Can beta be negative ?

[higaurav]

Hi David,

Given the formula of beta (slide 16), I was thinking hard that can beta be negative and if yes then what will be the impact on the expected return?

Considering the formula = Beta = Cov (m,i)/var m, if we use the formula for correlation then this will become Beta = correlation(m,i) * var i , now if this is correct, then in case if the security is having negative correlation then beta should be negative.

But how do we interpret negative beta, and are there any securities with negative beta...

In addition to this, is there any relation between CML and SML ?

Look fwd to your guidance on this..

thnx
OM

 

[David Harper CFA FRM]

Hi OM,

Yes, exactly, beta can be negative. Note, I get a slightly different ratio: Cov(i,M)/var(M) = correlation(i,M)*volatility(i)/volatility(M); i.e., it's just like the minimum variance hedge ratio. Since volatilities and variances are always (+), negative correlation implies negative beta.

Re interpretation: a regression of stock's excess returns against market excess return has negative slope. Higher market returns imply lower instrument returns. I don't have a security example *but* Andrew Lo's assigned hedge fund replication reading gives (as we'd expected) a negative beta for the dedicated short bias strategy; as market goes up, fund has some negative factor exposures. (that's in a multifactor model, but it's the same idea). So, the use could be for a hedge.

David

 

[higaurav]

Thanx David for the explanation and correction in the derivation of the beta, in a way now I can link the concept with the min variance hedge ratio as well...

 

[David Harper CFA FRM]

sure thing, i forgot to share link to screencast on beta last week . David

 

[dxeath]

Hi everyone,

I have a question about negative beta and Capital asset pricing model (CAPM)

I understand it is possible to get a negative beta, mathematically, and it is logical. But in CAPM we have the following formula,

Ri = Rf + Beta_i * (Rm - Rf)

where
Ri: return on asset i,
Beta_i : beta of asset i
Rf : return of risk free asset,
Rm : return of market.

When beta negative, asset is less risky and we expected less maybe negative return ??
How we will interpret this equation with a negative beta..
I appreciate any explanation
Thanks a lot

 

[koolrud]

Hi im new to the world of derivatives, and this is what i understand so far.
I understand the relationship between the minimum variance hedge ratio = beta
I understand beta can be negative, and it implies that the stock moves inversely with respect to the market.
However when cross-hedging in the commodities market what does a negative hedge ratio mean or did i overlook something very fundamental?

thank you for you patience and understanding

 

[David Harper CFA FRM]

Hi koolrud,

As hedge ratio = beta = correlation * (vol spot returns/vol futures returns), a negative hedge ratio implies (is implied by) a negative correlation between the change in forward and spot prices. Note that is *different than* an inverted forward curve (e.g., oil traditionally would be inverted/backwardation). Rather, negative hedge ratio says an increase in (change in) spot is correlated with a decrease in the forward; I don't know of a commodity where this should be expected? The correlations are theoretically quite positive. But for arguments sake, where a typically positive MV hedge ratio implies that a commodity consumer (e.g., airlines buys jet fuel) would hedge by going long futures contracts (buying forward), if the hedge ratio were negative it would imply the airline would instead hedge by shorting futures contracts.

Hope that helps, David

 

[koolrud]

Hi David, Thx for your speedy response.

I don't mean to sound uneducated so I am sorry if my question is a dumb one, but our book, options, futures, and other derivatives, is quite complex to understand.

By shorting futures, the commodity in question would be hedging against risk by selling futures contracts in the hope that the price would go down in the futures market?

I'm sorry i get confused with the taking a short position and long position.
I have it as taking a short position as selling and taking a long position as buying.

