Does financing a portfolio on margin affect the variance of a portfolio?

Question

For example, if the portfolio is financed with 50% margin, does it affect the portfolio variance?

Answer 1

Yes, more leverage increases the variance of your individual portfolio (variance of your personal net worth). The simple way to think about it is that if you only own only 50% of your risky assets, then you can own twice as many risky assets. That means they will move around twice as much (in absolute terms). Expected returns and risk (if risk is variance) both go up. If you lend rather than borrow, then you might have only half your net worth in risky assets, and then your expected returns and variation in returns will go down.

Note, the practice of using leverage differs from portfolio theory in a couple important ways.

1. There's a threshold or cliff danger, which is when the broker makes you force-sell your assets, or you have to declare bankruptcy, or whatever. Then you are just stuck with a loss and you can't hold and wait for prices to go back up. Leverage thus allows you to lose 100% of your assets, permanently. Another way to put it is that margin calls can create a cash crisis or lack of liquidity, destroying you.
2. the interest rate charged by brokerages is pretty far above the risk-free rate, if you're just an average joe individual. This lowers expected return for a given risk, so the CML bends at the point where leverage begins and leverage isn't as useful as it would be. http://www.investing-in-mutual-funds.com/asset-allocation.html has an illustration of this.

Answer 2

Financing a portfolio with debt (on margin) leads to higher variance. That's the WHOLE POINT. Let's say it's 50-50.

On the downside, with 100% equity, you can never lose more than your whole equity. But if you have assets of 100, of which 50% is equity and 50% is debt, your losses can be greater than 50%, which is to say more than the value of your equity.

The reverse is true. You can make money at TWICE the rate if the market goes up. But "you pay your money and you take your chances" (Punch, 1846).

Answer 3

Variance of a single asset is defined as follows:

σ² = Σ_i(X_i - μ)²

where X_i's represent all the possible final market values of your asset and μ represents the mean of all such market values.

The portfolio's variance is defined as

σ_p² = Σ_iw_i²σ_i²

where, σ_p is the portfolio's variance, and w_i stands for the weight of the i^th asset.

Now, if you include the borrowing in your portfolio, that would classify as technically shorting at the borrowing rate. Thus, this weight would (by the virtue of being negative) increase all other weights. Moreover, the variance of this is likely to be zero (assuming fixed borrowing rates). Thus, weights of risky assets rise and the investor's portfolio's variance will go up.

Also see, CML at wikipedia.