UTS16657 Chap.3 Measuring Return on a Single Asset
Measuring Return on a Single Asset
Return is the basic yardstick of how an investment has performed, and its job is to let you compare alternatives that are otherwise nothing alike. It has two components. Income is the cash the asset throws off along the way, whether coupons, dividends, distributions, interest or net rent, and may be fixed or variable.
Capital gain or loss is the difference between what you paid and what you receive on sale, after transaction costs.
Four questions change the number before any calculation begins. Is the gain realised, or only a valuation on paper that can reverse? Is the figure before or after tax, which depends on the investor's own position and, for capital gains, on the holding period and carried-forward losses?
Have management fees been deducted, given that they vary widely with strategy and with whether the manager shares in success? And is the series geared, which makes it a return on equity rather than on assets?
Compounding the problem, some assets publish total returns, some publish price returns that omit income, and some are appraisal-based rather than traded.
Averaging a run of yearly outcomes can be done two ways and they give different answers.
The arithmetic mean adds the yearly percentages and divides by their number, and it is the right answer to what a typical year looked like, which is why it is the usual forward-looking estimate. The geometric mean multiplies the growth factors and takes the n-th root, and it is the only rate that reproduces the actual ending wealth, because a loss is suffered on a base that a later gain has to rebuild from.
The geometric mean is never above the arithmetic one and the gap widens with volatility. Two conversions complete the toolkit: annualising a below-annual rate by compounding it, and extracting the compound annual return implied by an index moving between two levels. Rebasing a price series to a common starting value of 100 makes several assets visually comparable, which is what the individual report's chart requires.
What this chapter covers
- 01
The two components of a period return and what is deducted from each
- 02
Total return series against price return series
- 03
The arithmetic mean and when it is the right answer
- 04
The geometric mean and why it is never higher
- 05
Converting a monthly or weekly rate to an annual one
- 06
The compound annual return between two index levels
- 07
Rebasing a price series so several assets can be compared
Two averages on one five-year return path, and the sentence that distinguishes them
- +1Arithmetic mean: (9.4 + 13.1 − 2.8 + 7.6 + 11.2) ÷ 5 = 38.5 ÷ 5 = 7.70% per annum. Carrying the minus sign through the sum is where most of the lost marks are.
- +1Convert every return to a growth factor before anything is multiplied: 1.094, 1.131, 0.972, 1.076 and 1.112. The negative year becomes a factor below one, which is the whole point.
- +1Multiply them: 1.094 × 1.131 × 0.972 × 1.076 × 1.112 = 1.439008, so $1 grew to $1.439008 over the five years.
- +1Take the fifth root and subtract one: 1.4390081/5 = 1.075506, so the geometric mean is 7.5506%. It sits 0.1494 points below the arithmetic mean, as it must whenever the returns are not all equal.
Key terms
- Income return
- The cash an investment throws off during a period, expressed against the price at the start of the period. It covers coupons, dividends, distributions, interest and net rent.
- Capital return
- The change in the price of an investment over a period, expressed against the price at the start of it, measured after transaction costs and whether or not the gain has been realised.
- Arithmetic mean return
- The sum of the period returns divided by their number. It answers what a typical period looked like and is the usual forward-looking estimate, and it never sits below the geometric mean.
- Geometric mean return
- The n-th root of the product of the growth factors, minus one. It is the constant rate that reproduces the actual ending wealth, so it is the honest description of what an investor earned.
- Compound annual return
- The constant annual growth rate implied by a series moving from a beginning value to an ending value over a stated number of years, found as the ratio raised to one over the years, minus one.
- Rebased index
- A price series rescaled so that its first observation equals a common base, conventionally 100, so several series with different starting prices can be compared on one chart.
- Price return series
- A return series reflecting capital growth only, with income excluded. Comparing one against a total return series flatters the latter for no reason other than that it counted the income.
Measuring Return on a Single Asset FAQ
Why do the two averages give different answers on the same data?
Because one adds and the other multiplies. Arithmetic averaging treats every period as independent and equally weighted, so a loss and a gain of the same size cancel. Geometric averaging respects the fact that a loss is suffered on the accumulated base and a subsequent gain has to rebuild that base before any new ground is made.
The wedge between them grows with the spread of the series, which is why a manager quoting an arithmetic average on a volatile fund is quoting a number no investor in it ever experienced.
How do I annualise a monthly return?
Compound it rather than multiplying by twelve. Raise one plus the monthly rate to the twelfth power and subtract one, so an average monthly return of 0.72% annualises to 8.9905% rather than the 8.64% that multiplication would give. The same form works at any frequency: use fifty-two for weekly data and four for quarterly. This matters because the individual report asks for annual figures built from monthly data.
What does rebasing actually do to the numbers?
Nothing to the returns, and a great deal to the readability. Rebasing sets every series to the same starting value, conventionally 100, and scales every later observation by the same factor, so two assets that started at very different prices can be plotted together and compared by eye.
In a spreadsheet the rebased value in any month is the base multiplied by that month's price divided by the first month's price, with the first month's price anchored as an absolute reference so it does not drift as you fill down.
Exam move
This is the chapter where careless arithmetic costs the most, so rehearse the conversions rather than the concepts. Practise turning percentages into growth factors and back until it is automatic, and always finish a compound annual return by compounding your answer forward to check it reproduces the ending value.
Then build the rebasing formula once in a spreadsheet with the absolute reference in place, because Question 1 of the individual report asks for exactly that and a broken reference silently corrupts every row below the first.
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