Advanced Portfolio Management. Giuseppe A. Paleologo

Advanced Portfolio Management

target="_blank" rel="nofollow" href="#ulink_3c8f0f81-2736-50b2-8539-ed74efbc0c62">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

StartLayout 1st Row 1st Column r equals alpha plus beta times m plus epsilon 2nd Column Blank EndLayout

We want to compute the volatility of Synchrony's stock return, vol(r). The alpha term is a constant, while the terms beta times m and epsilon are random. The constant alpha 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

StartLayout 1st Row 1st Column dollar-sign 10 upper M times 1.2 times 0.8 percent-sign equals dollar-sign 96 upper K 2nd Column Blank EndLayout

The idiosyncratic volatility is

StartLayout 1st Row 1st Column dollar-sign 10 upper M times 1.3 percent-sign equals dollar-sign 130 upper K 2nd Column Blank EndLayout

The dollar variance of Synchrony returns is 9 6 squared plus 13 0 squared equals 26116 normal upper K squared 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 less-than GO greater-than ), 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.

Schematic illustration of the variance of the sum of two independent random variables is equal to the sum to the variances of the two random variables.

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:

StartLayout 1st Row 1st Column PnL Subscript port Baseline equals 2nd Column NMV Subscript SYF Baseline times r Subscript SYF plus NMV Subscript WMT times r Subscript WMT plus NMV Subscript SPY times r Subscript SPY EndLayout

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 1.2 times dollar-sign 10 normal upper M plus 0.7 times dollar-sign 5 normal upper M plus 1 times dollar-sign 10 normal upper M equals dollar-sign 25.5 normal upper M . 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 ,

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