 

[David Harper CFA FRM]

not dumb, it is confusing

this is true: "I have it as taking a short position as selling and taking a long position as buying.
long forward is buying in the future, short forward is selling commodity in the future."

the hedge is situational, it depends on who is hedging and whether they are "hurt" by price increase or descrease.
Traditionally, we say it depens on whether they are producer or consumer:

producer of commodity (e.g., corn farmer) hedges with short futures contract.
spot market: he will be selling corn tomorrow (next year), so to hedge he must:
Go short the futures contract
so that, if price declines his loss in spot market will be offset by gain in futures

but if consumer of commodity (e.g., airline), the hedge is long futures
spot market: airline will be buying fuel tomorrow (next year), so to hedge must:
go long futures contracts
so that, if price increases, loss in spot market (higher fuel prices) offset by gain in futures contract

one thing i've learned about hedging: it's easy to forget the hedge is two things. One, the underlying spot transaction (buying a commodity in future? selling in future?). Two, the other different thing is the hedge instrument where the point is to make an offsetting profit.

David

 

[koolrud]

Thanks a lot, again, David you have been very helpful with my understanding of the hedgers position.

I am glad i found this website.

I will be around reading and researching.

Thx again

 

[koolrud]

Ok so back with the beta again.

if a fund manager wants to hedge all exposure to the market, does this not simply mean we would want to change the beta of the portfolio to zero so that he would be earning the risk free rate.

Its mine understanding that a portfolio with a beta of zero and the treasury bill beta being zero, that there is a no arbitrage opportunity. Therefore, the portfolio should receive the same return as the treasury bill.

assuming my assumptions are correct and the maturity of the futures contracts are close to the maturity of the hedge. Lets say the index level is 1250 and one contract costing 250 times the index and the portfolio has a cost of 50mil and a beta of 0.87. We would short.....

0.87*(50mil/312,500)=139.2 approx.~139 contracts for the optimal hedge.

However since we are trying to hedge all exposure, and assuming again that I am on the right path, we want to achieve a beta of zero. But this is where i get lost since the book states that if your BETA > the BETA you wish to achieve the you should subtract your BETA by the BETA you wish to achieve. But the beta i wish to achieve is simple zero. Does this mean i just omit the beta from the formula and take a short position in 160 contracts? or is the optimal hedge actually what i calculate for hedging against all exposure to the market?

I wish this book provided more examples to extrapolate more meaning and information.

thanks you for your patience and understanding, Rudy

 

[David Harper CFA FRM]

Hi Rudy,

I frankly don't memorize that formula...i think it's an example of where memorization is too much trouble (IMO). Just unpack it:

portfolio beta +/- transaction = target Beta

So, your formula is correct, I would use your formula regardless (long, short, +/-). It says: 139 contracts have beta impact of +/- 0.87. At this point, signage isn't critical.

Okay, then going long will increase the net portfolio beta and going short will decrease is.
Going long will make it 2*0.87 and short will make it zero.

e.g., if target beta is 1.0, then I'd just us: 1.0 - 0.87 = 0.13 beta impact needed. Then, after solving for contracts, as i need to increase the net portfolio beta, i know that they should be long.

so, all you are doing here is adding a futures transaction (with + beta impact for long, - beta impact for short) to a preexisting portfolio beta.


"is the optimal hedge actually what i calculate for hedging against all exposure to the market?"
great question, no way. first, the beta is linear (it's really a transformed correlation, which is linear). In this regard, it's only an *approximate* hegde for the linear exposure. Second, related, beta is only one factor; surely the portfolio has other risks. Third, the beta is probably historically measured; it may change going forward (time varying).

David

 

[koolrud]

Thanks a lot David, I actually got the concept last night after reviewing the book like a million times.

Now I am confused with Roll over hedging.

I do not even know where to begin. Well i take that back,i do not know what questions to ask since i am so confused with it. I understand that Roll over hedging has to do with an expiration date of the hedge being later than the delivery dates of all futures contacts that can be used. So as the hedger, for examples sake, i take a short position at time 1 and then close it out, and take the same position again at time 2, and so on and so on...

My other understanding is that the differences in the futures prices from open to close is calculated as the basis risk. or as my teacher so comically puts it "your level of happiness."

once again the going short or long confused me because as i understand it in rolling the hedge, if the company gains when the price of the asset increases and loses when the price of the asset decreases, then a short hedge is appropriate but if its the other way around then a long hedge is appropriate.

so after all the rolling and calculating "my level of happiness" in the end i am given a number that represents what? either the gain or lose per what ever i was hedging against?


and as always David, thanks for your help