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The portfolio return

1. The portfolio return

Analyzing the portfolio weights reveals the investment bets.

2. Portfolio returns: relative value

The larger the weight of an asset in the portfolio, the more influential it will be in determining the future value of the portfolio. When studying this impact, investors typically do not analyze the change in the investment value in absolute terms, but in relative terms. This leads them to compute simple returns, defined as the change in value over the period, relatively to the initial value. The simple return is thus the final value minus the initial value, divided by the initial value.

3. Portfolio returns: relative value

As an example, suppose the

4. Portfolio returns: relative value

initial value is 100 dollars

5. Portfolio returns: relative value

the final value

6. Portfolio returns: relative value

is 120 dollars.

7. Portfolio returns: relative value

Then the return on that investment

8. Portfolio returns: relative value

equals 20%, obtained by

9. Portfolio returns: relative value

taking the difference between 120 and 100 and dividing it by 100.

10. Three steps

In the slide, I show you how we can apply this definition for computing portfolio returns. This involves three steps. First, for the initial date, we need to compute the total value invested as the sum of

11. Three steps

the values of the different investments. Second, for the final date,

12. Three steps

we have to sum the final values of the individual

13. Three steps

investments to obtain the final portfolio value.

14. Three steps

Then, we can

15. Three steps

compute the

16. Three steps

portfolio return as the percentage change of the final value compared to the initial value.

17. Example: two assets

As an example, let us consider a 2-asset portfolio

18. Example: two assets

that invests 200 dollars in asset 1 and 300 dollars

19. Example: two assets

in asset 2.

20. Example: two assets

The end value is 180 dollars and 330 dollars.

21. Example: two assets

If we sum the values, we find

22. Example: two assets

that the

23. Example: two assets

total initial value of the portfolio is 500 dollars, while the total final value is 510 dollars. It follows that the

24. Example: two assets

simple return on the portfolio is the 10 dollar change in value,

25. Example: two assets

divided by the initial 500USD invested, which gives us a return

26. Example: two assets

of 2%.

27. Example: two assets

28. Portfolio returns: weighted average return

A disadvantage of this calculation method is that it does not show how the portfolio weights determine the portfolio return. Let us, therefore, consider a different formula,

29. Portfolio returns: weighted average return

in which the portfolio return is computed as the

30. Portfolio returns: weighted average return

weighted average of the returns of the underlying assets.

31. Three steps

Its calculation proceeds also in three steps.

32. Three steps

First, the initial

33. Three steps

weights of the positions are computed.

34. Three steps

Secondly, the return on each of the

35. Three steps

individual positions is determined. Then, in the third step,

36. Three steps

the portfolio return is computed as the

37. Three steps

sum over the products between the initial weights and the corresponding returns.

38. Example: two assets

In the slide, you see how to apply this formula to compute the return for our example portfolio with two assets. We first compute the initial portfolio weights.

39. Example: two assets

Since the initial value of the first asset is 200 dollars, and the total value invested is 500 dollars, the initial weight of asset 1 is 40 percent.

40. Example: two assets

The remainder of 60 percent

41. Example: two assets

is the weight of asset 2.

42. Example: two assets

Then, in a second step, we need to compute the returns for each of the assets. For asset 1, we obtain that the individual return

43. Example: two assets

minus 10 percent, while for asset 2, the return is

44. Example: two assets

plus 10 percent. Finally,

45. Example: two assets

we can combine those results and compute the portfolio

46. Example: two assets

return by summing over the weights multiplied by their respective returns. The first term is the weight of 40 percent times the returns of minus 10 percent, which gives us minus 4 percent. The second term is the weight of asset 2, 60 percent times its return of 10 percent, which gives us plus 6 percent. Adding minus 4 percent and plus 6 percent gives us the portfolio return of 2 percent, which is exactly the same number as obtained before.

47. Let's practice!

The next interactive exercises put this theory in practice.