Calculating a Treasury Bill Zero Rate from Purchase Price

Summary

The document clarifies how to calculate a one-year zero rate from the purchase price of a Treasury bill that pays its face value at maturity. The annual holding-period return is the difference between the maturity payment and the bill’s price, divided by the price paid. With a face value of 100 and a price of 89, the discount of 11 produces a return of 11 divided by 89, or 12.36% for the year.

The same logic explains the six-month example: a bill bought for 94 and redeemed for 100 earns 6 divided by 94 over six months. The initial question compares annual compounding with annualizing a six-month rate under semiannual or continuous compounding; the answer focuses on the one-year bill’s actual dollar gain rather than that convention comparison. The figures are illustrative, and the document does not discuss day-count conventions, taxes, or other market details that can affect rate calculations.

Key ideas

• A zero-coupon bill’s holding-period return is its maturity payment minus its purchase price, divided by the purchase price.
• A bill priced at 89 and redeemed at 100 earns a one-year return of 11 divided by 89.
• A six-month bill priced at 94 and redeemed at 100 earns a six-month return of 6 divided by 94.
• The one-year example uses the bill’s full maturity gain and does not compare alternative rate conventions in detail.

Tags

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


# 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?

## Answer by ocstl (score 4, accepted)

https://quant.stackexchange.com/a/21049

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:

$\frac{100 - P}{P} = \frac{100 - 94.0}{94.0} = \frac{6}{94} = 6.38 \%$.

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.