Lesson 3 Redux

Why is standard deviation used as a measure of risk?

Standard deviation measures the dispersion of returns around the expected return. Greater dispersion implies greater uncertainty and therefore greater investment risk.

What does a Sharpe Ratio of 1.5 indicate?

The investment generates 1.5 units of excess return for every 1 unit of risk taken. A higher Sharpe Ratio indicates superior risk-adjusted performance.

Why is the normal distribution attractive in portfolio theory?

The normal distribution allows returns to be fully described using only the mean and standard deviation, making risk measurement, forecasting and portfolio optimisation more manageable.

What are Expected Returns?

Expected returns are the probability-weighted average return expected from an investment across all possible future scenarios.

What is the most common measure of risk in Modern Portfolio Theory?

The standard deviation of returns.

What is the risk-free rate and what is usually used as its proxy?

The risk-free rate is the rate of return that can be earned with certainty. It is commonly proxied using Treasury Bills (T-bills) or short-term government securities.

What is a risk premium?

A risk premium is the additional expected return above the risk-free rate that investors require as compensation for bearing risk.

What is excess return?

Excess return is the difference between the actual return earned and the risk-free rate.

What is the difference between risk premium and excess return?

Risk premium refers to the expected additional return required for taking risk.

Excess return refers to the realised return earned above the risk-free rate.

What is risk aversion?

Risk aversion is the tendency of investors to prefer lower risk to higher risk for a given level of expected return and to require compensation in the form of a risk premium for accepting additional risk.

Why is the Sharpe Ratio useful?

The Sharpe Ratio allows investors to compare investments on a risk-adjusted basis by measuring excess return earned per unit of risk.

Give four reasons why investment management is more manageable when returns can be approximated by the normal distribution.

• The distribution is symmetric.

• The distribution is stable.

• Only the mean and standard deviation are required to describe returns.

• Relationships between returns can be summarised using a single correlation coefficient.

What is a criticism of using normal distributions in Modern Portfolio Theory?

Real-world financial returns are often not perfectly normally distributed. Extreme market events and crashes occur more frequently than predicted by the normal distribution, which can lead to an underestimation of risk.

How realistic does the normality assumption need to be before we use it?

The normal distribution is useful because it greatly simplifies portfolio analysis and forecasting. However, investors must recognise that real-world returns may deviate from normality and exhibit more extreme outcomes than predicted by the model.

What is the difference between expected return and holding period return?

Expected return is the probability-weighted average return expected in the future.

Holding period return is the actual return realised over the investment holding period.

Why is standard deviation used as a measure of risk?

Standard deviation measures the dispersion of returns around the expected return. Greater dispersion implies greater uncertainty and therefore greater investment risk.

What does a Sharpe Ratio of 1.5 indicate?

The investment generates 1.5 units of excess return for every 1 unit of risk taken. A higher Sharpe Ratio indicates superior risk-adjusted performance.

Why is the normal distribution attractive in portfolio theory?

The normal distribution allows returns to be fully described using only the mean and standard deviation, making risk measurement, forecasting and portfolio optimisation more manageable.

What is the equation for Holding Period Return (HPR)?

HPR = (P₁ − P₀ + D) / P₀

Where:

P₀ = Initial price

P₁ = Final price

D = Dividends received

Suppose you bought XYZ shares at £100 three years ago. You sold the shares today for £250 and received £10 in dividends. Calculate the HPR.

HPR = (250 − 100 + 10) / 100

HPR = 160 / 100

HPR = 1.60

HPR = 160%

What is the equation for Expected Return?

E(r) = Σ [p(s) × r(s)]

Where:

p(s) = Probability of scenario s

r(s) = Return in scenario s

s = Scenario

Calculate the expected return using the following data:

Scenario | Probability | Return
Boom | 30% | 20%
Normal | 50% | 10%
Recession | 20% | -5%

A:

E(r) = 0.30(20) + 0.50(10) + 0.20(-5)

E(r) = 6 + 5 - 1

E(r) = 10%

What is the equation for variance?

• σ² = Σ [ p(s) × ( r(s) − E(r) )² ]

Where:

• σ² = Variance

• p(s) = Probability of scenario s

• r(s) = Return in scenario s

• E(r) = Expected return

• Σ = Sum across all possible scenarios

What is the equation for standard deviation?

σ = √Var(r)

What is the equation for Risk Premium?

A:

Risk Premium = E(r) − rf

Where:

E(r) = Expected return

rf = Risk-free rate

Calculate the risk premium.

Expected return = 10%

Risk-free rate = 3%

Risk Premium = 10% − 3%

Risk Premium = 7%

Meaning investors require 7% additional expected return for accepting risk.

What is the equation for Excess Return?

Excess Return = Actual Return − Risk-Free Rate

What is the equation for the Sharpe Ratio?

Sharpe Ratio = (E(r) − rf) / σ

Where:

E(r) = Expected return

rf = Risk-free rate

σ = Standard deviation of returns

Interpretation:

Measures excess return earned per unit of risk.

