Question

"One-year Treasury bills currently earn 3.25%; You expect that one year from now, 1-year Treasury bill rates will increase to 3.55% and that two years from now, one-year Treasury bill rates will increase to 4.15%; If the unbiased expectations theory is correct, what should the current rate be on three-year Treasury securities?"

3.56%

5.84%

3.42%

3.65%

3.93%

Answer 1

Solution:

The answer is option 4 i.e. 3.65% which is explained below:

The unbiased expectations theory holds that the long term interest rate is an indication of what the short term rates should be. This means that an investor can earn the same interest on two consecutive investments in 1 year bonds as that of a single investment in a two year bond; while the 1 year bonds would definately have lower interest rate the theory assumes that compounding would put the two 1 year bond investments at par with the single 2 year investment.

In the given problem we have three consecutive 1 year bond investments that earn 3.25%, 3.55% and 4.15% one, two and three years from now.

Thus if we invest a dollar in the bond for the first year, at the end we get $1 + ($1 x 3.25/100) = 1.0325

This $1.0325 invested in the bond for the 2nd year, at year end gives us $1.0325 + ($1.0325 x 3.55/100) = 1.06915375

This $1.06915375 invested in the bond for the 3rd year, at end of third year gives us $1.06915375 + ($1.06915375 x 4.15/100) = 1.11352363062

Thus every $1 invested in the three 1 year bonds gives us $1.11352363062 at the end of 3 years

We need to find what rate of interest compounded annually gives us this ($1 - $1.11352363062) or $0.11352363062 interest.

The compounding formula is given below:

FV = PV (1+r)ⁿ

Here FV = 1.11352363062, PV = 1, n = 3

Substituting

1.11352363062 = 1 x (1+r)³

= 1+r

1+r =

1+r = 1.03649325266 ....... (using the function x^y on a calculator, we have to find 1.11352363062^1/3 or 1.113523630620^0.33333333)

r = 1.03649325266 - 1

r = 0.03649325266 or

r = 0.0365 or 3.65%