Question

1. Consider the one-year interest rates known at the following dates:

year 0: 2%

year 1: 2.5

year 2: 3%

year 3: 3.5%

year 4: 4%

year 5: 4.5%

year 6: 5%

year 7: 5.5%

year 8: 6%

year 9: 6.5%

Using the Expectations Theory, find the interest rates of maturities 1 through 10. Use the arithmetic average method. What do they suggest about the shape of the yield curve?

Answer 1

Expectations theory states that interest rates in the short term will be calculated based on long term interest rates prevailing currently. These rates are generally those of Government bonds, zero coupon bonds etc.

The theory suggests that an investor earns the same amount of interest if he invests in 2 consecutive one-year bonds and if he invests in one two-year bond today.

The formula is given below:

Interest rate with maturity n = ((1+YTM of nth year)^n/(1+ YTM of n-1th year)^(n-1)) -1

In the question, we are given the one-year Interest rates (YTM) which will be used to calculate the interest rates of maturities 1 through 10 using the above mentioned formula.

Maturity	Interest rates (using formula)	Rate
1	((1+0.025)^1/(1+0.02)^0))-1	0.025	2.50%
2	((1+0.03)^2/(1+0.025)^1))-1	0.035	3.50%
3	((1+0.035)^3/(1+0.03)^2))-1	0.045	4.50%
4	((1+0.04)^4/(1+0.035)^3))-1	0.055	5.50%
5	((1+0.045)^5/(1+0.04)^4))-1	0.065	6.50%
6	((1+0.05)^6/(1+0.045)^5))-1	0.075	7.50%
7	((1+0.055)^7/(1+0.05)^6))-1	0.086	8.60%
8	((1+0.06)^8/(1+0.055)^7))-1	0.096	9.60%
9	((1+0.065)^9/(1+0.06)^8))-1	0.106	10.60%
10	((1+0.07)^10/(1+0.065)^9))-1	0.116	11.60%