#### 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 B0(h + m). Assuming \$1 face value: B0(h + m) = 1 - r0d(h + m) x (h + m) / 360, where r0d(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 f0(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 f0(h). As f0(h) is known on day 0, this transaction is risk-free.

The futures price f0(h) should be the price of the (h + m)-day T-bill compounded at the h-day risk-free rate:

f0(h) = B0(h + m) x [1 + r0(h)]h/365

where r0(h) is the h-day risk-free rate on day 0.

Alternatively, f0(h) should earn the same return per dollar invested as would a T-bill purchased on day 0 that matures on day h: f0(h) / B0(h + m) = 1 / B0(h)

f0(h) = B0(h + m) / B0(h)

where B0(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, B0(h) = 1 - r0d(h) x (h) / 360, where r0d(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:f0(h) = 1 - r0df(h) x (m/360), where r0df(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.

r0df(h) = [1 - f0(h)] x (360/m)

Note that a futures contract does not pay an interest. The r0df(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.

r0(h)* = [f0(h)* / B0(h + m)]365/h - 1

where f0(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.
• r0d(h) = r0d(45) = 0.055.
• r0d(h + m) = r0d(135) = 0.0595.
• The price of the h-day T-bill is B0(h) = B0(45) = 1 - r0d(h) x (h) / 360 = 1 - 0.055 x 45 / 360 = \$0.9931.
• The price of the (h + m)-day T-bill is B0(h + m) = B0(135) = 1 - r0d(h + m) x (h + m) / 360 = 1 - 0.0595 x 135 / 360 = \$0.9777.
• The price of a futures contract is therefore f0(h) = f0(45) = B0(h + m) / B0(h) = 0.9777 / 0.9931 = 0.9845.
• The discount rate implied by the futures price would be r0df(h) = r0df(45) = [1 - f0(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%.
What is the implied discount rate on the futures?

First find the prices of the 60- and 150-day bonds:
B0(60) = 1 - 0.0515 (60/360) = 0.9914.
B0(150) = 1 - 0.0545 (150/360) = 0.9773.
The futures price is f0(60) = 0.9773/0.9914 = 0.9858.
The implied discount rate is r0df(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.
III. Lend money in the repo market.
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.

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.
C. Short selling the spot asses.
D. Borrowing funds as opposed to investing.

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%.
What is the implied discount rate on the futures?

A. 0.0436.
B. 0.0465.
C. 0.047.

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

B0(40) = 1 - 0.049 (40/360) = 0.9946.
B0(130) = 1 - 0.045 (130/360) = 0.9838.
The futures price is f0(40) = 0.9838/0.9946 = 0.9891.
The implied discount rate is r0df(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.
III. Lend money in the repo market.
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 VI

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 II

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%.
Assume \$1 par value. The observed price of the 60-day futures contract is 0.9853. What is the implied repo rate?

A. 0.076.
B. 0.064.
C. 0.063.