The term structure of spot rates in this example is flat at 10%. Very simply, investors are willing to lock in 10% for two or three years because they assume that the one-year rate will always be 10%.
Now let’s assume that the one-year rate is still 10%, but that all investors forecast the one-year rate next year to be 12% and the one-year rate in two years to be 14%. Using the forward rate concept from FRM Part 1, the two-year spot rate, 𝑟̂ 2 , is such that
Solving, 𝑟̂ 2 =10.995% . Similarly, the three-year spot rate, 𝑟̂ 3 , is such that
Solving, 𝑟̂ 3 = 11.998%. Hence, the evolution of the one-year rate from 10% to 12% to 14% generates an upward sloping term structure of spot rates: 10%, 10.995%, and 11.988%.
In this case, investors require rates above 10% when locking up their money for two or three years because they assume one-year rates will be higher than 10%. No investor, for example, would buy a two-year zero at a yield of 10% when it is possible to buy a one-year zero at 10% and, when it matures, buy another one-year zero at 12%.
𝑟̂(1) = 10% , 𝑟̂ 2 = 8.995%, and 𝑟̂(3) = 7.988%.
0.5 × 8% + 0.5 × 12% = 10%
0.25 × 14% + 0.5 × 10% + 0.25 × 6% = 10%
Dividing both sides by 1.10 gives :-
The left-hand side of this inequality is the price of the two-year zero-coupon bond today. Hence, the price of the two-year zero is greater than the result of discounting the terminal cash flow by 10% over the first period and by the expected rate of 10% over the second period. It follows immediately that the yield of the two-year zero, or the two-year spot rate, is less than 10%.
To summarize, Jensen’s inequality is true because the pricing function of a zero-coupon bond, 1/ 1+r, ,is convex rather than concave.