Question

A one-year zero-coupon bond yields 4.0%. The two- and three-year zero-coupon bonds yield 5.0% and 6.0% respectively.

The one-year spot rate r(1) = 4%, the forward rate for a one-year loan beginning in one
year is 6%, and the forward rate for a one-year loan beginning in two years is 8%. The three-year spot rate is closest to __________.

Answer 1

The three-year spot rate (r(3)) can be calculated using the given one-year spot rate and the forward rates over the next two years, by equating the compound returns of consecutive one-year investments to the compound return of a three-year bond.

Step-by-step explanation:

To find the closest value for the three-year spot rate, we can use the given one-year and two-year spot rates (r(1) and r(2)) and forward rates for the one-year loans starting in one and two years (f(1,1) and f(2,1)). Given these rates, the third year spot rate (r(3)) can be calculated using the formula that equates the total return from consecutive one-year investments to the total return from directly investing in a three-year bond:

• 
  (1 + r(1))(1 + f(1,1))(1 + f(2,1)) = (1 + r(3))^3
  

Substituting the given rates:

• 
  (1 + 0.04)(1 + 0.06)(1 + 0.08) =
  (1 + r(3))^3
  

When we solve this equation, we will obtain the three-year spot rate r(3). It is important to express percentages as decimals (4% as 0.04, for example) when performing these calculations.