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**Subject 7. Pricing Treasury bill futures**

This subject is not required by the Level II study guide. It is for reference only.

Suppose we buy an (h + m)-day T-bill and sell a futures expiring on day h, which calls for delivery of an m-day T-bill. m is traditionally 90.

The amount we invest to buy the bond is denoted as B

_{0}(h + m). Assuming $1 face value: B_{0}(h + m) = 1 - r_{0}^{d}(h + m) x (h + m) / 360, where r_{0}^{d}(h + m) is the discount rate in effect on day 0 of a (h + m)-day T-bill.On day 0 the price of the futures contract is denoted as f

_{0}(h). On day h, we are required to deliver an m-day T-bill. The bill we purchased now has m days to maturity. We therefore deliver that bill and receive the original futures price f_{0}(h). As f_{0}(h) is known on day 0, this transaction is risk-free.The futures price f

_{0}(h) should be the price of the (h + m)-day T-bill compounded at the h-day risk-free rate:

**f**

_{0}(h) = B_{0}(h + m) x [1 + r_{0}(h)]^{h/365}where r

_{0}(h) is the h-day risk-free rate on day 0.Alternatively, f

_{0}(h) should earn the same return per dollar invested as would a T-bill purchased on day 0 that matures on day h: f_{0}(h) / B_{0}(h + m) = 1 / B_{0}(h)

**f**

_{0}(h) = B_{0}(h + m) / B_{0}(h)where B

_{0}(h) is the price of an h-day spot T-bill on day 0.This shows that the futures price is the ratio of the longer-term bill price to the shorter-term bill price. This price is in fact the same as the forward price from the term structure.

Note that assuming $1 face value,

**B**, where r_{0}(h) = 1 - r_{0}^{d}(h) x (h) / 360_{0}^{d}(h) is the discount rate in effect on day 0 of a h-day T-bill.In the futures market, traders often refer to the futures contract in terms of

**implied discount rate**rather than the price, although the price is the more important variable:**f**, where r_{0}(h) = 1 - r_{0}^{df}(h) x (m/360)_{0}^{df}(h) is the implied discount rate on day 0 of futures contract expiring on day h and the deliverable instrument is an m-day T-bill.

**r**

_{0}^{df}(h) = [1 - f_{0}(h)] x (360/m)Note that a futures contract does not pay an interest. The r

_{0}^{df}(h) is imbedded in a futures price.Another important term is the

**implied repo rate**, which is the rate of return from a cash-and-carry transaction that is implied by the futures price relative to the spot price.

**r**

_{0}(h)* = [f_{0}(h)* / B_{0}(h + m)]^{365/h}- 1where f

_{0}(h)* is the observed futures price in the market.The implied repo rate tells a trader what rate of return to expect from the cash-and-carry transaction. It can be compared with the rate in the actual repo market to determine the attractiveness of an arbitrage transaction.

- If the financing rate available in the repo market is less than the implied repo rate, the cash-and-carry strategy is worthwhile and would generate an arbitrage profit: a trader could borrow in the repo market at a lower rate and implement this strategy (which generates a higher rate) in the futures market.
- If a trader could lend in the repo market at greater than the implied repo rate, the appropriate strategy would be to reverse the transaction - selling the T-bill short and buying the futures - turning the strategy into a source of financing that would cost less than the rate at which the funds could be lent in the repo market.

*Example*A 45-day T-bill has a discount rate of 5.5%. A 135-day T-bill has a discount rate of 5.95%. Assume the par value is $1. Calculate the price of a futures contract that expires in 45 days, and the implied discount rate.

In this case:

- h = 45.
- h + m = 135.
- r
_{0}^{d}(h) = r_{0}^{d}(45) = 0.055. - r
_{0}^{d}(h + m) = r_{0}^{d}(135) = 0.0595. - The price of the h-day T-bill is B
_{0}(h) = B_{0}(45) = 1 - r_{0}^{d}(h) x (h) / 360 = 1 - 0.055 x 45 / 360 = $0.9931. - The price of the (h + m)-day T-bill is B
_{0}(h + m) = B_{0}(135) = 1 - r_{0}^{d}(h + m) x (h + m) / 360 = 1 - 0.0595 x 135 / 360 = $0.9777. - The price of a futures contract is therefore f
_{0}(h) = f_{0}(45) = B_{0}(h + m) / B_{0}(h) = 0.9777 / 0.9931 = 0.9845. - The discount rate implied by the futures price would be r
_{0}^{df}(h) = r_{0}^{df}(45) = [1 - f_{0}(h)] x (360/m) = [1 - 0.9845] x 360 / 90 = 6.2%.

In other words, in the T-bill futures market, the rate would be stated as 6.2%, which would imply a futures price of 0.9845.

A trader would buy the 135-day T-bill for $0.9777, and sell the futures at a price of $0.9845. Then, 45 days later, the T-bill would be a 90-day T-bill and would be delivered to settle the futures contract. The return per dollar invested would be 0.9845/0.9777 = 1.006955.

If the trader had purchased a 45-day T-bill at the price of $0.9931 and held it for 45 days, the return per dollar invested would be 1/0.9931 = 1.006948.

The two risk-free strategies would be equivalent in terms of return per dollar invested, with a slight difference due to rounding.

