GARP.FRM.PQ.P1 Increasing Beta question

[marque]

A fund manager has a USD 100 million portfolio with a beta of 0.75. The manager has bullish expectations for the next couple of months and plans to use futures contracts on the S&P500 to increase the portfolio´s beta to 1.8. Given the following information, which strategy should the fund manager follow.

• The current level of the S&P index is 1250
• Each S&P futures contract delivers USD 250 times the index
• The risk-free interest rate is 6% per annum

a. Enter into a long position of 323 S&P futures contracts
b. Enter into a long position of 336 S&P futures contract
c. Enter into a long position of 480 S&P futures contracts
d. Enter into a short position of 240 S&P futures contracts

CORRECT: B
Since the desired beta (1.8) is greater than the current beta (0.75), a long position in S&P
futures contracts is needed. The number of contracts needed is:
(β* - β) * Portfolio_Value / Futures_Value = (1.8 - 0.75) * 100,000,000 / (1250 * 250)
= 336 contracts.

Hi all, in the first attempt, I set X as dollar amount of index, in order to make the total portolio increase to 1.8, so 0.75*100,000,000+1*X=1.8*(X+100,000,000), then X resolved to -$131,250,000, then 131250000/(1250*250), so my answer is d.

I used weighted avg beta, it seems like answer seems like not taking weighted average beta into consideration. Does anyone know the logic behind the official answer? why no weighted avg of beta? thanks.

 

[ShaktiRathore]

Hi,
Let the S&P changes by x% then the portfolio value shall change by .75*x%*100 million.
Let N be the No. of futures contracts on the S&P500 to increase the portfolio´s beta to 1.8 so that futures position changes by 1250*250*1*x%*N(beta of futures=1),
Net portfolio position changes by .75*x%*100 million+1250*250*x%*N=1.8*x%*100 million
=>.75*100 million+1250*250*N=1.8*100 million
=>1250*250*N=1.8*100 million - .75*100 million
=>1250*250*N=(1.8- .75)*100 million
=>N=(1.8- .75)*100 million/(1250*250)
=>N=(1.8- .75)*100,000,000/(1250*250)
=>N=336
thanks

 

[David Harper CFA FRM]

Source is here (in paid) https://forum.bionicturtle.com/threads/question-21-portfolio.3540

 

[marque]

Hi,
Let the S&P changes by x% then the portfolio value shall change by .75*x%*100 million.
Let N be the No. of futures contracts on the S&P500 to increase the portfolio´s beta to 1.8 so that futures position changes by 1250*250*1*x%*N(beta of futures=1),
Net portfolio position changes by .75*x%*100 million+1250*250*x%*N=1.8*x%*100 million
=>.75*100 million+1250*250*N=1.8*100 million
=>1250*250*N=1.8*100 million - .75*100 million
=>1250*250*N=(1.8- .75)*100 million
=>N=(1.8- .75)*100 million/(1250*250)
=>N=(1.8- .75)*100,000,000/(1250*250)
=>N=336
thanks

Thanks, but I just wondered why the equation is not like this .75*x%*100 million+1250*250*x%*N=1.8*x%*(100 million+1250*250*N) ? Because after we buy futures, the total portfolio size increased to (100 million+1250*250*N).

 

[ShaktiRathore]

Hi,
we are seeing the effect on the portfolio value only,we just net out the loss/gain of futures from portfolio.
If S&P changes by 1% then portfolio value changes by 1%=.75*.01*100 million=750,000.
The Futures value changes by 1250*250*.01*336=1,050,000 just add this profit of futures into the portfolio to get net portfolio profit of 750,000+1,050,000=1,800,000
Thus the portfolio value changes by 1,800,000/100,000,000=.018 or 1.8% which is 1.8 times 1% so that if S&P changes by 1% portfolio value changes by 1.8% so that the portfolio beta is effectively 1.8. Beta=1.8%/1%=1.8.
thanks

 

[marque]

Hi,
we are seeing the effect on the portfolio value only,we just net out the loss/gain of futures from portfolio.
If S&P changes by 1% then portfolio value changes by 1%=.75*.01*100 million=750,000.
The Futures value changes by 1250*250*.01*336=1,050,000 just add this profit of futures into the portfolio to get net portfolio profit of 750,000+1,050,000=1,800,000
Thus the portfolio value changes by 1,800,000/100,000,000=.018 or 1.8% which is 1.8 times 1% so that if S&P changes by 1% portfolio value changes by 1.8% so that the portfolio beta is effectively 1.8. Beta=1.8%/1%=1.8.
thanks

So it seems like it is all this 100 million portfolio matters, the base value for all the profit and loss is always this 100 million. Thanks.

 

[David Harper CFA FRM]

Hi @marque The beta is a weighted average:

100%*0.75 + x%*1.0 = 1.80, such that x% = 1.80 - 0.75 = 1.05 or 105% weight on the futures contract. And 105% weight on futures contract = $100.0 million * 105% = 105.0 million in notional, which is 105.0/(1250*250) = 336 contracts. So the maybe confusing part is that our weights here end up being 100% principal + 105% notional.

So the beta is weighted but the futures contract is a notional amount with 1.0 beta such that, after then leverage:
weighted beta = 1.80 = ($100 mm * 0.75 + $105 mm * 1.0 beta)/$100 mm

which is why I prefer to approach this in terms of beta dollars. In this case, our goal is $100 mm * 1.8 = $180 million beta dollars. We are currently at 0.75 * 100 = 75 million beta dollars, so we want to add 105 million beta dollars: 105.0/(1250*250) ... which is just a way of acknowledging the goal is a weighted beta. I hope that's helpful!

 

[marque]

Hi @marque The beta is a weighted average:

100%*0.75 + x%*1.0 = 1.80, such that x% = 1.80 - 0.75 = 1.05 or 105% weight on the futures contract. And 105% weight on futures contract = $100.0 million * 105% = 105.0 million in notional, which is 105.0/(1250*250) = 336 contracts. So the maybe confusing part is that our weights here end up being 100% principal + 105% notional.

So the beta is weighted but the futures contract is a notional amount with 1.0 beta such that, after then leverage:
weighted beta = 1.80 = ($100 mm * 0.75 + $105 mm * 1.0 beta)/$100 mm

which is why I prefer to approach this in terms of beta dollars. In this case, our goal is $100 mm * 1.8 = $180 million beta dollars. We are currently at 0.75 * 100 = 75 million beta dollars, so we want to add 105 million beta dollars: 105.0/(1250*250) ... which is just a way of acknowledging the goal is a weighted beta. I hope that's helpful!

Thanks David, this makes more sense