For money managers (and for our personal investments), rate of return is the most important number that is expressed in percentage terms. But how do we calculate it? Do we use the simple and intuitive total rate of return formula? Not so fast…

## A distinction without a difference

In the absence of **intraperiod external cash flows**, there’s no difference between the two concepts. Both converge to the basic **total rate of return** calculation.

where

= value at the beginning of the measurement period

= value at the end of the measurement period

In other words: total rate of return is the percentage change in market value over a defined period of time. This captures both income from dividends or coupons and capital appreciation.

Pretty basic, isn’t it? Funnily enough, total rate of return wasn’t really used until the 1960s or so. Lack of computational power was one hurdle, and also in the old days investors used income as their return yardstick.

## A distinction with a difference

The distinction between these two concepts is relevant when **intraperiod external cash flows** are involved, otherwise, the basic total rate of return formula is sufficient.

External cash flows are contributions or withdrawals, as opposed to internal cash flows such as dividend and interest payments.

## What do we want to evaluate?

One of the two measurement methods will be the appropriate one to use depending on what aspect of portfolio management we want to evaluate.

For example, imagine we have 100M to invest and we hire a portfolio manager that specializes in large-cap equities. The manager is expected to add value through stock selection and industry rotation.

On day one, we give the manager 10M and keep 90M in cash. We don’t allocate the entirety of our funds to the manager because we believe that there will come a better time to invest in large-cap equities.

Six months later, we allocate the remaining 90M to our large-cap equity strategy.

At year-end, the value of the portfolio is 110M. It evolved as follows:

- Day 1: cash infusion of 10M
- Day 180: portfolio value is 20M (i.e., 2x return in 180 days)
- Day 181: cash infusion of 90M ← intraperiod external cash flow
- Day 360: portfolio value is 110M (i.e., zero appreciation in days 181 to 365)

## Time-weighted return: evaluating the portfolio manager

Let’s say we want to evaluate what was the return of the large-cap equity strategy. If we blindly apply the basic total return formula, depending on what we consider to be the beginning value (V_{b}), we may get a rate of return of:

or

Both results are intuitively wrong. The first one amplifies the portfolio manager’s return by counting the intraperiod cash flow as part of the return generated through investing. The second one penalizes the manager by counting the intraperiod cash flow as if it had been available to invest since day one.

This is where the time-weighted return method comes to the rescue. It corrects the distortion caused by external cashflows by breaking the investment period into discrete sub-periods, estimating the return for each sub-period, and then linking the sub-period returns.

In our example:

- Sub-period 1
- Time range: day 1 – day 180
- = 10M (the amount invested with the portfolio manager on day 1)
- = 20M (the value of the portfolio immediately before the 90M external cash flow)
- Return = = 100%

- Sub-period 2
- Time range: day 181 – day 360
- = 110M (value of the portfolio on day 180 + 90M external cash flow)
- = 110M (value of the portfolio on day 360)
- Return = = 0

- Total return for the full period:

*Generic formula at the end of this post.*

So, the value added by investing in large-cap equities was 100%. From here, we can start the evaluation of our portfolio manager. Did he add or subtract value when compared to a large-cap index (presumably shared with the manager in advance to serve as his benchmark)?

## Money-weighted return

If, instead, we want to measure how much value was created with our money (by investing or not investing it), then the money-weighted return is the more useful method. For instance, in a situation in which the manager has control over the timing and amount of cash flows into and out of the portfolio.

In the example above, the money-weighted return is calculated as follows:

The money-weighted return is the rate R that solves the equation above. It’s essentially an internal rate of return (IRR) calculation, and R equals 0.05% or 18.9% annualized.

*Generic formula at the end of this post.*

## Time-weighted vs. money-weighted return

The time-weighted return represents the growth of a **single unit of money** invested, while the money-weighted return represents the growth of **all money** invested.

For the most part, portfolio managers do not have control over external cash contributions or withdrawals, and therefore the time-weighted return is the appropriate method to measure performance.

### Appendix: formulas

**Time-weighted return (R _{tw})**

where

= total return for sub-period

**Money-weighted return (R _{mw})**

where

= number of time units in the evaluation period

= the th cash flow

= number of time units between the th cash flow and the beginning of the period

Solve for R_{mw} through iterative calculation.

Hasta la próxima.

BTW, if you like these posts, and think someone else might too, please forward it on.