The Return Measures, and When Each One Is the Right Answer
A fund can honestly report 6.67% and 4.59% for the same three years, and an investor in it can honestly report 0.00% and −7.88% for the same two. None of those four numbers is wrong. They are answers to four different questions, and the work is knowing which question has been asked.
That is what this topic is really testing. The arithmetic behind each measure takes a line or two. Choosing between them is where the marks sit, and it is also where a fact sheet can mislead without stating anything false.
Key Takeaways
- The holding period return is price change plus income, divided by the amount invested, and it carries no time unit until the length of the period is stated.
- The arithmetic mean estimates a single future period; the geometric mean is what an investor who stayed invested actually earned.
- On returns of +30%, −20% and +10%, the arithmetic mean is 6.67% and the geometric mean 4.59%, and the gap widens with volatility.
- The money weighted return is the internal rate of return on the investor’s own cash flows, so it includes the consequences of when money was added.
- The time weighted return strips those cash flows out, which is why performance standards require it for judging a manager.
- One investment gave a time weighted return of 0.00% and a money weighted return of −7.88%, and the gap is the cost of the timing decision.
- Annualising is a unit conversion rather than a forecast: a 1% week annualises to 67.8%, which says nothing about the year.
- A 9.30% headline return became 0.74% after trading costs, fees, 30% tax and 5% inflation, and tax is charged before inflation is removed.
Holding period return, the base case
Every other measure on this page is built from the holding period return, so it is worth stating precisely. It is the total gain over one period, price change plus income received, divided by what was invested at the start.
HPR = (P1 − P0 + income) / P0
P0 is the price paid, P1 the price at the end of the period, and income is any dividend or coupon received during it
A share is bought at 40.00, pays a dividend of 1.20 during the year, and is worth 44.00 at the end of it. What is the holding period return?
- Price gain = 44.00 − 40.00 = 4.00.
- Total gain = 4.00 + 1.20 = 5.20.
- HPR = 5.20 / 40.00 = 0.130, that is 13.0%.
Answer: 13.0%, of which 10.0 percentage points came from the price and 3.0 from the dividend. Two features carry into everything that follows. The return has no time unit attached to it, so a holding period return of 13% means nothing until the length of the period is stated. And the dividend is credited in full at the end of the period, with no assumption made about reinvesting it, which is a simplification that matters when periods are chained together.
Averaging across periods: arithmetic and geometric
Given several yearly returns, there are two defensible averages and they answer different questions.
The arithmetic mean is the simple average. It is the best estimate of the return in a single randomly chosen year, which makes it the right input when a future one-period return is being forecast.
The geometric mean is the constant rate that would have produced the actual ending wealth. It is the right answer whenever the question is about what an investor who stayed invested actually earned.
Geometric mean = [(1 + R1)(1 + R2) … (1 + Rn)]1/n − 1
the returns are compounded, then the nth root converts the total back into a per-period rate
A fund returns +30%, then −20%, then +10%. What are the arithmetic and geometric mean returns, and what happened to 100 invested at the start?
- Arithmetic mean = (30 − 20 + 10) / 3 = 6.67%.
- Compounding: 1.30 × 0.80 × 1.10 = 1.144, so 100 grew to 114.40.
- Geometric mean = 1.1441/3 − 1 = 0.0459, that is 4.59%.
- Check: 100 × 1.04593 = 114.40, which reproduces the actual ending value.
Answer: 6.67% arithmetic and 4.59% geometric, a gap of over two percentage points across three ordinary years. The investor earned 4.59% a year. Quoting 6.67% would overstate the ending wealth after ten years by about 34 per 100 invested, which is why performance is reported geometrically and why the arithmetic mean appears in advertising more often than in fact sheets.
The geometric mean is never above the arithmetic mean, and the two are equal only when every period return is identical. The size of the gap grows with the volatility of the returns, which gives it a second reading: the difference between the two averages is a rough measure of how bumpy the ride was. A fund with a 6% arithmetic mean and a 5.9% geometric mean was steady. One with 6% and 3% was not.
Money weighted return
The money weighted rate of return is the internal rate of return on the investor’s actual cash flows. Every deposit and withdrawal is treated as a cash flow, and the rate that sets their present value to zero is the return.
It answers a specific question: what did this investor earn on the money that was actually at work, given when it was put in and taken out. That makes it the correct measure for judging an individual portfolio where the investor controls the timing of contributions.
An investor buys one share at 100. A year later the share is worth 125, and the investor buys a second share at that price. At the end of the second year the price has fallen back to 100 and both shares are sold. What is the money weighted return?
