that will mature in two years has a face value of $1000 and 20% coupon rate. The one year spot market interest rate is 13% and the expected second period's forward rate is 12%. According to the expectation hypothesis, the two year spot rate is:
A) 11.50%.
B) 9.12%.
C) 12.74%.
D) 7.00%.

Question

Quizplus · Accepted Answer

The expectation hypothesis suggests that the future long-term interest rates can be approximated by the current short-term interest rates and the expected future short-term rates. To find the two-year spot rate under the expectation hypothesis, we use the formula that equates the present value of cash flows from holding two one-year bonds sequentially to the present value of holding one two-year bond. Given a 20% coupon rate on a $1000 face value bond, the annual coupon payment is $200. The one-year spot rate is 13%, and the expected second period's forward rate is 12%.The calculation involves finding the rate $r$ that satisfies the equation:$$1000 = \frac{200}{(1+0.13)} + \frac{1200}{(1+r)^2}$$However, the above calculation is not directly needed for this question. Instead, we use the relationship between spot and forward rates to find the two-year spot rate. The formula to link one-year spot rate ($S_1$), the two-year spot rate ($S_2$), and the one-year forward rate one year from now ($F_{1,2}$) is:$$(1 + S_1)(1 + F_{1,2}) = (1 + S_2)^2$$Given $S_1 = 0.13$ and $F_{1,2} = 0.12$, we solve for $S_2$:$$(1 + 0.13)(1 + 0.12) = (1 + S_2)^2$$$$1.13 \times 1.12 = (1 + S_2)^2$$$$1.2656 = (1 + S_2)^2$$Taking the square root of both sides:$$1 + S_2 = \sqrt{1.2656}$$$$S_2 = \sqrt{1.2656} - 1$$$$S_2 \approx 0.1274 \text{ or } 12.74\%$$Therefore, according to the expectation hypothesis, the two-year spot rate is approximately 12.74%.

Suppose That a Bond That Will Mature in Two Years