# Price of a T Bill futures contract

Hi,

A 60 day t bill is quoted @ 6% and a 150 day bill is prices @ 6.5%. Calculate the no arbitrage price of a 60 day futtures on a 90 day T Bill.

Can someone please solve the equation and explain it.

Thanks.

Recall that T-Bills are priced at a discount. The value (as a fraction of par) of the 60-day is:

1 – 6%(60/360) = 0.9850

The value (as a fraction of par) of the 150-day is:

1 – 6.5%(150/360) = 0.9729

Thus, the value (as a fraction of par) of the 90-day, 60 days from today, is:

0.9729/0.9850 = 0.9877

To get the price, solve:

1 – d(90/360) = 0.9877

d(90/360) = 1 – 0.9877 = 0.0123

d = 0.0123(360/90) = 0.0491 = 4.91%

This is confusing.

I understand this part:

Thus, the value (as a fraction of par) of the 90-day, 60 days from today, is:

0.9729/0.9850 = 0.9877

However,this should the value of a 90 day bill today. How do you get this as the value 60 days from today.

For a no arbitrage condition to exist, the value of the 150 day T-bill must be the same as:

Investing in a 60-day T-bill today, when it matures, taking those proceeds and investing in a 90-day T-Bill. This should be the equivalent return as buying a 150 day T-bill today, otherwise aribitrage would exist. So, the question is, what must be the value of that 90-day T-bill, starting in 60-days, for no arbitrage to exist?

Mathetmatically this is:

0.9850 * 90-day T-bill starting in 60 days = 0.9729

Solve for X to get the value above.

Now it makes sense! Thanks!

Great explanations!

My pleasure.

As with many other bond questions, this one benefits from drawing a time line.

S2000 - how’s 1 – 6%(60/360) = 0.9850

Is this a typo? I think it should be 1 - .06*(60/360) = 0.99

Another way to look at this problem:

Recall that no arbitrage price of the future is F = S(1+R)^t

We need S - spot price of 150 day T-bill as it will be a 90 days T-bill after 60 days, and

R - the applicable interest rate for the 60 days period

150 day T-bill is available today at 6.5% discount (I suppose the customary way is to say T-Bill is available at 100 - 6.5 = 93.5%)

So the discount is 0.065*150/360 = 0.02708; or the spot unit price of a 150 day T-bill is 1 - 0.02708 = 0.97292 = S

To compute 60 days interest rate R first calculate the spot price of a 60 days T-bill

60 days T-bill is available at 6% discount (or at 94%)

So the discount is 0.06*60/360 = .01; or the unit spot price of a 60 days T-bill is 1 - 0.01 = 0.99

This 60 days T-bill will mature at 1 dollar after 60 days. So the holding period return is (1 - 0.99)/0.99 = 0.0101 or 1.01%

This is the value of R we need to use in the Future price calculation. t = 1, as we are talking about one 60-days period.

F= 0.97292(1+0.0101) = 0.98275

Since T-bills are described as a discount of the face value of 100, the discount%, say “d”, needs to calculated as below

1 - d*90/360 = 0.98275, or d = 0.069, or 6.9%. The T-bill should be quoted as 100 - 6.9 = 93.1%