- The cash flows are −100 at time 0, −125 at time 1, and +200 at time 2.
- Setting the present value to zero: −100 − 125 / (1 + r) + 200 / (1 + r)2 = 0.
- Writing x for (1 + r) and multiplying through gives 100x2 + 125x − 200 = 0, which simplifies to 4x2 + 5x − 8 = 0.
- Solving, x = 0.9212, so the money weighted return is −7.88% a year.
Answer: −7.88% a year. The share ended exactly where it started, so the investment itself went nowhere. The loss is entirely a timing effect: the investor held one share through the good year and two shares through the bad one, so more money was exposed to the fall than to the rise.
Time weighted return, and when each is the right question
The time weighted return removes the effect of those cash flows entirely. The period is broken at every contribution and withdrawal, a holding period return is computed for each sub-period, and the sub-period returns are compounded.
Take the same two years. What is the time weighted return?
- Year one return: 125 / 100 − 1 = +25%.
- Year two return: 100 / 125 − 1 = −20%.
- Compounding: 1.25 × 0.80 = 1.00.
- Time weighted return = 1.001/2 − 1 = 0.00%.
Answer: 0.00% a year, against a money weighted return of −7.88% on the same investment over the same two years. Neither figure is wrong. The manager delivered a return that netted to zero, and the investor lost money by adding capital before the decline. The two measures are answering different questions and the gap between them is exactly the value, in this case the cost, of the timing decision.
The rule that follows is a rule about responsibility. A portfolio manager who does not control when clients add or withdraw money should be judged on the time weighted return, which is why performance presentation standards require it. An investor who does control that timing should look at the money weighted return, because it includes the consequences of decisions that were theirs.
Reading a large gap between the two figures as an error in one of them. The gap is information. When the money weighted return is well below the time weighted return, capital was added before weak periods, and the standard behavioural pattern of buying after a rise produces exactly that signature. It shows up in the aggregate too, in the persistent difference between fund returns and the returns their investors actually receive.
Annualising, and what the convention hides
Returns over different lengths of time cannot be compared until they are put on the same basis, and the standard basis is one year.
Annualised return = (1 + Rperiod)n − 1
n is the number of those periods in a year, so n is 2 for a half year, 4 for a quarter, and 1/3 for a three-year cumulative return
| Observed return | Period | Annualised | Simple scaling |
|---|---|---|---|
| 4.0% | Six months | 8.16% | 8.00% |
| 2.0% | One quarter | 8.24% | 8.00% |
| 33.0% | Three years | 9.97% | 11.00% |
| 1.0% | One week | 67.8% | 52.0% |
The last row is the warning. Annualising a one-week return is arithmetically correct and practically meaningless, because it asserts that the week is representative of the other fifty-one. The convention is a unit conversion, not a forecast, and the shorter the observation window the more aggressively it exaggerates. Treat any annualised figure built from less than a full quarter as a statement about the past period rather than about a year.
Gross and net, nominal and real, and leveraged
The measures so far differ in how the return is computed. The remaining ones differ in what has been taken out of it before it is reported.
| Measure | Deducted | Not deducted |
|---|---|---|
| Gross return | Trading expenses | Management and administrative fees, tax, inflation |
| Net return | Trading expenses, management and administrative fees | Tax, inflation |
| After-tax nominal return | The above, plus tax on income and gains | Inflation |
| Real return | The above, plus the effect of inflation | Nothing further |
Gross return is the cleanest measure of the manager’s skill, because trading expenses are unavoidable in implementing any decision while fees are a commercial matter. Net return is what the fund actually delivered to the investor. Neither is more correct; they are answers to different questions, and a comparison that mixes one fund’s gross return with another fund’s net return is the most common way to reach a wrong conclusion about relative performance.
A portfolio returns 9.30% before any deductions. Trading expenses are 0.30%, management and administrative fees are 0.75%, tax on the return is 30%, and inflation over the year is 5.0%. Work down to the real after-tax return.
- Gross return = 9.30 − 0.30 = 9.00%.
- Net return = 9.00 − 0.75 = 8.25%.
- After-tax nominal return = 8.25 × (1 − 0.30) = 5.775%.
- Real after-tax return = (1.05775 / 1.05) − 1 = 0.738%.
Answer: 0.74%. A headline of 9.30% became less than one percent of purchasing power. Note the order of operations, because it is examinable: tax is charged on the nominal return, then inflation is removed from what survives. Removing inflation first and taxing the real return would give a different and higher figure, and it is not how tax works.
