The yield to maturity on one-year zero-coupon bonds is 8%, the yield to maturity
ID: 2726331 • Letter: T
Question
The yield to maturity on one-year zero-coupon bonds is 8%, the yield to maturity on two-year zero-coupon bonds is 10%.
a. What is the forward rate of interest for the second year?
b. If you believe in the expectations hypothesis, what is your best guess as to the expected value of the short-term interest rate next year?
c. If you believe in the liquidity preference theory, is your best guess as to next year’s short-term interest rate higher or lower than in (b)?
Note: the yield to maturity of a zero coupon bond is equal to the r0t implied by that bond
Explanation / Answer
a. Forward rate of interest for second year
A basic formula for calculating forward rates looks like this:
Forward = (((1 + (spot rate for year "x")^"x") / ((1 + (spot rate for year "y")^"y")) - 1
In the formula, "x" is the end future date (2 years), and "y" is the closer future date (1 years), based on the spot rate curve.
Calculation for the given case where two-year bond is yielding 10% while a one-year bond is yielding 8%. The return produced from the two-year bond is the same as if an investor receives 8% for the one-year bond and then uses a rollover to roll it over into another one-year bond at 12.04%. That hypothetical 12.04% is the forward rate of the investment.
b.
The expectations hypothesis of the term structure of interest rates (whose graphical representation is known as the yield curve) is the proposition that the long-term rate is determined purely by current and future expected short-term rates, in such a way that the expected final value of wealth from investing in a sequence of short-term bonds equals the final value of wealth from investing in long-term bonds.
To do the calculation, first add 1 to the two-year bond's interest rate, which in this case gives us 1.1 (or 110%).
Next, we take this result and square it: 1.1 squared gives us 1.21.
The next step is to divide this number by the current year's one-year interest rate plus one. In this example, that means 1.21 divided by 1.08 (8% + 1 = 1.08), which yields 1.12.
The final step is to subtract 1 from that 1st calculation, giving us the predicted one-year interest rate for next year, of 12% in this case.
Therefore , expected value of the short-term interest rate next year is 12%
c. Liquidity Preference theory
The liquidity preference theory is the idea that investors demand a premium for securities with longer maturities, which entail greater risk, because they would prefer to hold cash, which entails less risk. The more liquid an investment, the easier it is to sell quickly for its full value. Because interest rates are more volatile in the short term, the premium on short- versus medium-term securities will be greater than the premium on medium- versus long-term securities. For example, a three-year Treasury note might pay 1% interest, a 10-year treasury note might pay 3% interest and a 30-year treasury bond might pay 4% interest.
As per this theory, next year’s short-term interest rate should be higher than in (b).
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