# Level 1 CFA® Exam:

Measuring Portfolio Performance

In this lesson, we're going to discuss portfolio return measurement and its tools:

- the money-weighted rate of return measure, and
- the time-weighted rate of return measure.

When you put your money to work, sooner or later you simply want to know how successful the investment has turned out to be. To evaluate the investment merits, you need to measure the rate of return and assess the performance. The money-weighted rate of return measure and the time-weighted rate of return measure will come in handy then.

You must know that portfolio measurement is not as easy a concept as it may seem and especially in the exam it might turn out to be quite challenging or even problematic.

The basic concept connected with the return on an investment portfolio is the holding period return (HPR):

\(HPR=\frac{P_1+D_1}{P_0}-1=\frac{P_1-P_0+D_1}{P_0}=\frac{P_1 - P_0}{P_0}+\frac{D_1}{P_0}= CG+R_d\)

- \(HPR\) – holding period return
- \(P_1\) – price received at the end of the holding period
- \(D_1\) – income (e.g. dividend) paid by the investment at the end of holding period,
- \(P_0\) – initial investment
- \(CG\) – capital gain
- \(R_d\) – dividend yield

On June 1, you purchased 100 shares for USD 50 each and on September 30 you sold them for USD 52 each. On September 30, you also received USD 3 dividend per share. What is the holding period return?

(...)

HPR is a valuable tool when you want to calculate the rate of return of an investment over one period assuming that no additions or withdrawals of money occur meanwhile.

But, how should we compute the rate of return on the portfolio over many periods if there are cash inflows and outflows?

In such a case, we have two alternative measurement tools at our disposal, that is:

- the money-weighted rate of return, and
- the time-weighted rate of return.

The money-weighted rate of return (MWRR) is simply an internal rate of return (IRR). However, we use the term internal rate of return in the context of capital budgeting. In portfolio management, this measure is called money-weighted rate of return.

How was this term coined? Well, the money-weighted rate of return accounts for the value of cash flows in given periods. So, logically the values of particular cash flows affect the value of the money-weighted rate of return.

The money-weighted rate of return is calculated through equating discounted cash inflows to discounted cash outflows:

\(\Sigma^{n}_{t=0}\frac{CF_{t}}{(1+IRR)^{t}}=0\)

- \(CF_{t}\) - after-tax cash flow at time "t"
- \(IRR\) - internal rate of return
- \(t\) - moment of time

Let’s illustrate the problem with an example of a two-year investment.

An investor purchased 10 shares of a technological company for USD 40 each. After a year, he received a per-share dividend of USD 5 and sold 2 shares for USD 45 each. The investor decided not to reinvest the dividends. At the end of the second year, he sold the remaining 8 shares for USD 50 each. What is the money-weighted rate of return on this 2-year investment?

(...)

An investor purchased 10 shares of a technological company for USD 40 each. After a year, he received a per-share dividend of USD 5 and sold 1 share for USD 45. The investor decided not to reinvest the dividends. At the end of the second year, he sold the remaining 9 shares for USD 50 each. What is the money-weighted rate of return on this 2-year investment?

(...)

An investor purchased 10 shares for USD 600. After a year, he got a USD 100 dividend, sold the shares for USD 80 and purchased more shares for USD 250. At the end of the second year, he received a dividend of USD 104 and sold the shares for USD 520. At the end of the third year, so at the end of the investment period, he received a dividend of USD 35 and sold the remaining shares for USD 350.

What is the money-weighted rate of return?

(...)

For practical reasons, we often apply the time-weighted rate of return. The time-weighted rate of return differs from the money-weighted rate of return as it does not depend on the value of particular cash flows. The time-weighted rate of return (TWRR) is a geometric mean return over the whole investment period.

We've invested USD 100,000 in a stock fund for 5 years. After 5 years, the value of the investment equals USD 200,000. The time-weighted rate of return will show us how much on average the value of the investment must grow every year in order to get us USD 200,000 after 5 years.

(...)

You would like to calculate the time-weighted rate of return on a 3-year stock fund. The fund's performance in previous years was as follows:

HPR for the first year was 19%, it was 5% for the second year and -15% for the third year.

What is the time-weighted rate of return?

(...)

An investor purchased 10 shares of a technological company for USD 40 each. After a year, he received a per-share dividend of USD 5 and sold 2 shares for USD 45 each. The investor decided not to reinvest the dividends. At the end of the second year, he sold the remaining 8 shares for USD 50 each. What is the time-weighted rate of return on this 2-year investment?

(...)

When should we apply the money-weighted rate of return and when time-weighted rate of return?

The money-weighted rate of return gives different weights to different periods, while the time-weighted rate of return gives the same weights to different periods.

Therefore, we should conclude that if an investment manager has full control of the timing of cash inflows and outflows, we should use the money-weighted rate of return. It would be appropriate in the case of our private investments when it is us who decides what amounts we invest in every period.

When the portfolio manager has little influence on the invested amounts, we should apply the time-weighted rate of return. This is the case with a fund manager, who doesn’t have control over how much the clients add to or withdraw from the fund.

Since we usually have different weights when calculating the money-weighted rate of return and the time-weighted rate of return, they are rarely equal.

- The money-weighted rate of return (MWRR) is simply an internal rate of return (IRR).
- The time-weighted rate of return (TWRR) is a geometric mean return over the whole investment period.
- The money-weighted rate of return gives different weights to different periods, while the time-weighted rate of return gives the same weights to different periods.
- If an portfolio manager has full control of the timing of cash inflows and outflows, we should use the money-weighted rate of return. When the portfolio manager has little influence on the invested amounts, we should apply the time-weighted rate of return.