Leverage is the last variation, and it is the one that changes the shape of the outcome rather than its level. Borrowing to invest applies the asset return to a larger base while the financing cost is fixed, so both the gains and the losses are magnified against the equity actually committed.
An investor puts up 100 of equity, borrows another 100 at 5%, and invests 200. What is the return on equity if the assets return 8%, and if they return −8%?
- Assets return 8%: gain = 200 × 0.08 = 16, interest = 100 × 0.05 = 5, so the return on equity is (16 − 5) / 100 = 11.0%.
- Assets return −8%: loss = 200 × 0.08 = 16, interest is still 5, so the return on equity is (−16 − 5) / 100 = −21.0%.
Answer: +11.0% against −21.0%. A symmetric 8% move in the assets produced an asymmetric outcome for the investor, because the financing cost is paid in both states. That asymmetry, not the multiplication of the gain, is what makes leverage a risk decision rather than a sizing decision.
Most questions on this material test the choice of measure rather than the arithmetic. A question describing a manager evaluated on a portfolio with client contributions is asking for the time weighted return. One describing an individual investor is asking for the money weighted return. One giving several yearly returns and asking what an investor earned wants the geometric mean, while one asking for an estimate of next year wants the arithmetic mean. Identify the question before starting the calculation, because all four numbers will be among the options.
The short version
- Holding period return is the base case, and every other measure is built from it.
- Arithmetic mean for a forecast of one period, geometric mean for what was actually earned.
- Money weighted return judges the investor, because it includes the timing of contributions. Time weighted return judges the manager, because it excludes them.
- A large gap between the two is information about timing decisions, not an error in either figure.
- Annualised figures built from very short windows exaggerate aggressively and should be read as statements about the period observed.
- Gross, net, after-tax and real are the same return with successively more deducted, and comparing one fund’s gross with another fund’s net is the standard way to reach a wrong conclusion.
- Leverage makes the outcome asymmetric rather than simply larger, because the financing cost is paid in both good and bad states.
Frequently Asked Questions
What is the difference between the arithmetic and geometric mean return?
The arithmetic mean is the simple average of the period returns and is the best estimate of the return in a single future period. The geometric mean is the constant rate that reproduces the actual ending wealth, so it is what an investor who stayed invested actually earned. On returns of plus 30%, minus 20% and plus 10%, the arithmetic mean is 6.67% and the geometric mean is 4.59%. The geometric mean is never higher, and the gap widens as returns become more volatile.
What is the difference between money weighted and time weighted return?
The money weighted return is the internal rate of return on the investor’s actual cash flows, so it reflects when money was added and withdrawn. The time weighted return breaks the period at every cash flow and compounds the sub-period returns, which removes the timing effect entirely. In the worked example both describe the same investment over the same two years, and give 0.00% and minus 7.88%.
Which return measure should be used to evaluate a portfolio manager?
The time weighted return, because a manager does not usually control when clients add or withdraw money and should not be credited or penalised for it. That is why performance presentation standards require it. An individual investor who does control the timing should look at the money weighted return, since it includes the consequences of decisions that were theirs.
How do you annualise a return?
Raise one plus the period return to the power of the number of such periods in a year, then subtract one. A 4% half-year return annualises to 1.04 squared minus 1, which is 8.16% rather than 8.00%. A 33% return over three years annualises to 9.97%, not 11%. The convention is a unit conversion, so annualising a very short window exaggerates: a 1% week comes out at 67.8%.
What is the difference between gross return and net return?
Gross return is after trading expenses but before management and administrative fees. Net return has those fees deducted as well, so it is what the investor actually received. Gross return is the cleaner measure of the manager’s decisions, because trading expenses are unavoidable in implementing them while fees are a commercial matter. Comparing one fund’s gross return with another fund’s net return is the most common way to reach a wrong conclusion about relative performance.
In what order are tax and inflation removed from a return?
Tax first, then inflation. Tax is charged on the nominal return, and the real return is what the remainder is worth in purchasing power. On a portfolio returning 9.30% before deductions, with 0.30% of trading expenses, 0.75% of fees, tax at 30% and inflation at 5%, the sequence gives a gross return of 9.00%, a net return of 8.25%, an after-tax nominal return of 5.775%, and a real after-tax return of 0.74%.
Why does leverage make returns asymmetric?
Because the financing cost is paid whether the investment gains or loses. An investor putting up 100 of equity, borrowing 100 at 5% and investing 200 earns 11.0% on equity if the assets return 8%, and loses 21.0% if they return minus 8%. A symmetric move in the assets produces an asymmetric outcome for the investor, which is what makes leverage a risk decision rather than a sizing decision.
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