#### Practice Question 1

A futures contract on a T-bill expires in 60 days. The T-bill matures in 150 days. The discount rates on T-bills are:- 60-day T-bill: 5.15%
- 150-day T-bill: 5.45%.

First find the prices of the 60- and 150-day bonds:

B

_{0}(60) = 1 - 0.0515 (60/360) = 0.9914.

B

_{0}(150) = 1 - 0.0545 (150/360) = 0.9773.

The futures price is f

_{0}(60) = 0.9773/0.9914 = 0.9858.

The implied discount rate is r

_{0}

^{df}(60) = (1 - 0.9858) (360/90) = 0.0568.

#### Practice Question 2

If a futures contract on a T-bill is trading in the market at an implied discount rate 20 basis points higher than is appropriate, what would a trader do to take advantage of this opportunity?I. Sell the T-bill short.

II. Buy the futures contract.

III. Lend money in the repo market.

IV. Buy the T-bill.

V. Borrow money in the repo market.

VI. Sell the futures contract.Correct Answer: I, II and III

#### Practice Question 3

If a current Treasury bill futures contract is quoted at 94.57, what would be settlement price once the futures contract expires?A. $9,547.

B. $945,700.

C. $986,425.Correct Answer: C

The T-bill contract is always based on the 90-day T-bill and the contract size is $1 million.

Step 1. Determine the discount yield (DY). Futures Price = 100 - DY. 94.57 = 101 - DY => DY = 5.43%.

Step 2. Compute settlement price. 1,000,000 - DY ($1,000,000) (Days to Maturity / 360) = $986,425.

#### Practice Question 4

Which of the following would not be a step in a series of trades designed to earn an arbitrage profit when the implied repo rate exceeds the actual repo rate for a stock index futures?A. Selling the futures contract.

B. Buying the spot assets.

C. Short selling the spot asses.

D. Borrowing funds as opposed to investing.Correct Answer: C

If the actual repo rate (or the financing cost of carrying the asset) is less expensive than expected, it is worthwhile to carry the asset, because the futures price is overcompensating the investor for holding on to the asset. In other words, buy the underlying asset in the spot market and deliver it in the future. Thus, sell the futures when the implied repo rate is too high.

Proper steps: 1. Sell the futures. 2. Borrow the funds in order to finance the cost of acquiring the underlying assets. 3. Buy the underlying assets today. 4. At maturity, sell the assets, pay off the loan and settle the futures position

#### Practice Question 5

A futures contract on a T-bill expires in 40 days. The T-bill matures in 130 days. The discount rates on T-bills are:- 40-day T-bill: 4.9%
- 130-day T-bill: 4.5%.

A. 0.0436.

B. 0.0465.

C. 0.047.Correct Answer: A

First find the prices of the 40- and 130-day bonds:

B

_{0}(40) = 1 - 0.049 (40/360) = 0.9946.

B

_{0}(130) = 1 - 0.045 (130/360) = 0.9838.

The futures price is f

_{0}(40) = 0.9838/0.9946 = 0.9891.

The implied discount rate is r

_{0}

^{df}(40) = (1 - 0.9891)(360/90) = 0.0436.

#### Practice Question 6

If a futures contract on a T-bill is trading in the market at an implied discount rate 20 basis points lower than what is appropriate, what would a trader do to take advantage of this opportunity?I. Sell the T-bill short.

II. Buy the futures contract.

III. Lend money in the repo market.

IV. Buy the T-bill.

V. Borrow money in the repo market.

VI. Sell the futures contract.

A. II, IV and V

B. I, II and III

C. IV, V and VICorrect Answer: C

The strategy would produce a risk-free return.

#### Practice Question 7

A futures contract on a 125-day T-bill expires in 30 days. John is considering an investment strategy to sell the futures contract and buy the 125-day T-bill. Which of the follow strategy/strategies is (are) equivalent to John's strategy, given that the market is efficient and all options are available?I. purchase a 30-day T-bill and hold it to maturity.

II. short sell the 125-day T-bill, invest the proceeds in repo market for 30 days, and buy the futures contract.

III. purchase a 95-day T-bill and hold it to maturity.

IV. purchase a 125-day T-bill and short sell a 95-day T-bill, invest (borrow) the difference in T-bill prices in (from ) the repo market.

A. I and IV

B. II and III

C. I and IICorrect Answer: C

Either of these two strategies would produce the same risk-free rate of return as that of John's strategy.

#### Practice Question 8

A futures contract on a T-bill expires in 60 days. The T-bill matures in 150 days. The discount rates on T-bills are:- 60-day T-bill: 6.1%.
- 150-day T-bill: 6.35%.

A. 0.076.

B. 0.064.

C. 0.063.Correct Answer: A

First find the prices of the 150-day T-bill: B0(150) = 1 - 0.0635 (150/360) = 0.9735. The repo rate is the annualization of the return per dollar:

r

_{0}(h)* = [f

_{0}(h)* / B

_{0}(h + m)]

^{365/h}- 1 = [0.9853/0.9735]

^{365/60}- 1 = 0.076.

Note that we don't need to find the price of 60-day bond and the theoretical futures price in this case.

B

_{0}(60) = 1 - 0.061 (60/360) = 0.9898.

The theoretical futures price should be f

_{0}(60) = 0.9735/0.9898 = 0.9835.

### Study notes from a previous year's CFA exam: