Monday, April 18, 2011

MPT gives thumbs up to Garth Turner

As I noted in my last post, Stephen Gordon has got it right on corporate taxation. Unfortunately he has it wrong, however, when he calls for former MP and financial advice blogger Garth Turner to be corrected with "the math."

I won't say that Garth's latest morsel of financial advice is "right" in all its details, never mind that he's right about the "goal of life" (the grand title of Garth's latest post) being related to achieving some level of financial wealth. But I can say that Dr Gordon is the mistaken party when he says that "variances and covariances and CAPM and stuff" will expose the errors of the former Parliamentarian.

In the comments to Gordon's post, Andy Harless takes a stab at "the math" by offering an example of how adding an asset that is less than fully correlated reduces the variance of a portfolio. In the world of "modern portfolio theory" or MPT, variance and risk are one and the same; a dubious assumption in my view but one I'll just run with for the purposes of this post. So far so so good. But then Dr Harless says, "Now free up a and b so that you can use leverage, i.e., take away the constraint that a+b=1." Sorry, but one cannot take away the constraint that the coefficients add up to 1 without taking away the equation. A weighted average means the coefficients must sum to 1 by definition.

When Garth suggests that people who have a $400K portfolio consisting solely of a house take out a home equity line of credit secured against the house for $200K and use the money for investing in a variety of other assets like financial instruments, he is indeed recommending diversification. The former portfolio contained a sole asset, the house (we'll call this asset X), and its weight coefficient was 100%, ie a = 1. In the new portfolio, a is still 1 ($400K) while the coefficient (say, "b") of the new assets (which we'll call asset Y) is 0.5 ($200K) and the coefficient (say, "c") of the HELOC (asset Z) is -0.5. The coefficient for the loan is negative because one is short the security. Thus a + b + c = 1 + 0.5 - 0.5 = 1.

Since variance is the square of standard deviation ("σ"), the variance of the portfolio is given by

where the CORR functions are the correlations between the subscripted terms (e.g. the first CORR term is the correlation between asset X and asset Y). Now suppose the standard deviation for asset X is 5% and 10% for asset Y, while the expected returns are 4% and 8% respectively. This would mean the house is expected to appreciate at just half the average annual rate of the new assets excluding the loan, but with just half of the volatility as well. Let's also assume the correlation between X and Y is 0.5. The standard deviation of asset Z would be zero if it's deemed a risk-free asset, which is an important concept in MPT. A risk-free asset returns the risk-free rate, which we will assume to be 3% for this example. The loan here may be reasonably defined as risk-free because it is secured by the home: the lender is accordingly guaranteed to be paid. As noted earlier, MPT defines risk as being variance, so the standard deviation of Z is zero.

Plugging these numbers in means the deviation for the new portfolio is 8.66%:

So Stephen Gordon is correct that total risk has been increased. A standard deviation of 8.66% is higher than 5%, which was the house alone. But can it be said conclusively that Garth Turner has "not got it right"? No, because the expected rate of return is also higher, and not just higher, but would be higher after adjusting for the increased risk. MPT uses what's called the Sharpe ratio to measure excess return per unit of risk. It subtracts the risk-free rate from the portfolio's expected return and then divides that by the portfolio's standard deviation. The portfolio's expected return is simply the weighted average of the expected return on its components, ie:

Another way to calculate this would be to take the dollar value of the expected return on the house (4% of $400K or $16K), add the additional $16K one would expect on the $200K investment (that returns 8% per annum), subtract the $6K one would have to pay on the $200K HELOC, and divide the resulting $26K by one's $400K net equity interest in the new portfolio. The Sharpe ratio for the new portfolio is 6.5% - 3% divided by 8.66% or 0.404. For the old portfolio of the house alone the Sharpe ratio was 4% - 3% divided by 5% or 0.2. In sum, while Garth's recommendation does increase risk, it more than compensates in higher expected return.

Now someone may object that the particular numbers I chose produced this result. Before one quibbles too much about that, I could make some observations about some of them such as noting as Garth does that the interest paid on the HELOC is tax deductible because it is considered money borrowed to invest and the investment is not in a tax-shelter. But the full answer is that the proposal is well-founded as a matter of theory and what I've provided is just an example.

MPT is primarily concerned with building mean-variance efficient portfolios. This means finding a portfolio mix on the "efficient frontier." Graphically, the efficient frontier (for a portfolio not including a risk-free asset) can be represented by the left boundary of a hyperbola sometimes called the "Markowitz bullet." One can think of the individual points along the frontier as portfolios of different risky assets in different proportions. The addition of a risk-free asset to the portfolio creates a new efficient frontier called the Capital Asset Line - or Capital Market Line (CML), which is the best possible Capital Asset Line - tangent to the hyperbola at the point where the Sharpe ratio is highest. Shorting the risk-free asset, or using leverage, is represented by the line above this point, which I have indicated in red:

Call the red part of the line the "Garth zone," if you like. All points on the CML have the maximum Sharpe ratio. It shows that just adding cash to a portfolio (represented by the part of the CML that is not red) or deleveraging can improve expected return for a given level of variance just as leverage can.

There are, of course, problems with this model like the fact it assumes that the variance of the assets has a neatly defined probability distribution. But almost all financial models have this problem, which can be loosely described as the "fat tails" problem. Garth Turner is in any case right in a more general sense in my view, since he appreciates the fact that the people who become truly wealthy in their own right through investments almost always use leverage to get there. Garth sums up, "The holy grail isn’t living in a place your friends covet. Then they’re not friends. The object is to posses enough wealth with liquidity to give you options. Freedom, choices." I think this is sound observation; a big house just ties one down, such that what the former Liberal (and former Conservative) politician proposes provides not just diversification benefits but liquidity benefits.

No comments: