Understand the math but need help conceptually. There’s an example in the text that leverages up a portfolio beta of 0.8 to a target beta of 1.1 by going long on 6 contracts that have a beta of 1.05. How does does going long on futures with a beta of 1.05 get you to a target beta of 1.1? Seems like the beta of the future would need to be greater than the target in order to increase the overall portfolio beta to 1.1.
How does this work? Thanks for any help
It is Dollar Duration you need to look at, not at the duration alone.
Leverage.
Example, you need a $100 portfolio to have a 1.1 beta. Futures contracts are 1.05 Beta and are $2 (including multiplier).
The mathematical formula as you know: 1.1 - 0 / 1.05 * (100/2) = 52 contracts (rounded).
52 contracts at $2 = $104 dollars of notional value. Therefore, you technically have $4 of leverage to achieve your 1.1 beta.
Seems like in order to raise the portfolio beta to 1.1, then you’d need a futures contract with a beta greater than the target of 1.1, no?
Not if you use leverage. Y our equity is still only $100. When you buy 52 futures contracts you are essentially essentially borrowing an extra $4.
From the previous example:
Assume market moves 1%. The futures, which have a 1.05 beta will move 1.05%.
Considering a notional principal of $104, this translates into a dollar increase of $104 * 1.05% = $1.092.
The equity in the portfolio is $100 (remember the $4 is essentially borrowed because we are leveraged)
% Change of equity based on a $1.092 increase is: $1.092/$100 = 1.09%
The equity portfolio move divided by market move (1.09%/1%) implies a beta of 1.09 (since we rounded down the contracts its not exactly a 1.1 beta).
What happens to the $4 though, it’s not like you have to pay that back?
Fixed that for you.
If you’re trying to change the beta on a portfolio with nothing but coke stock, would you/should you use a future on coke stock? What determines what the contract is written on?
The point of using futures is that you don’t invest any additional money, so you’re leveraging your existing investment. If you buy the stock you have to scrape up some more money, or you have to sell some of your existing portfolio; presumably you can’t or don’t want to do either. Furthermore, as you pointed out in your original post, if you buy KO you can only change the beta to something between your existing beta and KO’s beta; using futures you can adjust the beta to any value.
You’re thinking about this as if the beta of the new portfolio were a weighted average of the betas of the original portfolio, and the sum of the weights is one. The point of using futures is that the sum of the weights is greater than one.
For example, suppose that your existing portfolio is $10 million, with a beta of 0.8. You want a beta of 1.2, using $100,000 futures contracts with a beta of 1.0. The formula would tell you to go long 40 futures contracts. The beta of the new portfolio is this weighted average:
($10,000,000/$10,000,000) × 0.8 + ($4,000,000/$10,000,000) × 1.0 = 1.2
Notice that the sum of the weights is 1.4 (= 1.0 + 0.4).
Many many thanks for the help!! So what determines then what the futures contract is written on? Seems like they generally stick with indexes but is there a rule behind what you should use?
You’re quite welcome.
Recall that beta is the product of two factors: the correlation of the asset’s returns and the market’s returns, and the relative volatility of the asset’s returns compared to the market’s returns.
If I were using futures to adjust beta, I’d want to use a futures contract whose correlation of returns with the market is very close to +1.0 to ensure a very high R² (= ρ²); that way, the change in the futures’ return will track the change in the market’s return extremely closely, without a lot of noise. Index futures fit the bill quite nicely.