Читать книгу Advanced Portfolio Management - Giuseppe A. Paleologo - Страница 14

The first use of risk models is for risk management. The questions we are asking are:

 What is the risk (i.e., the volatility) of my current portfolio?

 Where does the risk come from?

 How does the risk change as my portfolio changes?

We are going to answer them by proceeding in small steps. Start with the risk of a single stock. The risk model gives the pieces of information about Synchrony shown⁵ in Table 3.4: What is the daily volatility of owning $10M of SYF stocks? Recall that for Synchrony, and any other stock, the return formula is

Table 3.4 Synchrony's risk parameters for the year 2019.

Field	Value (%)
Beta	1.2
Daily Market Vol (%)	0.8
Daily Idio Vol (%)	1.3
Net Market Value	$10M

We want to compute the volatility of Synchrony's stock return, vol(r). The alpha term is a constant, while the terms and are random. The constant does not contribute to volatility, but the two random terms do. What is the volatility of the sum of two random variables? The square of the volatility (or standard deviation) is also called the variance of a random variable. You may remember from a Statistics or Probability class that the variance of the sum of two independent random variables is equal to the sum of their variances. This is easier to show in pictures, because it is nothing else than Pythagoras's Theorem! The volatilities of two random variables are the two legs of a right triangle. The volatility of their sum is equal to the length of the hypotenuse; see Figure 3.3.

Now, we can apply this formula to Synchrony. The volatility of the market term is

The idiosyncratic volatility is

The dollar variance of Synchrony returns is and finally the volatility is $162K.

I hope I bored you with this calculation, because it is boring. But this is a back-of-the-envelope calculation that helps for many tasks. For example: you can pull SYF and SP500 returns from a website or Bloomberg for the past year, estimate their vols, then perform a quick regression in Excel to estimate the beta of SYF to SP500 and then use the same calculation above to derive the idiosyncratic vol (do it!). Yes, there is a Bloomberg function for that (BETA GO), and yes, even Yahoo Finance reports the trailing 3-year beta. But suppose that you don't want a three-year beta. Or suppose that you don't want to include a certain date range in SYF returns, for example, the day of an idiosyncratic event that resulted in a one-off large return, and is not representative of the future volatility of the stock. There are many valid reasons for wanting to customize the estimation of vols and betas. Once you have understood the principle, you are the master of your own destiny, even if you pull the data from a commercial model.

Figure 3.3 The variance of the sum of two independent random variables is equal to the sum to the variances of the two random variables. You can interpret the standard deviations as sides of a right triangle.

Now that we have understood the risk decomposition of a single stock into market and idio, let's extend to a whole portfolio. The parameters from the risk model and the portfolio are in Table 3.5.

Table 3.5 Synchrony, Wal-Mart and SP500 risk parameters, together with holdings for each asset.

Field	SYF	WMT	SPY
Beta	1.2	0.7	1
Daily Market Vol (%)			1.4
Daily Idio Vol (%)	1.2	0.5	0.0
Net Market Value	$10M	$5M	$10M

The daily PnL (“Profit and Loss”) of the portfolio is defined as the sum of the holdings times their respective returns:

Now, we can use Equation (3.1) to replace returns with their components, and rearrange the terms:

(3.2)

The performance of the portfolio can be split into the contribution of two terms: a market term and an idiosyncratic one. This is a simple example of performance attribution.

The beta of the portfolio is . The important thing to notice is that the beta of the portfolio is the sum of the betas of the individual holdings. This overall portfolio beta is expressed in dollars, and usually is called the dollar beta of the portfolio. This dollar beta, multiplied by the market return , gives the 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.