# Performance Measurement

The measurement of the performance of a financial portfolio is based on the concept of *return*: the increase or decrease of value over a specific period. For example, your portfolio starts at 100 EUR and grows to 104 EUR by the end of the period, resulting in a performance of + 4%. Equation 1 offers three versions of the base formula; also named the *Simple Rate of Return* (ROR) or *Return on Investment* (ROI).

$$\mathrm{r = \frac{(MVE - MVB)}{MVB} \quad \Leftrightarrow \quad r = \frac{MVE}{MVB} -1 \quad \Leftrightarrow \quad MVB \times (1 + r) = MVE \quad (Eq 1)}$$

where MVE = the market value of the investment at the end of the reporting period, and MVB = the market value of the investment at the beginning of the period. The *Market Value* (MV) of a deposit account refers to the total balance of the account on a particular date. For a security or a security account, such as stocks, the MV is calculated by multiplying the number of shares by the closing price. The MV of a portfolio is determined by summing the market values of all accounts on that specific date.

Regarding Eq 1, please note that:

- The rate of return percentage applies to the entire period between MVE and MVB (3 years, 6 months, ...). It is
*not*an annual rate. - There are
*no*additional transactions within the specified period. The MVE is solely influenced by the MVB and time. - The equation can be applied to calculate the performance of the entire portfolio, a specific (security or deposit) account, an individual security, or a particular trade.

To address the above concerns, PP employs two distinct approaches to calculate the rate of return when additional transactions occur within the reporting period: the money-weighted or time-weighted rate of return. It also differentiates between measuring performance on the portfolio, security, or trade level.

Note

The simple Rate of Return (ROR) isn't adequate for describing the performance of a portfolio when transactions occur in the reporting period. Let's take the example from above. Your portfolio already contains one share valued at 100 EUR at the beginning of the year (MVB=100). Due to the favorable track record of the company issuing the shares, its price has increased to 104 EUR per share by the end of the year. Let's now compare the following scenarios:

**No additional transactions**: You take no actions throughout the year. According to Equation 1, r = (104-100)/100 = 4%. This seems intuitively correct. You gained 4 EUR, which is 4% of the initial value.**Additional transactions**: You acquire a second share at 100 EUR; right at the beginning of the year. MVB is still 100 EUR; while MVE becomes 2 * 104 = 208 EUR. The performance becomes a staggering 108% or (208-100)/100. This does not feel OK. You*know*that the growth rate of the company's share is 4%. Certainly, your total investment of 200€ has gained 8 EUR or 4% on 200 EUR; but not on the MVB of 100 EUR. The additional transaction has distorted the simple rate of return formula.

## The money-weighted rate of return

The money-weighted rate of return (MWR) employs the **Internal Rate of Return (IRR)** technique commonly used in project management. This calculation takes into account both the timing (when) and the size (how much) of cash, flowing into or out of the portfolio within the reporting period, effectively weighting the return by the amount of money invested at different times.

The money-weighted rate of return or IRR is the annual interest rate that is necessary to bring the beginning market value of the investment (MVB) and all subsequent cash flows to the end value (MVE). Your portfolio must grow (or shrink) each year by a percentage equal to the IRR to generate the specified cash flows within the given time period.

If you find the concept of IRR challenging, please start by first reading the money-weighted section. The calculation method is thoroughly explained by formulas and numerous examples, ranging from a simple single-share investment to multiple transactions including dividends. The examples are based on our demo portfolio to solidify your understanding.

## Time-weighted Rate of Return

The time-weighted rate of return (TWR) is not influenced by the amount invested. Whether you invest one EUR or 100 EUR, you will achieve the same TWR. The reporting period is divided into several holding periods, and for each holding period, a return value is calculated and then compounded into an overall return. Each period carries equal weight; hence the name time-weighted rate of return. In the True Time-Weighted Rate Of Return methodology (TTWROR), performance is calculated using simple returns without any consideration for the total amount invested.

The TTWOR method is explained in depth in the time-weighted section of this chapter.

A nice video about the calculation and difference between the money-weighted and time-weighted approach is given at the Finance and Risk Corner.