1. Dimensions of portfolio performance
How
2. Interpretation of portfolio returns
well did a portfolio perform in
3. Interpretation of portfolio returns
the past and how good to we expect its
4. Interpretation of portfolio returns
performance to be in the future? This question can be answered in a data-driven way by using statistical functions in R to analyze portfolio returns. This allows us to make sensible conclusions about the portfolio’s past performance and to make reliable predictions about future portfolio performance.
It is important to understand that portfolio performance is a general concept, which cannot be summarized by one number.
5. Risk vs. reward
Broadly speaking, there are two dimensions, namely: reward and risk. The rewards tell us the success in terms of reaching high levels of portfolio value, while risk has to do with the performance in terms of avoiding extreme losses.
For doing a data-driven performance analysis in R, we thus need to first translate the concept of reward and risk into a formula.
6. Need for performance measure
In this video, I will focus on the portfolio mean return as the measure of reward, and the portfolio volatility as the measure of risk.
7. Arithmetic mean return
The formulas I will show assume that we have a sample of T portfolio return observations: R1, R2, R3 up to RT.
Then the portfolio mean return can be estimated as the average value of those T return observations. This is also called the arithmetic mean return. It indicates how large the portfolio return is on average. The averaging is important: You win some, you lose some, but on average, the portfolio return needs to be high enough to compensate for the investment risk.
8. Risk: portfolio volatility
That risk originates from the fact the returns can deviate from the average return. I will use the name “de-meaned return” to refer to the difference between the return and its mean value. If the demeaned return is positive, then the return is higher than average.
If we take the average of the squared demeaned portfolio returns, then we obtain the portfolio variance. The variance has large values when the portfolio return can deviate a lot from its mean value.
Most often, we do not use the portfolio variance itself, but its square root called the portfolio standard deviation, or also, the portfolio volatility. The higher the volatility, the higher is the probability of a large positive or negative return on your portfolio.
9. No linear compensation in return
The volatility thus causes risk. It also causes a mismatch between the average return and the actual investment return.
Suppose for example that the investor makes a 50% gain
10. No linear compensation in return
and a 50% loss. Then the arithmetic mean value of those returns
11. No linear compensation in return
is thus the average of plus 50% and minus 50%, which is zero. The zero mean return is in contrast with the actual outcome for the investor, since, as you can see on the slide, the final value of the investment is only 75% of the initial value.
Because of this mismatch between average return and the actual investment return, investors prefer using the geometric mean return
12. Geometric mean return
to take into account that there is no linear compensation in the returns.
The geometric mean return is obtained by first multiplying the total returns, and then raising that number to the power of one divided by the number of observations. If you then subtract one from that number, you obtain the geometric mean return.
In our example of a plus 50% return and a minus 50% return, the geometric mean return is minus 13.4%, and this matches, by definition, with the average effective investment return.
Once we have the average return and volatility, we can start interpreting portfolio performance.
13. Application to the S&P 500
In the next exercises, we do this for the S&P 500 portfolio, which is invested in the 500 largest publicly listed US stocks, with weights that are proportional to the stocks' market capitalization. The S&P 500 portfolio is generally considered as the most important benchmark portfolio for investors in US stocks.
14. Let's practice!