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Calculate the Sharpe Ratio.

Expected Return = 12%

Risk-Free Rate = 3%

Standard Deviation = 15%

Sharpe Ratio = (0.12 − 0.03) / 0.15

Sharpe Ratio = 0.09 / 0.15

Sharpe Ratio = 0.60

Which portfolio is better according to the Sharpe Ratio?

Fund A:
Return = 15%
Risk = 30%

Fund B:
Return = 12%
Risk = 10%

Assuming a risk-free rate of 0%:

Sharpe A = 15/30 = 0.50

Sharpe B = 12/10 = 1.20

Fund B is preferred because it generates more return per unit of risk.

An investment has the following possible returns:

Scenario | Probability | Return
Boom | 0.25 | 20%
Normal | 0.50 | 10%
Recession | 0.25 | -4%

Calculate:

1. Expected return

2. Variance

3. Standard deviation

Expected Return:

E(r) = 0.25(20) + 0.50(10) + 0.25(-4)

E(r) = 9%

Variance:

Var(r) = 0.25(20 − 9)² + 0.50(10 − 9)² + 0.25(-4 − 9)²

Var(r) = 72

Standard Deviation:

σ = √72

σ = 8.49%

How do risk-averse investors react to a fair game, and what is a fair game?

A risk-averse investor will reject a fair game because it provides no compensation for bearing risk.

A fair game is a risky investment with a risk premium of zero.

Mathematically:

E(r) = rf

The expected return is equal to the risk-free rate.

In terms of risk and return, what makes a portfolio more attractive?

A portfolio becomes more attractive when:

• Expected return increases

• Risk decreases

How can risk aversion be characterised?

Risk aversion can be characterised through the utility function, which quantifies the trade-off between expected return and risk.

In the following example, which investment would a risk-averse investor choose?

Investment | Return
Treasury Bill | 4%
Risky Share | Expected 4%

A risk-averse investor would choose the Treasury Bill because the risky share provides no additional expected return to compensate for the extra risk.

If Portfolio A has a higher return but also higher risk than Portfolio B, which portfolio is better?

There is no universal answer.

Different investors have different attitudes toward risk. Utility scores are used to determine which portfolio is preferred by a particular investor.

What is utility?

Utility is a numerical measure of investor satisfaction or welfare.

Investors prefer portfolios with higher utility values.

Why is utility used in portfolio theory?

Utility is used to quantify the trade-off between risk and return and to compare portfolios with different levels of risk and expected return.

What is the relationship between utility, expected return and volatility?

Utility increases as expected return increases and decreases as volatility (risk) increases.

What is the equation for the utility function?

U = E(r) − ½Aσ²

Where:

U = Utility

E(r) = Expected return

A = Risk aversion coefficient

σ² = Variance

What is the relationship between utility, expected return, risk and risk aversion?

Utility:

• Increases with expected return

• Decreases with risk (variance)

• Decreases more rapidly when the investor is highly risk averse

What is meant by a high, small and zero value of risk aversion coefficient (A)?

High A:
Indicates a highly risk-averse investor who places a large penalty on risk and prefers safer portfolios.

Small Positive A:
Indicates a less risk-averse investor who is more willing to accept risk for higher expected returns.

A = 0:
Indicates a risk-neutral investor who only cares about expected return and is indifferent to risk.

In terms of A, give a numerical representation and description of a risk-averse investor.

A > 0

Risk-averse investors require compensation in the form of a risk premium before accepting risk.

In terms of A, give a numerical representation and description of a risk-neutral investor.

A = 0

Risk-neutral investors care only about expected return and consider risk irrelevant.

In terms of A, give a numerical representation and description of a risk-loving investor.

A < 0

Risk-loving investors may prefer riskier investments even if expected returns are lower.

What does a risk-neutral investor care about?

A risk-neutral investor cares only about expected return and is indifferent to risk.

What does a risk-loving investor care about?

A risk-loving investor derives satisfaction from taking risk and may prefer riskier investments even when expected returns are lower.

What is the risk-return trade-off?

The risk-return trade-off is the principle that higher expected returns generally require accepting higher levels of risk.

Investors demand additional expected return as compensation for bearing additional risk.

Why do risky assets generally offer higher expected returns?

Because investors require compensation in the form of a risk premium for bearing risk.

What is the Mean-Variance Criterion?

The Mean-Variance Criterion is the selection of portfolios based solely on expected return and variance.

Investors choose:

• The highest expected return portfolio for a given level of variance, or

• The lowest variance portfolio for a given level of expected return.

What are the requirements for Portfolio A to dominate Portfolio B?

E(rA) ≥ E(rB)

and

σA ≤ σB

with at least one strict inequality.

Meaning Portfolio A offers:

• Higher return for the same risk
  OR

• Lower risk for the same return
  OR

• Both

What does it mean for Portfolio A to dominate Portfolio B?

Portfolio A dominates Portfolio B if it offers:

• Higher expected return for the same level of risk, or

• Lower risk for the same expected return,

with at least one strict advantage.

