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# change beta , shorting sell FNB

#### haroun

##### New Member
I have one question :
1) we want to decrease the beta of our portfolio to 0.75 without selling one of our positions current , your colleague suggests selling short the spy exchange traded fund (ETF) SPDR S&P 500 ETF trust . How many SPY shares must be sold ?
The value of my portfolio is 271478,18 $The beta of my portfolio is 0,96 Thank you . #### haroun ##### New Member hey @David Harper CFA FRM ,I have one question : 1) we want to decrease the beta of our portfolio to 0.75 without selling one of our positions current , your colleague suggests selling short the spy exchange traded fund (ETF) SPDR S&P 500 ETF trust . How many SPY shares must be sold ? The value of my portfolio is 271478,18$
The beta of my portfolio is 0,96
Thank you .

#### David Harper CFA FRM

##### David Harper CFA FRM
Staff member
Subscriber
hi @haroun to alter net beta we use Hull's Δβ = Value(portfolio)/Value(futures contract), which in this case can be adapted to: Δβ = value(portfolio)/value(ETF share) = (0.96 - 0.75)*271478.18 / 267.50 = 213.12, but I had to lookup the SPDR EFT price (and infer that it's 10% of the index; I actually didn't know that as I construct my portfolio with individual stocks) at https://us.spdrs.com/en/etf/spdr-sp-500-etf-SPY.

So if your portfolio value is about $271,478, then I get a short of about 213 ETF shares. As a gut check on my math, we're always seeking to change (as a special case of neutralizing) beta-dollars, β*Value. We start with 271,478*0.96 =$260,619 beta dollars then the short subtracts 213*267.50 = 57,010 beta dollars such that this trade implies our net portfolio beta = (\$260,619 - 57,010) / 217,478 = 0.75, which does looks right. But the question doesn't provide the ETF price of 267.50. So unless i'm missing something (entirely possible, I did this quickly), I'm not sure the question can be answered as-is. What's the source?

#### haroun

##### New Member
hey @David Harper CFA FRM , thank you for your answer , it's really what I thought to use this formula ( hull book ) , and for the source (Question) you will find the question that I really have a problem with it ( question 3 Risk Analysis - Beta) , excel file it's about question 1 and 2 that I have already calculated .

thank you again .

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#### David Harper CFA FRM

##### David Harper CFA FRM
Staff member
Subscriber
oh okay @haroun I realized a flaw in my calculation: by using Hull's future-based formula, implicitly my calculation above treats the short ETF as unfunded (a futures contract is unfunded). But the short position returns cash, and probably should therefore have a negative weight. In this way, Hull's future-based assumption that, beta being a simple value-weighted average, portfolio beta equates V(A)*β(target) = V(A)*β(current) + N*Value(future)*β(future) ... and this formula solves for the 213 shares above ... should be adjusted to: [V(A)+V(ETF)]*β(target) = V(A)*β(current) + N*Value(EFT)*β(EFT) in which case, instead, I calculate a short of about 852 EFT shares. It's the interesting, non-trivial issue of the difference in how we (can or should?) treat the weight of the short position; the futures is unfunded but the short is mathematically negatively weighted. I hope that's interesting.