# Rate of Return – Time is not Money

Today we’re going to take a look at Rate of Return.

Contents

###### Preamble

I had planned to write about rate of return back in April, when Monevator wrote an article explaining How to unitize your portfolio. The article generated a lot of debate, and The Investor^{1} pointed readers back to a Henry Wirth article on Return Rates.

Then we had the General Election, and I forgot about it. I was reminded last week by a post on Canadian Couch Potato called Calculating Your Portfolio’s Rate of Return. So here we are.

Rate of Return used to be a big thing for me, especially when I was just starting out in investment.

At that stage:

- you want to be sure that you’re making more money than you would without taking the investment risks that you do, and
- you need to be confident that you are on track to reach your long-term investment goals

As you move closer to those goals and your portfolio gets bigger, absolute rate of return becomes less important, at least at a portfolio level.

You become more focused on preserving what you already have, with maybe an annual check to see how you are performing against the various benchmarks that you have set yourself.

But Rate of Return is interesting, and confusing to many people. So let’s take a look.

###### Simple Rate of Return

When all your money is invested into a portfolio at the same time, and none is taken out, calculating the rate of return is simple. You just need to find the compound return that will get you from the starting value to where you are now.

You can get this via an Excel sheet,^{2} or by using a built-in function like IRR() – the Internal Rate of Return. You can even use LN() and EXP().

Things get more complicated when money has been added to, or withdrawn from, the portfolio during the time period under consideration. Then IRR() won’t work.

###### Time vs Money

There are two kinds of Rate of Return that can be calculated for a portfolio:

- Time-weighted rate of return (TWR or TWRR)
- Money-weighted rate of return (MWR or MWRR)

**Time-**weighted returns are not affected by the amount of money invested or withdrawn during each period (a year, say, or a day).

- All time periods are weighted equally, irrespective of how much money is invested when.
- TWRs are usually used by funds for reporting purposes, and are calculated using daily periods.
- A fund manager uses the same strategy for all investors, and so should not be rewarded or penalized for the effect of cash flows over which he had no control.
- The manager’s choice of underlying assets and their performance will determine returns, and this allows investors to assess how skillful the manager is.

**Money**-weighted returns are affected by the amount of money invested or withdrawn, and these make more sense for private investors.

- time periods in which more money is invested have more impact on returns than time periods in which less money is invested

Private investors should also calculate their TWR since a comparison between the two numbers will determine whether their market timing (should they believe in such a thing) is adding or subtracting value.

Studies have shown that private investors usually underperform the market, due to buying high and selling low.

- If MWR is greater than TWR then your market timing / rebalancing / regular saving is adding value
- If TWR is greater than MWR then your market timing is subtracting value

Comparing your TWR to a benchmark fund or index will show you how skilled you are in asset allocation / stock and fund selection.

To work out whether your returns are down to skill or simply risk taking, you would need to calculate something like the Sharpe Ratio.^{3} We’ll leave that for another day.

###### Some calculations

Using Henry Wirth’s example, the Return for One Period (Year, say) is:

This is both a MWR and a TWR, since no money was added or withdrawn during the year.

Extending this example into a second year, another £10,000 is added to the portfolio. By the end of the year, the value of the portfolio has dropped to 10,000.

This again is both a MWR and a TWR, since no money was added or withdrawn during the year.

The formula for calculating the multi-period TWR is (where the periods are equal):

Total Rate = ( 1 + Rate 1 ) ( 1 + Rate 2 ) ( 1+ Rate 3 ) … ( 1 + Rate N ) – 1

In this example:

Total Rate = 1.2 * 0.8929 – 1 = **7.15%**

Since the portfolio has lost £1,000 of the total £11,000 that has been invested, this positive rate of return is a counter-intuitive result.

For a fund manager this makes sense, since he can’t control when investors buy and sell his fund. For a private investor managing his own fund, it makes less sense.

To calculate the MWR, we use the Excel function XIRR. XIRR calculates the annualised internal rate of return.

XIRR matches a series of cash flows (1000 in, 10000 in, 10000 out) with a series of dates (Dec-01, Dec-02, Dec-03) and gives an MWR of **-8.39%**. Note that the time periods do not need to be equal for XIRR to work.

