1. Drivers in the case of two assets
Analyzing portfolios in R
2. Future returns are random in nature
goes beyond analyzing the portfolio returns. It is also about understanding how the portfolio can be optimized such that the future portfolio return has the desired properties in terms of mean and volatility.
Since we are talking about the future, I need to teach you first about the difference between computing averages on past returns and taking expectations of random variables.
In chapter 2, we took the observed returns as a given, and we thus used averages of the returns to describe the past performance.
But, when optimizing a portfolio,
3. Future returns are random in nature
we need to deal with the uncertainty of what the future return will be.
4. Future returns are random in nature
Since its future value is a random outcome,
5. Future returns are random in nature
the portfolio return is a random variable. The transition to working with portfolio returns as random variables has also implications for how we write the mean and variance of the portfolio return.
6. Past performance to predictions
The mean return is no longer the average of the past returns, but it is the best possible prediction of the portfolio return. This best possible prediction is called the expectation of the portfolio return, or also, the expected portfolio return. When we take the expectation of a random variable, we denote this by the capital letter
7. Past performance to predictions
E.
A
8. Past performance to predictions
similar result holds for the portfolio
9. Past performance to predictions
variance, which is no longer the sample variance of the past portfolio returns. The variance of a random variable is instead defined
10. Past performance to predictions
as the expected squared deviation of the portfolio return with respect to its mean.
11. Past performance to predictions
Let’s now turn to our main question:
12. Drivers of mean & variance
what are the drivers of the portfolio mean and variance?
I will work this out for the case of two assets in the portfolio: asset 1 has weight w1, and asset 2 has weight w2.
Then, the portfolio return is the weight of asset 1 times the return of asset 1, plus the weight of asset 2 times the return of asset 2.
Plugging the formula of the portfolio return into the definition of the expectation of the portfolio return, and using that the expected value of a sum is the sum of the expected values, we see that the expected portfolio return is the weight of asset 1 times the expected return of asset 1 plus the weight of asset 2 times the expected return of asset 2.
13. Portfolio return variance
The impact of the weights on the portfolio variance is slightly more complex because of the square. Working out the square of the portfolio de-meaned return, we find that the portfolio variance equals the sum of the squared weight of the assets times their individual variance plus two times the product of the weights and the expected value of the product between the demeaned return of asset 1 and the demeaned return of asset 2.
The expectation of this product between the demeaned returns of asset 1 and asset 2 is called the covariance of those two asset returns.
The term covariance may be new for you, but probably you have already heard about correlations. If you know about correlations, then you also know about covariances, since the covariance of two asset returns is the product of their standard deviations and the correlation between the asset returns.
14. Correlations
The correlation measures the intensity of the relationship between the asset returns.
If they are unrelated, then the correlation is zero. If there is, on average, a positive linear relationship between both, then the correlation will be positive. This means that, when one asset return is above average, then the return of the other asset also tends to be above average. In the case of a negative correlation, then it will be the case that, if one return is above average, the other one tends to be below average.
15. Take away formulas
To conclude, the drivers of expected portfolio returns are the expected return of the individual assets and their portfolio weights. The drivers of the portfolio variance are the individual variances of the asset returns, their covariance, and the portfolio weights.
16. Let's practice!