From individual assets to financial portfolios
When we invest in a financial asset over a period of time, we receive a return on this asset. This return can be positive, in which case we make a profit, or the return can be negative, in which case we make a loss.
When we hold more than one asset, we have a portfolio. The overall return on the portfolio will depend on the returns on the individual assets. But how can we compute the return on the portfolio on the basis of the returns on the assets?
Let’s consider an example. Suppose that you invest in two assets, let’s say asset \(A\) and asset \(B\). These could be stocks issued by two companies. We can denote the return on asset \(A\) by \(R_A\) and the return on asset \(B\) by \(R_B\). For instance, if the return on \(A\) is 10% then \(R_A\) = 10% = 0.10. If the return on asset \(B\) is negative, we would be making a loss on holding that asset: for instance, if \(R_B\) = –5% = –0.05, our loss on asset \(B\) will be 5 per cent of the amount which we have invested in the asset.
So, we are making a profit on asset \(A\) but a loss on asset \(B\). Are we making a profit or a loss on our portfolio which includes both \(A\) and \(B\)?
That depends on the composition of our portfolio, that is, on the relative proportions of \(A\) and \(B\) in our portfolio. Intuitively, if our portfolio mostly consists of asset \(A\) we would be making a profit. But if we are mostly holding asset \(B\) then we could be making a loss overall. If we hold \(A\) and \(B\) in equal proportions, we would be making a profit because 10% is greater than 5%.
Can we be more precise?
Let us denote the return on the portfolio by \(R_P\). If the assets \(A\) and \(B\) are held in equal proportions, then half of our investment is in \(A\) and half in \(B\). The proportion of each asset in the portfolio is ½. The return on the portfolio is therefore:\[R_P=\frac 12 R_A + \frac 12 R_B = \frac 12 \times 0.10 + \frac 12 \times (-0.05) = 0.025 = 2.5\%\]
The return on the portfolio is 2.5% and we would be making a profit.
In general, if we say the proportion of asset \(A\) is \(w_A\), and the proportion of asset \(B\) is \(w_B\), then the return on the portfolio is given by:\[R_P=(w_A \times R_A) + (w_B \times R_B)\]
In words, to calculate the return on the portfolio, the return on each asset is multiplied by the proportion of the asset in the portfolio. We also call \(w_A\) and \(w_B\) the weights on each asset in the portfolio.
For instance, if \(w_A\) = 0.75 and \(w_B\) = 0.25, then\[R_P=0.75 \times 0.10 + 0.25 \times (-0.05) = 0.0625 = 6.25\%\]
and we would still be making a profit. If instead \(w_A\) = 0.20 and \(w_B\) = 0.80, then\[R_P=0.20 \times 0.10 + 0.80 \times (-0.05) = -0.02 = -2\%\]
and we would be making a loss.
Finally, what do you notice about the proportions or weights of the assets in the portfolio? In each of our examples they always add up to one\[w_A + w_B = 1\]
Can you see why this is so? Use the comments section to share your thoughts with your fellow learners.