# 12-month rate calculation for Problem 4.23 in Hull's Options, Futures, and Other Derivatives

From Hull's Options, Futures, and Other Derivatives, 8th ed., problem 4.23:

Excerpt from Problem 4.23

The cash prices of six-month and one-year Treasury bills are 94.0 and 89.0 ... Calculate the six-month, one-year ... zero rates.

Working out the six-month zero rate first, I understand that

$$\frac {6}{94} = 0.06383 \text{, or } 6.383\text{% in six months}$$

Thus, the six-month zero rate is

$$2 \times 6.383 \cong 12.766\text{% per annum (semi-annual compounding)}$$ $$2 \times \ln{(1 + 0.06383)} \cong 0.1238 \text{, or }12.38\text{% per annum (continuous compounding)}$$

But it is the following calculuation for the 12-month rate that puzzles me:

$$\frac {11}{89} = 0.12360 \text{, or }12.36\text{% per annum (annual compounding)}$$

Why 11 for the numerator, and not 12, when calculating the 12-month rate?

The rate is the return on your investment. Since you'll receive 100\$after 12 months,$\frac{100 - P}{P} = \frac{100 - 89.0}{89.0} = \frac{11}{89} = 12.36 \%$. Same for the 6-month T-Bill:$\frac{100 - P}{P} = \frac{100 - 94.0}{94.0} = \frac{6}{94} = 6.38 \%\$.