# Valuing a T-Bill Futures Contract

A 60-day T-bill is quoted at 6% and a 150-day bill is priced at 6.5%. Calculate the No-Arbitrage price of a 60-day future? A. 0.9900 B. 0.9729 C. 0.0101 D. 0.9827 What’s your pick on this?? I tried to understand the solution to this, but in vain. It would be great if someone could list down the exact steps for solving such kind of a derivatives valuation problem that are going to flood the 15% of the L2 exam. Thanks in advance!!

(1 - 60/360*.06) = 0.99? Choice A

cpk123 Wrote: ------------------------------------------------------- > (1 - 60/360*.06) = 0.99? Choice A That is only the price for the 60 day T-Bill. I believe the answer is D, as follows: Bo(60) = 1 - .06(60/360) = .99 Bo(150) = 1 - .065(150/360) = .9729 Fo(60) = Bo(150)/Bo(60) = .9729/.99 = .9827 Similarly, the Futures price can be worked out by finding out the rate we need to compound the 150 day T-Bill at by using the price of the 60 day T-Bill: 1/.99 = 1.0101 Fo(60) = Bo(150)*ro(60) = .97292*1.0101 = .98274 This Futures price has a yield of 6.904% (1-.98274)*(360/90) = .06094 If the Future price was trading at a lower yield (higher price), you could short the Futures and buy the 150 day T-Bill and hold until maturity at which point you would earn a risk-free return.

thought you are 100% correct with answer D, darkhelmet !! And this is the exact same solution Schweser has given on Bk-5-Pg-220. But could you elaborate as what what’s going on??? Probably in Layman’s terms

The LOS does not ask you to calculate this. In fact, it’s all optional material in the CFAI texts.

dinesh, Essentially, you calculate the hpy for the shorter maturity bond and use it as the risk free rate. You then adjust the price of the longer maturity bond like you would any other spot rate to get to the futures price. This question’s a bit tricky because they don’t give you the price t-bills so you have to calculate the prices from the rates.

So darkhelmet’s solution is good and all but it’s really not exactly right. Just for fun - futures contracts are marked to market so gains and losses on the contracts earn interest rates that are highly correlated with the underlier for the contract. So darkhelmet’s solution works exactly for forward contracts. Is the futures contract worth more or less than the forward contract and why?

Guys, I think I need a break… I don’t understand a word of the calculation done above 1. Why is Bo(60) calculated as (1 - the adjusted rate) ??? 2. Why is Bo(150) calculated as (1 - the adjusted rate) ??? 3. How come this equation?? Fo(60) = Bo(150)/Bo(60) ??? 4. What this? 1/.99 = 1.0101 (How did this 1 come into picture all of a sudden??) ??? 5. The only thing I understood is this -> Fo(60) = Bo(150)*ro(60) ??? Joey to answer your question, Everything boils down to the correlation between the interest rates and the cash flows that you are going to receive due to this MTM daily activity (some investors like it, some don’t) When the MTM occurs on your margin account and the surplus cash flow generated can be invested at a bigger-better rate then MTM feature of the futures-contract is the preference and value of the future is higher than that of an equivalent forward. Similarly, any deficit in those margin accounts due to the MTM activities and there occurs a margin call, and you need to borrow cash to reach-back to the minimum maintenance level, if in that case if the borrowing rate is low, investors prefer the MTM feature of futures-contract and value of the future is higher than that of an equivalent forward. So we see a direct correlation between the cash-generated by the underlier and the prevailing interest rates – investor-decision boiling down to FUTURES contract. If we get cash, we would like to reinvest it at a higher IR. If we loose cash, we would like to borrow it at a lower IR to repay out debts. So when cash and IR seems to be positively related, then FUTURES preferred over FORWARD and when cash and IR seems to be negatively related, then FORWARDS preferred over FUTURES So LONG a FUTURE if positive correlation between the cash-generation & interest-rate And LONG a FORWARD if negative correlation between the cash-generation & interest-rate Am I talking BS here?

