# CFAI Question on Synthetic Cash

This is from a practice CFAI question. It goes something like this: Portfolio is currently invested in mid-cap equity with beta of 1.2 and value (say) \$100 M. Goal is to convert position to synthetic cash. Mid-cap equity futures contract available has price (say) \$200,000, beta 1.29, and maturity 3 months. Riskless rate is (say) 4% per annum. Now the CFAI answer says the number of futures contracts to be shorted is = 100 M * (1.04)^0.25 / 200,000 = 505 contracts Now the above basically matches values, that is once these contracts have been shorted, then the value invested in equities is balanced by the value invested in futures. However, the position ISN’T CASH! Why? Because it has a non-zero beta. Cash does not have any correlation with market movements. Obviously the portfolio equity and index equity are not based on the exactly the same stock weights, for if they were, then their betas would be equal. So can’t really hedge the portfolio with the index futures to create synthetic cash. In fact, Schweser also gives the above equation for creating synthetic cash… but, in another place for creating synthetic cash (while converting the equity position to a bond position) Scheweser says that we must make the beta to be zero. So you have two (non-similar) suggested methods of creating synthetic cash. First is by matching values (above equation). Second is by making beta zero. Which one to use in the exam? I would recommend try both and hope that only one gives an answer that matches the options. NC

If I had to choose between investing the total amount of cash, and setting beta to zero… I would go with the first option. NC

the general rule is: to change beta, or create synthetic cash from stock, or create synthetic stock from cash: No of contracts = (T-C)/F * value/size which means: No of contracts = (Target Beta - Current Beta)/Futures Beta * value/contract size. this general formula works for changing beta, duration and works with futures or swaps (with swaps it is (T-C)/S, etc.) (and with Duration you also multiply by CTD Conversion ratio, Yield Beta if hedging) In this case (T-C) = -1 because you are wanting to get out of the market altogether and remove all systematic risk, regardless of the current beta of the stocks you hold. Because you are wanting to remove all systematic risk the you need to calculate the futures beta on the denominator. Futures Beta = 1/(1+RFR)^T = 0.99024. So number of Futures contracts to remove all systematic risk = -1/0.99024 * \$100m/\$200k = sell 505 rounded. the alternate formula = value/size * (1+RFR)^T which is the same thing. but I prefer to start with the general form: (T-C)/F.

^ The two methods will give you different results because they totally different inputs (beta isn’t included in the synthetic cash formula). Use the “synthetic cash” formula when you are creating synthetic cash out of equity, or when you are creating a synthetic stock index out of cash. Use the “target beta” formula when your intent is to alter the portfolio beta, or rebalance between asset classes (ie EQ to FI). *Mock Exam Spoiler* There is a question on the mock exam which asks you to create an equity exposure from a cash holding. The answers for BOTH METHODS are listed among the three choices, but only the “synthetic cash” answer is correct. So it the choice is not of a trivial nature, as it was demonstrated on the mock exam.

we’re in violent agreement here. both formulas [synthetic cash = val/size * (1+RFR)^T] and the target beta formula [(T-C)/F *val/size] are the same concept. When creating synthetic cash: (T-C) = -1, and F = 1/(1+RFR)^T. Just different forms of the same concept.

I agree that conceptually they should be the same, but CFAI must have simplified things by mispricing the futures contract. The data given is as follows: *Spoiler again* 100,000,000 Euro portfolio in cash Rf = 2% Beta Europe EQ = 0.8 T = 0.25 years Futures = 3,000 Multiplier = 10 The correct answer was: using synthetic cash formula (100,000,000*(1.02)^0.25) / (30,000) = Go long 3,350 Europe EQ contracts My answer (incorrect) was ((0.8 - 0) / 0.8) * (100,000,000 / 30,000) = Long 3,333 Euro EQ contracts

I think the use of the formulae above depend on what is stated in the question , have they explicity used the word " Synthetic " ?

They do use the magic word: “Carly Dungan requests that her portfolio be reallocated to a synthetic index fund of European equities for the next three months.” and later “In order to reallocate Dungan’s portfolio, the number of European futures contracts that should initially be purchased is closest to:” If they said “How European futures contracts are required to change the portfolio beta to 0.8” , we would use the beta formula. Just need to pay closer attention to the wording I guess.

I guess we wd also consider the time factor …i.e " for the next 3 months " . Using the second formula ( target beta - current beta ) does not reflect the length of time .

Rudeboi Wrote: ------------------------------------------------------- > I guess we wd also consider the time factor …i.e > " for the next 3 months " . Using the second > formula ( target beta - current beta ) does not > reflect the length of time . Wouldn’t that be embedded in the price of the future?

I guess it would be , but in the absence of any information on the length of the contract … the use of the word " Synthetic" and if they provide the RFR , I would stick with the synthetic cash formula . Remember that CFAI is usually pretty clear in thier questions unlike some of the weird Schweser Q’s , atleast I hope that is the case .

I think I’ve got it. The real price of the future should be 3,003 euros, but they must have rounded at some point while making the question. Somebody was probably like “hey, lets round this so it looks better.” Since 3,333 is the answer using the synthetic index formula: 3,333 = [(0.8 - 0) / 0.8] * (100,000,000) / (10*X) >> solve for X and you get F = 3,003 Using the futures value (rounded) provided in the question you would get: 3,350 = [0.8 - 0 / 0.8] * (100,000,000 / (10*3,000) which is one of the incorrect answers Pretty cheap of them to leave the other answer as one of the choices.

can’t see what the problem is. you can get there either way: 1) no of contracts = 100m/30k * (1.02)^0.25 = 3,350 rounded or 2) (Target B of 1 less Current B of 0) = 1, and Futures B = 1/(1.02)^0.25 = 0.99506 so the no of contracts = 100m/30k * 1/0.99506 = 3,350 rounded. then once you have 100% systematic risk exposure to Europe, then you can fiddle with given futures betas to get the desired beta, etc. (Not sure what the given “Europe Beta” of 0.8 means - maybe that’s the beta to the world market - but so what? You want 100% exposure to Europe equity index, which has beta of 1, so the futures beta required would be a fraction less than 1: ie 1 discounted by the RFR for the given term)