# Number of contracts

Anybody feels like revising his/her studies and post ALL the different ways of calculating # of contracts to buy (gaining exposure) for Equity/Bond/Synthetic fund?

I just use (T-C)/F * Value/contract size as the general starting point. T = Target (beta or Dur) C = current (beta or dur) F = futures (beta or Dur) works for stocks, bonds - modifying duration or beta or reducing to synthetic cash - or creating synthetic stocks/bonds from cash. For bonds: also multiply by Conversion ratio of CTD bond - also mutiply by Yield Beta (if given) - usually only given if creating synthetic cash For equities - when creating sythetic cash from bond portf: - calc own Futures beta = 1/(1+RFR)^t. - - and then numerator is +1 to create synth stock, or -1 to create synthetic cash for swaps: it’s (T-C)/S * Value where S = Swap beta or duration that’s about it.

Thanks for the input. Good luck

null&nuller’s comments on the # contracts formula, and the variations looks correct to me. Can someone else confirm?

I like that way of thinking about it.

God this is de-moralising me I’ve done this only days ago…

I always use :

(E1):TargetBeta*TargetPortfolioValue= CurrentBeta*CurrentPortfolioValue+NumberofContracts*PriceFuture*BetaFuture

as the starting point in all situations (duration instead of beta depending on the situation)

TargetBeta*TargetPortfolioValue can be 0 or CurrentBeta*CurrentPortfolioValue can be 0 when one needs to change portfolio (equity to cash or vice versa)

i add conversion factor for the final number for ctd bond

It as always worked so far

Example (Q2 Application of Derivatives - Allison) :

Client B’s portfolio holds \$40 million of U.S. large-cap value stocks with a portfolio beta of 1.06. This client wants to shift \$22 million from value to growth stocks with a target beta of 1.21. Allison will implement this shift using S&P/Barra Growth and S&P/Barra Value futures contracts

Given:

Price of December S&P/Barra Growth futures contract

\$117,475

Price of December S&P/Barra Value futures contract

\$120,875

Beta of S&P/Barra Growth futures contract

1.15

Beta of S&P/Barra Value futures contract

1.03

When implementing the shift from value to growth stocks for Client B, the number of S&P/Barra Value future contracts Allison shorts will be closest to ?

From my pov, this kind of question is tricky because one doesn’t need all of the information given This client wants to shift \$22 million from value to growth stocks with a target beta of 1.21 , hence he needs first to covnert 22M of his porfolio to cash ( and then to buy Growth futures contract but that is not the question) In this situation , “Current portfolio”=22M , Beta = 1.06 “Target Situation” = 22M in Cash , Beta=0 Hence, (E1) would give 0=22*1.06+N*0.120875*1.03 => N=187

Example with change of Equity/Bond allocation (Application of Derivatives - Bing , Q3) The current market value of KPM Inc.'s pension portfolio is 950 million, with a 65% allocation to US midcap stocks and a 35% allocation to US bonds. The stock allocation has a beta of 1.45, and the bond allocation has a modified duration of 5.3. Bing's research indicates that midcap stocks are likely to underperform in the near term. Accordingly, she decides to reduce the allocation to midcap stocks to 60% and increase the bond allocation to 40%. Future Contract Mid cap , price =0.481900 (M) Beta =1.12 Future Bond Index , price = 0.165260 Duration =4.89 To carry out the proposed adjustment to the KPM Inc. pension portfolio, the number of S&P 400 MidCap futures Bing would need to sell and the number of Barclays US Aggregate Bond Index futures she would need to buy, respectively, are closest to: 128 S&P 400 MidCap Futures and 312 Barclays US Bond Futures 99 S&P 400 MidCap Futures and 287 Barclays US Bond Futures 76 S&P 400 MidCap Futures and 265 Barclays US Bond Futures Regarding Equity : Current portfolio = 617.5 M\$ (0.65*950), Beta= 1.45 New portfolio = 570 M\$ , Beta= 1.45 (E1) : 570*1.45=617.5*1.45+N*0.4819*1.12 => N=-128 or equivalently, 47,5M must be converted to cash Current portfolio = 47.5, Beta= 1.45 New portfolio = 47.5 M\$ , Beta= 0 (E1) : 47.5*0=47.5*1.45+N*0.4819*1.12 => N=-128 Regarding Bonds: Current portfolio = 332.5 M\$ (0.35*950), Duration= 5.3 New portfolio = 380 M\$ , Duration= 5.3 (E1) : 380*5.3=332.5*5.3+N*0.165260*4.89 => N=312 or equivalently, 47,5M must be converted from cash to bonds Current portfolio = 47.5, Duration=0 New portfolio = 47.5 M\$ , Duration= 5.3 (E1) : 47.5*5.3=47.5*0+N*0.165260*4.89 => N=312 Remarks that in this case, due to the proposed answers, one only needs the correct number of future equity contracts to sell to have the correct answer

I wrote a series of articles on this: http://financialexamhelp123.com/level-iii-risk-management-applications-of-derivatives/

Just finished reading through them. Good posts, thanks for sharing.

In the interest of not confusing myself, I think the second post here has it mathematically correct. If so, i’ll still to that formula.

Yes: the first formula is exactly what I have in my articles; the main difference is that I then go on to write them using dollar duration or dollar beta.

Very true. Thank you as always for your time. Won’t mess this concept up on the exam – I was a little confused with the CFAI presentation. Took me a good 2-3 hours to lock this in.

My pleasure.