... possibly the most important and least understood topic in finance

Transcription

1 Correlation possibly the most important and least understood topic in finance 2017 Gary R. Evans. This lecture is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License.

2 The Beta interpretations... One year data Correlation: Ratio of SD: Beta: We can see why we need to separate the Beta into the two components and leave them that way. 30 day data Correlation: Ratio of SD: B 2. The ratio of SD is really indicative of the relative volatilities, BUT Beta: The traditional Beta matters because if you do add an uncorrelated asset to a portfolio, cs c s the variance of the portfolio is reduced, which means the portfolio has less risk!! That is a big issue!

3 2-asset Portfolio Variance Sums Variance is purely additive if two variables are strictly independent: Vx y V x Vy 2 COVx, y remembering that Covariance is equal to the Correlation Coefficient (0 if no perfectly independent, 1 if perfectly correlated, -1 if perfectly polar) times the product of the standard deviations:,, COV x y CORREL x y SD x SD y

4 2-asset weighted portfolio variance V 2 2 ax by a V X b V ( Y ) 2abCOV X, Y and in a weights sum to one, the coefficients above are restricted to the condition i 1 so in the special case of portfolio P consisting of two completely independent stocks with exactly the same variance and each equally represented, then the variance of the portfolio will be... P 0.5 V ( X ) so if you only invested in one of the two stocks your volatility would be V V X but if you diversified your portfolio 50/50 your volatility would be V x

5 Simple example of diversification using our formula: Suppose you have two uncorrelated stocks, X (, ), X 1 (002003) (0.02,0.03) and X 2 (0.04,0.05) ) If you are risk-adverse, you may want to put all of your money in stock X 1 and accept the lower 2% yield. But what if you split your portfolio 50/50, giving you a 3% yield? What would your risk be?? V 1 = and V 2 = and each alpha equals 1/2. Therefore V 1,2 = 0.25 X ( ) = σ 1,2 = /2 = Therefore, by diversifying your portfolio you have raised your yield to 0.03, 50% more than the conservative stock, while lowering your risk to a level below the most conservative of the two stocks (which was at 0.03).

6 The risk-yield efficiency frontier Portfolio yield Efficient trade-off region X1a X2a PVar PVol Palpha Alpha Vol Var X X Portfolio risk

7 n-asset Portfolio Volatility If we have 'n' assets in the portfolio, then we calculate the variance using this additive formula: n n n1 n VAR x i VAR x COV x x i1 i1 i1 j( i1) i 2 i, j which is easy to program if* you have the data. What finally matters, of course, is the square root of this term, the standard deviation, which is our volatility measure. *this requires the calculation of all 'n' standard deviations and all paired correlations n1(15 for 6 stocks). i11 i For reference and discussion, see

8 Weight-adjusted n-asset Portfolio Volatility If you assign weights to your portfolio, represented here as alphas, which of course you would, then the variance formula is: n n n 1 n VAR ix 2 i VAR x COV x x i1 i1 i1 j( i1) 2, i i i j i j The volatility of this portfolio, the standard deviation, is the square root of this expression. Clearly, the greater the independence of your portfolio components, the smaller the risk. This shows the benefits of diversification into non-correlated stocks.

9 Help from David Coates '08 Coding the Covariance (prior 2 equations) For the covariance part of the equation only, for I = 1 to (n-1) do for J = (i+1) to n do COV(I,J) = CORREL(I,J)*SD(I)*SD(J); SUMCOV = SUMCOV + COV(I,J); end; end; S S S S C C C 0 0 C C C S S S S SUMCOV = 2*SUMCOV; and the weighted portfolio calculation would be the same except WCOV(I,J) = a(i)*a(j)*correl(i,j)*sd(i)*sd(j); Memo slide for sticklers for accuracy (a desirable trait), those of you who want to work in finance, and you coders who own a laptop and want to retire bf before age 35t trading off of any beach with a wireless setup.

10 The 2015 HW assignment... but this time we are doing it in Python but this time we are doing it in Python, but I am giving you the completed Excel application as a zip file, so you can look at the formula used here:

11 Why is this so important? All kinds of strategies, like pairs trading, general spread trades (trading on spread anomalies and expecting mean reversion) depend d upon correlation either being one or being stable. Pairs trading for stocks like Coke and Pepsi Playing spreads like the spread between WTI and Brent crude (the correlation of which broke down because of fracking gains in the US). The entire theory of portfolio diversification depends mathematically upon correlations being near zero for major clusters in the portfolio. When correlations break down, the benefits of diversity break down. You can see it in the math!! Is JWN supposed to be correlated with CSCO, and more generally, should tech be correlated with retail? Are the indexes like SPY, IWM, QQQ, and DIA correlated, and should they be? Will global fixed income, now diverse, become correlated in crisis?