Question

Consider the following quoted discounts currently posted for two Treasury Bills and a futures contract that delivers a 3-month Treasury Bill. Contract and its Maturity Quoted Discount (%)

90-day Treasury Bill 6.5%

180-day Treasury Bill 5.8%

90-day futures 15%

Answer the following: i) Is the futures contract priced as per the theoretical arbitrage valuation model? Show your calculations.

Answer 1

We have a forward contract to deliver treasury @15%.

Plus we have spot market rates of 6.5% 90 days and 5.80% for 180 days

i.e 1.63% for each quarter 90 days and 1.45% for each quarter for 180 days

Hence compounded return of first 90 days plus next 90 days should equal to compounded return of 180 days.

Let return for forward 90 day be X

i.e 1 (1+1.63%) (1+x%) = 1(1+1.45%)(1+1.45%)

1.0163 * (1+X%) = 1.0292

1+X% = 1.0292 / 1.0163

i.e X% = 1.27% for each quarter

i.e Forward rate 90 days= 5.08% p.a

As we observe forward rate is 15%, it is not a arbitrage free rate.

By just looking at the number also we could have determined that it is not a arbitrage free forward rate.

How so?

We observe that 180day spot is lower than 90day spot, hence 180 day spot will fall only if the forward rate falls after 90 days.

But the given situation showed otherwise where the forward rate was higher than 90 day spot. Hence it was not a balanced rate and not a arbitrage free rate.