Advanced Portfolio Management. Giuseppe A. Paleologo
contribution of the market to the portfolio PnL. The daily volatility of the portfolio deriving from the market is
An alternative way to quote the beta of a portfolio is in percentage beta, which is defined as the dollar beta divided by the net market value of the portfolio. If we netted out our positions, the percentage beta is the dollar beta per unit of dollar held long in the portfolio.6 In our case the percentage beta is
The other term is the idiosyncratic PnL. The volatility of the idiosyncratic PnL of the portfolio is the sum of three terms. As in the case of two variables, the variance of the sum is the sum of the variances:
And the volatility is
Finally, the variance of the portfolio is the sum of the variances, because idio and market returns are independent of each other. The volatility is the square root:
The procedure is described in Procedure Box 3.1. Let us go through another simple example.
Procedure 3.1 Compute the volatility of a portfolio.
1 Compute the dollar betas for the individual positions;
2 Compute the dollar portfolio beta as the sum of the individual betas;
3 Compute the market component of the volatility as (portfolio beta) (market volatility);
4 Compute the dollar idio volatility as the square root of the sum of the squared dollar volatilities.
Table 3.6 Portfolio example, with increasing number of stocks. Each stock has unit beta. The daily stocks' idio vol and market vol are both 1%.
| # Stocks | Idio Vol ($) | Market Vol ($) | Idio Var (% tot) |
|---|---|---|---|
| 1 | 1M | 1M | 50 |
| 10 | 316K | 1M | 9.09 |
| 100 | 100K | 1M | 0.99 |
| 1000 | 31.6K | 1M | 0.01 |
Say that you consider four long-only portfolios, each one with $100M of market value. The first one has one stock, the second one has ten stocks, the third has 100 stocks, the fourth one has 1000 stocks. Each stock has beta 1, and a daily idio vol of 1%. The market also has a daily vol of 1%. These are made-up numbers of course, but they simplify the calculation, and real portfolios are not that far from these values. What are the idio and market components of these portfolios? Table 3.6 has the numbers.7
Given this example, you can now understand better why SPY has zero percentage idio volatility in Table 3.5. The SPY is a long-only portfolio of 500 stocks. Each stock in the portfolio has a positive beta. As a percentage of the total risk, the idiosyncractic risk is very small, and is usually approximated to zero.
3.5 First Steps in Risk Decomposition
This very simple decomposition already has very powerful implications. Volatility either comes from a systematic source or a stock-specific one. Where does your skill lie? In going long or short the market, or rather in going long or short the company-specific returns? One attractive feature of being long the market is that its risk is accompanied by positive expected returns. The inflation-adjusted annualized historical return of the S&P 500 from 1926 (the inception year of the index) to 2018 is 7%; so it may seem a good idea to have a positive beta to the market. Indeed, a sizable fraction of the global assets under management are actively managed funds that track an index. This means that the portfolio has a positive percentage beta, often equal to 1, and a certain budget of volatility allowed to run in idiosyncratic PnL, sometimes called the tracking error. For example, our previous portfolio in Table 3.5 has a tracking dollar vol of $1.9M, or, in percentage of the portfolio's NMV, a 7.8% tracking error.8 Compared to commercial products, this is a relatively high value. Tracking errors range from 0% (i.e., a “passive” fund tracking the market) to 6 or 7%, which is not hard to achieve when the portfolio consists of hundreds of securities rather than three. One less attractive feature of carrying beta in your portfolio is that it is harder and harder to justify to investors. The principals who entrusted their capital to you also see the alpha-beta decomposition; and they usually have very inexpensive ways to invest in the market, either by buying e-mini SP futures, or by buying a low-expense ratio ETF like SPY, or by investing in a passive fund like Vanguard's Total Stock Market Index Fund. This decomposition, which is useful to you to understand your return, is available to them as well, and the beta component is easy to replicate. Therefore, if you still want to keep a significant beta in your portfolio, you better have a good argument. We will revisit one such argument in Chapter 8, which is devoted to performance. For the time being, it suffices to point out two facts. The first one is that any argument in favor of conflating beta and alpha is weaker than the simple argument in favor of decomposing them. Secondly, that it is easy to remove the market component from a portfolio. This is the subject of the next section.