That’s all correct but you don’t have the final piece. So for a T-bill contract which would you rather have, a forward or a futures since the T-bill itself is highly negatively correlated with the interest rate you are going to get on your MTM p/l? If nobody clarifies above 5, I’ll do it later (c’mon folks, especially darkhelmet, give it to him a little slower).

dinesh.sundrani Wrote: ------------------------------------------------------- > Guys, I think I need a break… I don’t > understand a word of the calculation done above > > > 1. Why is Bo(60) calculated as (1 - the adjusted > rate) ??? > 2. Why is Bo(150) calculated as (1 - the adjusted > rate) ??? > 3. How come this equation?? Fo(60) = > Bo(150)/Bo(60) ??? > 4. What this? 1/.99 = 1.0101 (How did this 1 come > into picture all of a sudden??) ??? > 5. The only thing I understood is this -> Fo(60) > = Bo(150)*ro(60) ??? > 1. and 2. you are essentially finding the Price of the T-Bill. T-Bills are quoted as 100 - annualized discount rate. Thus to find the price of Bo(60) and Bo(150) you use the formulae given by darkhelmet above or simply (1-adjusted rate) 3. Basically helps you price a 90 day T-Bill 60 days from now = The Futures Price for a 90 day T Bill in 60 days 4. 1/.99 = This is just the risk free return you are getting on your 60 day TBill. Essentially you are lending .99 and receiving 1.0 at the end of the 60 days, hence giving you an HPR of .0101 or 1.01%. That then becomes the risk free rate for a 60-day period 5. You claim you understood this…so it should fall into place now…

JoeyDVivre Wrote: ------------------------------------------------------- > That’s all correct but you don’t have the final > piece. So for a T-bill contract which would you > rather have, a forward or a futures since the > T-bill itself is highly negatively correlated with > the interest rate you are going to get on your MTM > p/l? forwards are more expensive in such situation.

T-bill is bit strange because it is discount instrument. If you are familiar with FRA, you just need to convert all the + to -, you know how to calculate T-bill price.

mumukada Wrote: ------------------------------------------------------- > dinesh.sundrani Wrote: > -------------------------------------------------- > ----- > > Guys, I think I need a break… I don’t > > understand a word of the calculation done above > > > > > > 1. Why is Bo(60) calculated as (1 - the > adjusted > > rate) ??? > > 2. Why is Bo(150) calculated as (1 - the > adjusted > > rate) ??? > > 3. How come this equation?? Fo(60) = > > Bo(150)/Bo(60) ??? > > 4. What this? 1/.99 = 1.0101 (How did this 1 > come > > into picture all of a sudden??) ??? > > 5. The only thing I understood is this -> > Fo(60) > > = Bo(150)*ro(60) ??? > > > > 1. and 2. you are essentially finding the Price of > the T-Bill. T-Bills are quoted as 100 - annualized > discount rate. Thus to find the price of Bo(60) > and Bo(150) you use the formulae given by > darkhelmet above or simply (1-adjusted rate) > > 3. Basically helps you price a 90 day T-Bill 60 > days from now = The Futures Price for a 90 day T > Bill in 60 days > > 4. 1/.99 = This is just the risk free return you > are getting on your 60 day TBill. Essentially you > are lending .99 and receiving 1.0 at the end of > the 60 days, hence giving you an HPR of .0101 or > 1.01%. That then becomes the risk free rate for a > 60-day period > > 5. You claim you understood this…so it should > fall into place now… mumu, need some more clarification to drive the question home!! So when I see a question like this… “Calculate the No-Arbitrage price of a 60-day future?” And I have a no-arbitrage futures pricing formula as “FP = So*(1 + RFR)^T” How do they all of a sudden come to the conclusion that this is a good thing to get the ans: “FP = B(150) * (1 + RFR-of-B(60))^T” I am surely missing something very big here, because if my math is good then 150-60 comes down to 90 and we also have standardized 90-day T-Bills. How to fit the pieces of this puzzle?

JoeyDVivre Wrote: ------------------------------------------------------- > That’s all correct but you don’t have the final > piece. So for a T-bill contract which would you > rather have, a forward or a futures since the > T-bill itself is highly negatively correlated with > the interest rate you are going to get on your MTM > p/l? > I would grab the LONG FUTURE position in that case…

“Calculate the No-Arbitrage price of a 60-day future?” And I have a no-arbitrage futures pricing formula as “FP = So*(1 + RFR)^T” How do they all of a sudden come to the conclusion that this is a good thing to get the ans: “FP = B(150) * (1 + RFR-of-B(60))^T” C’mon - you have it here. S0 = ? and RFR = ?.

