squared 2nd Row 1st Column Blank 2nd Column plus left-parenthesis NMV Subscript WMT Baseline times vol Subscript WMT Baseline right-parenthesis squared 3rd Row 1st Column Blank 2nd Column plus left-parenthesis NMV Subscript SPY Baseline times vol Subscript SPY Baseline right-parenthesis squared EndLayout"/>
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.
3.6 Simple Hedging
We revisit the example portfolio in Table 3.5. We had a portfolio with a large amount of market risk, but we also have $6.7M of idiosyncratic risk coming from SYF and WMT. In addition, we have high conviction that these two stocks will have returns in excess of the market in the next quarter, whereas we have very little idea of the direction of the market over the same time horizon. We can express this by saying that we have an edge in our specific stock selections, but not in the market. Even if we did not hold any SPY, we would risk marring our good insight about the underpriced SYF and WMT stock. A large market drawdown can result in negative returns for the portfolio. If we seek absolute returns from our portfolio, this may not be a wise choice. We can remedy this by taking a short position in SPY. We borrow the shares from an existing owner and we sell them at the market price. At some point in the future, we cover the position by purchasing the shares at the market price and returning them to their owner. Shorting has costs and risk. We pay a “borrow rate” to the lender, which can be high if the stocks are hard to find (or locate). In addition, shorted stocks can be recalled by the original owner if she wants to sell them, forcing us to cover the stock earlier than anticipated.9 By shorting SPY, we reduce the beta exposure of the portfolio. The dollar beta for SYF is $12M; for WMT
