FINS5513 Chap.2 Return, Risk and Investor Choice
Return, Risk and Investor Choice
Week 2 turns market observations into portfolio inputs. Holding-period return combines the price change and income earned over the actual investment interval. APR-style simple annualisation and effective annual return are different conversions, so the horizon must be identified before choosing one. Ex ante analysis uses possible states and their probabilities; ex post analysis uses observed returns and sample estimators.
Variance measures squared dispersion and standard deviation restores return units. The Sharpe ratio divides expected excess return by total risk. Quadratic utility adds investor risk aversion and ranks portfolios by expected return minus a variance penalty. Covariance records joint movement, while correlation standardises covariance to a value between minus one and plus one.
Portfolio expected return is a weighted average, but portfolio variance includes both weighted component variances and covariance cross terms. The calculation sequence should preserve units: compute means first, retain variance units through every term, take the square root only at the end, and interpret whether the relationship creates or limits diversification.
What this chapter covers
- 01
Holding-period return from ending value, beginning value and income
- 02
APR and effective annual return across different investment horizons
- 03
Ex ante expected return and probability-weighted variance
- 04
Ex post arithmetic mean and sample variance using n minus one
- 05
Sharpe ratio as excess return per unit of total risk
- 06
Quadratic utility and the effect of investor risk aversion
- 07
Covariance, correlation and the standardisation of joint movement
- 08
Two-asset portfolio expected return, variance and diversification
Worked example · free
Portfolio risk with an imperfect correlation
- +1Expected return is 0.60 times 10% plus 0.40 times 6%, which equals 8.40%.
- +1Covariance is 0.25 times 0.12 times 0.08, which equals 0.0024.
- +1Variance is 0.60 squared times 0.12 squared plus 0.40 squared times 0.08 squared plus the covariance cross term, which equals 0.00736. Risk is 8.58%.
- +1Sharpe ratio is 6.40% divided by 8.58%, or about 0.746. Correlation below one lowers portfolio risk.
Key terms
- Holding-Period Return
- Total investment return over the actual holding interval, including price change and income.
- Expected Return
- The probability-weighted average return across possible future states.
- Sample Variance
- Average squared sample deviation using n minus one in the denominator.
- Sharpe Ratio
- Expected excess return divided by total standard deviation.
- Risk Aversion
- The preference parameter determining how strongly variance reduces quadratic utility.
- Correlation
- Standardised covariance bounded between minus one and plus one.
Return, Risk and Investor Choice FAQ
When is holding-period return annualised?
Annualise only after calculating the total return over the stated horizon. Use the effective compounding conversion when the question asks for an annual compound rate, and retain the holding period in years.
Why does sample variance use n minus one?
Observed returns estimate an unknown population mean from the same sample. Dividing by n minus one corrects the downward bias that would arise from using deviations around the fitted sample mean.
What does a higher Sharpe ratio mean?
It means the portfolio offers more expected excess return per unit of total standard deviation under the stated inputs. It does not prove future performance or capture skewness and tail behaviour.
How does correlation create diversification?
When correlation is below plus one, the covariance cross term is smaller than under perfect co-movement. Portfolio risk can then fall below the weighted average of component standard deviations.
Why can investors choose different portfolios?
Investors with different risk-aversion coefficients apply different variance penalties in the utility function. They can rank the same risk and return combinations differently even when market inputs are identical.
Exam move
Build a formula card with units and data type. Put HPR beside the actual horizon, then list the simple and compound annual conversions. Separate ex ante probability formulas from ex post sample formulas. For variance and covariance, calculate the mean before deviations and do not annualise the observation series inside the estimator unless the question explicitly requests a scale conversion.
For portfolio problems, check that weights sum to one, convert correlation to covariance, retain squared units and take one final square root. Use two directional checks: lower correlation should not change weighted expected return, and it should not raise risk while every other input is fixed. When comparing portfolios, state whether dominance is clear or whether risk aversion is required to decide.
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