That’s what I wanted to know why is this, S0 = B(150) and why is RFR = RFR-of-B(60)? and why not this, S0 = B(60) and RFR = RFR-of-B(150) That’s what software development makes you - “A DUMB BOZO”

I don’t think you’re a dumb bozo which is why I didn’t tell you the answer. I think that F = S0*(1 + rfr)^T is the forward price formula, but let’s say they are the same. The reason for that formula is that if I own a stock I should be indifferent to selling it now (for S0) and putting the money in risk-free deposits for a year (leaving me with S0*(1+rf)^T) or entering into a forward contract to deliver it in a year at the price at S0*(1+rf)^T (and T = 1 yr here and rf is the 1-yr risk-free rate). Ignoring counterparty risk, both give me a risk-free way of getting the same amount of money in a year. The same then => the same now. So forget stocks for a second and say I own a 150 day T-bill. I should be indifferent to selling it to you now for B(150) and depositing my money in risk free deposits for 60 days earning RFR of B(60) thus giving me B(150)*(1 + RFR of B(60))^T or enetering into a forward contract at that price. It’s just the same as in the stock example except for that thing that I added that the T-bill price is much more correlated with the margin deposit interest rate than a stock is likely to be.

Thanks for a very lucid explanation Joey. So when we say we have a T-Bill futures contract, it is implicitly assumed that we are talking about delivering a 90-day T-Bill at the contract expiration??? ***I think I missed this bit from the book completely (and I don’t have them now to reconfirm) *** Yet another problem on the similar lines below, A 45-day T-bill has a discount rate of 5.5%. A 135-day T-bill has a discount rate of 5.95%. Calculate the price of a futures contract that expires in 45 days. Here are want to price a futures contract that expires in 45-days and delivers a 90-day T-Bill. So h = 45 (contract expiration) m = 90

dinesh.sundrani Wrote: ------------------------------------------------------- > Thanks for a very lucid explanation Joey. > > So when we say we have a T-Bill futures contract, > it is implicitly assumed that we are talking about > delivering a 90-day T-Bill at the contract > expiration??? ***I think I missed this bit from > the book completely (and I don’t have them now to > reconfirm) *** > Yes although a) Nobody trades T-Bill futures anymore b) The untraded T-bill futures contract is now cash-settled but nobody told CFAI. c) Today the volume in Eurodollar futures was 1.6 million contracts and the volume in T-bills was 0. d) I have no idea why anybody would ask questions about a security that isn’t traded anymore. Why not have some questions about GNMA futures? > Yet another problem on the similar lines below, > A 45-day T-bill has a discount rate of 5.5%. A > 135-day T-bill has a discount rate of 5.95%. > Calculate the price of a futures contract that > expires in 45 days. > > Here are want to price a futures contract that > expires in 45-days and delivers a 90-day T-Bill. > > So > h = 45 (contract expiration) > m = 90 the problem > Par Value of the bonds = \$1 h + m = 135 (T-Bill to be delivered is a 135-Day > T-Bill) > > r(45) = 0.055 > r(135) = 0.0595 > r-adjusted(45) = 0.055*45/360 = 0.006875 > r-adjusted(135) = 0.0595*135/360 = 0.0223125 > Price of a 45-day T-Bill = B(45) = 1 - 0.006875 = > 0.993125 > Price of a 135-day T-Bill = B(135) = B(45 + 90) = > 1 - 0.0223125 = 0.9776875 > HPY(of the 45-day Tbill) = 1/0.993125 - 1 = > 0.006922 = 0.6922% Price of the futures contract is F(45) = B(135)*(1 > + 0.006922) > F(45) = 0.9776875*1.006922 = 0.984455052875 > > > This also means that we can solve a problem like > this “A 45-day T-bill has a discount rate of 5.5%. > A 173-day T-bill has a discount rate of 5.95%. > Calculate the price of a futures contract that > expires in 45 days” … because [173 – > 45] is not equal to 90

Thanks Joey, as always, learnt a lot from you.