To test this result, Henry compounds the initial 1K at -8.39% for 2 years, and the second 10K at -8.39% for 1 year. The results tally.

In this example the 2-yr MWR is much worse than the 2-yr TWR, so our hypothetical investor is a terrible market timer.

###### Unitisation

To calculate the TWR for your portfolio when the periods between cash flows are uneven, you need to unitise your portfolio, as suggested by the Monevator article in April.

This is the process used by managers of open-ended funds (now called OEICs, but previously known for this reason as unit trusts).

The basic procedure is quite simple:

- each time you add money to your portfolio, you “buy” a certain number of units, at the prevailing market price

The process for withdrawals is the reverse:

- each time you take out money, this is generated by selling (cashing-in) a certain number of units, at the prevailing market price

Adding or withdrawing money doesn’t change the price of a unit.

Dividends, splits, trading commissions and stamp duty can all be ignored, so long ass they are funded from within the portfolio. Only external cash flows (in and out) need to be recorded and their impact calculated.

The main issue for the private investor is diligently recording the value of the portfolio (and the units within it) at the time of each cashflow in or out.

###### The unitisation process

Retro-fitting unitisation to an existing portfolio is difficult, so it’s best to begin at the beginning.

- Choose a starting value for units. £1 is the obvious choice, but £1K might work well for larger portfolios.
- Convert the starting value of the portfolio into a starting number of units (a simple division).
- eg. a £100K portfolio might have 100 units of £1K

- When you add money, you need to first calculate the current price of a unit:
- simply divide the current value of the portfolio by the last number of units you have
- if the portfolio is now worth £125K, then a unit is now worth £1.25K
- if you are adding another £10K, this is equal to 8 units
- you now have a portfolio worth £135K
- this is made up of 108 units of £1.25K
- you need to keep a record of the new number of units for the next transaction

- Withdrawing money is the same process in reverse:
- the portfolio is now worth £162K
- each of the 108 units is worth £1.5K
- to withdraw £30K in cash, you need to “sell” 20 units
- you now have a portfolio worth £132K
- this is made up of 88 units of £1.5K
- you need to keep a record of the new number of units for the next transaction

- To work out you unitised return, you simply look at the value of a unit:
- a unit is now worth £1.5K
- units started off at £1K
- you are up 50%
- this number – the percentage gain – is useful when comparing to other funds, or to an index

- To find out what the TWR is, you simply work out what rate of compounding is needed to increase a unit from £1K to £1.5K over the period you’ve held the portfolio
- if the portfolio had taken 3 years to get from £1K units to £1.5K units, then the annual TWR is 14.5% (since 1.145 * 1.145 * 1.145 = 1.5)

###### Conclusions

- There are two ways to calculate returns – money-weighted and time-weighted.
- Time-weighted returns are not affected by the amount of money invested or withdrawn during each period.
- Money-weighted returns are affected by the amount of money invested or withdrawn, and these make more sense for private investors.
- Private investors should also calculate their time-weighted returns, since a comparison between the two will determine whether their market timing is adding or subtracting value.
- To calculate time-weighted returns for a portfolio with uneven time periods between cash flows, a unitisation approach is best.
- Comparing your time-weighted returns to a benchmark fund or index will show you how skilled you are in asset allocation / stock and fund selection.

Until next time.

###### Sources

- Return Rates – Henry Wirth
- How to unitize your portfolio – Monevator
- Calculating Your Portfolio’s Rate of Return – Canadian Couch Potato

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This was a nice analysis, thank you. It is a source of some surprise to me that UK online brokers as far as I can tell do not advertise that they offer or maybe don’t offer rate of return analyses on their client’s portfolios. Certainly I cannot find this for Hargreaves Lansdowne. Yet Vanguard in the USA does this routinely. It would take little computer time for the brokers to offer this and would be very helpful – at least to the clients! Would it be possible to rattle the cages of the brokers a bit and point out that some sorts of portfolio analysis would be very value added for their clients. Unless of course they don’t want their clients to see their true rates of return!