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:
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)
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.
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.
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.
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:
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.
First find the prices of the 60- and 150-day bonds:
I. Sell the T-bill short.
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.
A. Selling the futures contract.
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
First find the prices of the 40- and 130-day bonds:
B0(40) = 1 - 0.049 (40/360) = 0.9946.
I. Sell the T-bill short.
A. II, IV and V
The strategy would produce a risk-free return.
I. purchase a 30-day T-bill and hold it to maturity.
A. I and IV
Either of these two strategies would produce the same risk-free rate of return as that of John's strategy.
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:
r0(h)* = [f0(h)* / B0(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.
B0(60) = 1 - 0.061 (60/360) = 0.9898.