Forward Rates: Implied Forward Rate Calculation
Suppose that an investor observes these prices and yields to maturity on zero-coupon government bonds.
| Maturity | Price | Yield to Maturity |
|-----------|-------|-------------------|
| 1 year | 96.70 | 3.384% |
| 2 years | 92.25 | 4.074% |
| 3 years | 87,80 | 4.384% |
| 4 years | 83.20 | 4.651% |
The prices are per 100 of par value. The yields to maturity are stated on a semi-annual bond basis.
The "1y1y" implied forward rate, stated on a semiannual bond basis, is _closest to_:
Incorrect
When the yield curve is upwardly sloping, forward rates are higher than current spot rates.
Incorrect.
This answer is an arithmetic average of the one- and two-year yields.
Correct.
The "1y1y" implied forward rate is the implied one-year forward yield, one year into the future. Since these are bond yields, you must divide the annual yield by 2, recognizing that there are two semiannual periods per year.
The "1y1y" implied forward rate "fills the gap" between the one-year and two-year bonds. First compute a semiannual, six-month rate, and then double it to express the "1y1y" implied forward rate as a bond yield.
$$\displaystyle \left( 1 + \frac{0.03384}{2} \right)^2 (1+IFR_{2,2})^2= \left( 1 + \frac{0.04074}{2} \right)^4$$
$$\displaystyle IFR_{2,2} = \left[ \frac{(1+\frac{0.04074}{2})^4}{(1 + \frac{0.03384}{2})^2 } \right]^\frac{1}{2} - 1 \approx 0.023835$$.
Now, multiply by 2 to produce the semiannual bond basis yield:
$$\displaystyle IFR_{1,1} = 0.023835 \times 2 = 0.04767 = 4.767 \%$$.
2.725%.
3.729%.
4.767%.
