FIN521 Chap.6 Risk, Return and the Historical Record
Risk, Return and the Historical Record
What a record of past returns can establish
This topic sits in the fourth teaching week and the midterm covers it, and one of the five learning outcomes names the risk and return relationship directly. The question the chapter answers is narrow and easy to overreach: given what an asset has done, what may you now say about what it will do?
The building block is the holding period return, which is income received plus the change in price, all over the price paid. Two components and one denominator.
Everything else in the chapter is built from many of those, and the first discipline is to keep a fact about a period that has happened apart from a belief about one that has not.
Two averages that answer different questions
Given a sequence of returns there are two defensible summaries and they disagree by an amount that grows with volatility.
The arithmetic mean adds and divides, and it answers what to expect in a single future year. The geometric mean compounds and takes a root, and it answers what constant rate would have produced the actual ending wealth. Neither is wrong and a summary that does not say which it used cannot be checked.
The gap between them is a mathematical consequence of dispersion rather than a fee: a thirty per cent fall needs a forty-three per cent rise to recover, so a volatile sequence always compounds to less than its arithmetic average suggests.
Risk measured as dispersion, and the objection to that
The standard measure is the standard deviation of returns, which counts an unexpectedly good year as risk in exactly the same way as an unexpectedly bad one.
That is a real limitation and the course expects you to be able to state it, because few investors experience upside dispersion as a problem. The defence is practical rather than philosophical: dispersion is roughly symmetric for diversified portfolios, it is easy to estimate, and it aggregates across assets in a way that downside-only measures do not.
What makes it useful is placing it next to a return, because a mean alone ranks assets in an order the market does not recognise.
The premium, and two cautions about the record
The risk premium is the expected return on a risky asset less the risk-free rate, and it is the compensation the market offers for bearing risk.
The historical record in developed equity markets shows a substantial positive premium over long horizons. Two cautions belong in any answer that uses that record. It is an expectation, so realised returns fall below the risk-free rate in many individual years and some long ones.
And the record available to be measured is the record of markets that survived to be measured, which flatters the average in a way no care with the arithmetic can remove.
What this chapter covers
- 01
Holding period return and its two components
- 02
Why a percentage is not comparable until the period is stated
- 03
Arithmetic mean as an expectation for one period
- 04
Geometric mean as the constant rate behind realised wealth
- 05
Why dispersion forces the two apart
- 06
Standard deviation as a symmetric measure of risk
- 07
Reward per unit of dispersion as the basis for comparison
- 08
The risk premium, and survivorship in the historical record
- 09
Nominal against real return, and why the exact relation divides
Reconcile two correct averages with one account balance
- 2Compute the arithmetic mean and say what question it answers.
- 4Compound the three years and give the ending wealth.
- 3Give the geometric mean and explain the gap in one sentence.
Key terms
- Holding Period Return
- Income received plus the change in price, divided by the price paid, measured over whatever period the question specifies.
- Arithmetic Mean
- The simple average of a set of returns, which answers what return to expect in a single future period.
- Geometric Mean
- The constant compound rate that would have produced the actual ending wealth over a sequence of periods.
- Standard Deviation
- The square root of the average squared deviation of returns from their mean, used here as the measure of risk.
- Risk Premium
- The expected return on a risky asset less the risk-free rate, which is compensation offered for bearing risk rather than a promised return.
- Real Return
- The nominal return adjusted for inflation, describing what the money will buy rather than what the statement says.
- Survivorship
- The distortion in a historical record caused by measuring only the markets or funds that lasted long enough to be measured.
Risk, Return and the Historical Record FAQ
Which average should I quote in an examination answer?
The one that matches the verb in the question. If it asks what an investor earned or what an account is now worth, compound the returns and give the geometric mean alongside the ending wealth. If it asks what to expect next year, the arithmetic mean is the right estimate. Quoting either without naming which it is leaves the answer unverifiable, and a single clause saying what your number answers protects the whole response.
Is standard deviation a good measure of risk?
It is the measure this course uses and it has a known weakness worth stating. Because it is symmetric, it treats an unexpectedly large gain as risk in the same way as an unexpectedly large loss, which does not match how investors experience outcomes. It is retained because it is simple to estimate, roughly symmetric for diversified holdings, and it aggregates across assets in a way that downside-only measures do not.
Naming the limitation and the defence together is worth more than choosing a side.
Why is a long historical average of equity returns not a forecast?
Two reasons, and both belong in the answer. It is an expectation, so individual years and even long stretches fall below the risk-free rate, and an investor who must sell in a bad stretch never receives the average.
And the record we can measure comes from markets that survived to be measured, so markets that closed or were expropriated are missing from the sample, which flatters the number regardless of how carefully it is computed.
Exam move
Take any three-year return sequence and compute both averages and the ending wealth, then widen the dispersion while keeping the arithmetic mean fixed and watch the ending wealth fall. Doing that twice makes the relationship intuitive rather than memorised.
Then write the two-sentence caution about expectations and survivorship in your own words and keep it, because it converts a description of the historical record into the evaluation the rubric is asking for.
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