Contents of this article
After a series of articles, you will have a basic picture of risk and the core concepts we usually discuss around it – variance, standard deviation, correlation and so on. At a more applied level, understanding risk helps us plan and take measures to make sure our investments earn the highest possible return.
One way to judge whether an investment is working is to compare its actual return with the return we expected at the outset. Everyone’s expectations differ, of course, but the most important thing is to have a target expectation against which to monitor and adjust the portfolio – just as you should always have a clear purpose for anything you do in life.
In this article, we look at a portfolio’s expected return and estimate its likely range in practice. Starting with:
What is expected return?
Expected return is the rate of return an investor hopes to earn from an investment in the future. For a portfolio of n stocks with different weights, the portfolio’s expected return is calculated as follows:
= (Average return of stock 1 × Weight of stock 1) + (Average return of stock 2 × Weight of stock 2) + … + (Average return of stock n × Weight of stock n).
We need to pay attention to the time period in the formula above. If the average returns are daily returns, the portfolio’s expected return will be a daily return. Likewise, if the returns are monthly averages, the result will be a monthly expected return.
If you have the portfolio’s daily expected return and want its annual expected return, simply multiply the daily figure by the number of trading days in the year, usually about 252.
To make it more concrete, let’s go through a specific example with the familiar five-stock portfolio from earlier articles:
As you can see, the first three columns of the table are fairly self-explanatory. The last column is calculated by multiplying each stock’s daily expected return by 252 (the number of trading days in a year).
For example: VPB – 0.06% × 252 = 15.12%
This means that if, at some point, I invested all my available capital in VPB (a 100% weight), the return we could expect from this stock would be 15.12% a year.
Besides VPB, however, we have a whole portfolio of other stocks, so now weights come into play. In fact, I invest only 10% of my capital in VPB and allocate the rest to the other stocks in the portfolio. The expected return of VPB, and of the other stocks, must therefore be weighted. Specifically:
= Weight × Expected return = 10% × 15.12% = 1.51%
This is one stock’s annual expected return, adjusted for its weight. After doing the same for the other stocks in the portfolio and adding up the results, we get the portfolio’s one-year expected return, as follows:
Portfolio expected return
The result is:
At this point, we have the portfolio’s expected return of 51.16% a year, together with the portfolio variance (Var) of 1.57% per day calculated in earlier articles.
We can convert the daily figure into an annual figure by multiplying it by the square root of the number of trading days (252 in this example), as follows:
VAR (annual) = 1.57% × √252 = 24.92%
We now have two of the most important parameters commonly used to assess a portfolio, and their most prominent use is to estimate the portfolio’s range of fluctuation at a given confidence level.
Estimating the portfolio’s range of fluctuation
To do this, we first calculate the portfolio’s return adjusted up and down (+/-) by its variance. In other words, we determine how much the portfolio might rise or fall over the next year – its range of fluctuation. The accuracy of these estimates comes at three levels:
- Level 1: within 1 standard deviation, with 68% confidence
- Level 2: within 2 standard deviations, with 95% confidence
- Level 3: within 3 standard deviations, with 99% confidence
Note that standard deviation is derived from variance. So when the portfolio’s figure is 24.92%, that is also one band of fluctuation for the portfolio. In our example, 24.92% represents one standard deviation, so two standard deviations are 24.92% × 2 = 49.84%, and three are 74.76%.
Finding the portfolio’s range of fluctuation means finding its limits (an upper and a lower bound). With a one-year variance of 24.92% and an expected return of 51.16%, the portfolio’s upper bound is 24.92% + 51.15% = +76.06%. Conversely, the lower bound is 51.15% – 24.92% = +26.23%.
We can see that, however it fluctuates, our portfolio does not lose money relative to its starting point; it always grows by between +26.23% and +76.06% a year.
At this point, many of you will have questions about this result, such as:
First, does the portfolio always grow regardless of the risks that occur?
=> Answer: no. The portfolio above is only an illustrative example for explaining how to calculate portfolio expected return; its results are not representative.
Second, if we keep holding this portfolio the following year, what will the return be?
=> Answer: the portfolio’s return will still fluctuate between +26.26% and +76.06%.
Third, on what basis can we say the portfolio will always fluctuate between +26.26% and +76.06%?
=> Answer: in a normal distribution there are three confidence levels, and the result above is at level 1, with 68% confidence. So the actual result can still deviate from it.
Fourth, how can we increase the confidence level?
=> Answer: we have to widen the band to more standard deviations. Put simply, using more standard deviations widens the portfolio’s range (a larger interval), so the actual result is more likely to fall within the predicted limits. Specifically:
For 95% confidence, the range must be two standard deviations. We calculated two standard deviations earlier as 49.84%. The upper and lower bounds are therefore 100.99% and 1.31% respectively.
If we want 99% confidence, we do the same with a range of three standard deviations. The upper bound is then 125.91% and the lower bound -23.61%.
By applying some basic knowledge of the normal distribution, this article has shown concretely how we can estimate a portfolio’s return over the short term. Errors are inevitable, since no one knows for sure how the market and individual stocks will move, but these calculations give us a better overall view of our investments and help us make more appropriate adjustments when volatility strikes, for optimal investment results.
Happy trading !
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