A rational investor would always prefer Portfolio A.

What is portfolio dominance?

Portfolio dominance occurs when one portfolio offers a superior risk-return combination compared with another portfolio, making it preferable to all rational investors.

What is shown by an indifference curve?

An indifference curve shows all combinations of risk and expected return that provide the same level of utility to an investor.

What is an indifference curve?

An indifference curve connects all portfolios that provide the same utility value to an investor.

Every point on the curve is equally preferred.

If you pick two different points on the same indifference curve, how would you calculate the difference in utility?

You would not calculate a difference because all points on the same indifference curve have identical utility values.

Difference in utility = 0

Why does an indifference curve slope upwards?

An increase in risk must be compensated by an increase in expected return to maintain the same level of utility.

Why does the gradient of an indifference curve increase as risk increases?

As risk increases, investors require increasingly larger increases in expected return to maintain the same level of utility.

This reflects risk aversion.

Why does the indifference curve become steeper as risk increases?

As risk increases, investors demand increasingly large increases in expected return to compensate for additional risk and maintain utility.

What happens to utility when variance increases?

Utility decreases as variance increases because higher variance represents greater risk.

The reduction in utility is larger for investors with higher levels of risk aversion.

What does the coefficient A represent in the utility function?

The coefficient A represents the investor's degree of risk aversion.

Larger values of A indicate greater risk aversion.

Smaller values of A indicate lower risk aversion.

Why would a risk-averse investor reject a fair game?

A risk-averse investor rejects a fair game because it provides no risk premium.

The investor bears risk without receiving any additional expected return above the risk-free rate.

Give three ways to estimate an investor's degree of risk aversion.

• Risk tolerance questionnaires

• Observing changes in portfolio composition over time

• Using average risk aversion estimates from groups of investors

Three different investors have utility scores of 2, 3.5 and 5 respectively. Using a utility table, how would you identify the most appropriate portfolio for each investor?

Choose the portfolio that provides the highest utility score for that particular investor.

Different investors may prefer different portfolios depending on their level of risk aversion.

(Note: this question requires a utility table to calculate a specific answer.)

How does a portfolio manager control portfolio risk?

A portfolio manager can control portfolio risk by changing the proportion of funds invested in risky assets versus risk-free assets.

What is a complete portfolio?

A complete portfolio is a combination of a risky portfolio and a risk-free asset.

What is the investment opportunity set?

The investment opportunity set is the set of all feasible combinations of expected return and standard deviation available to investors.

It represents every possible portfolio that can be constructed.

What is shown by the Capital Allocation Line (CAL) and how is it formed?

The Capital Allocation Line (CAL) is a graph showing all attainable combinations of risk and return formed by combining a risky portfolio with a risk-free asset.

What do different points along the CAL represent?

Each point represents a feasible portfolio with a different value of y, the proportion invested in the risky portfolio.

What is the slope of the CAL?

The slope of the CAL is equal to the reward-to-volatility ratio, also known as the Sharpe Ratio.

Why is the slope of the CAL equal to the Sharpe Ratio?

The slope measures the increase in expected return obtained for each additional unit of risk taken.

This is exactly the definition of the Sharpe Ratio:

Sharpe Ratio = (E(rP) − rf) / σP

What does y represent?

‘y’ represents the proportion of wealth invested in the risky portfolio.

What does (1 − y) represent?

(1 − y) represents the proportion of wealth invested in the risk-free asset.

What does y > 1 imply?

It implies that the investor is borrowing money at the risk-free rate and using leverage to invest more than 100% of their wealth in the risky portfolio.

How can an investor invest more than 100% in risky assets?

By borrowing money and using leverage so that y > 1.

What is leverage?

Leverage is the use of borrowed funds to increase investment exposure beyond the investor's own capital.

What is the relationship between optimal allocation to risky assets and risk premium?

As risk premium increases, the optimal allocation to risky assets increases.

What is the relationship between optimal allocation to risky assets and risk aversion?

As risk aversion increases, the optimal allocation to risky assets decreases.

What is the relationship between optimal allocation to risky assets and portfolio risk?

As portfolio risk increases, the optimal allocation to risky assets decreases.

What happens to optimal allocation to risky assets when risk aversion doubles?

The optimal allocation to risky assets decreases.

More risk-averse investors invest less in risky assets.

What happens to optimal allocation to risky assets when risk premium rises?

The optimal allocation to risky assets increases.

What factors increase y*?

The optimal allocation to risky assets (y*) increases when:

• Risk premium increases

• Risk aversion decreases

• Portfolio variance decreases

What factors decrease y*?

The optimal allocation to risky assets (y*) decreases when:

• Risk premium decreases

• Risk aversion increases

• Portfolio variance increases

What is the difference between an indifference curve and the CAL?

Indifference Curve:
Represents investor preferences and utility levels.

CAL:
Represents available investment opportunities.

How does one identify the optimal portfolio?

The optimal portfolio occurs where the highest attainable indifference curve is tangent to the CAL.

What is the Capital Market Line (CML)?

The Capital Market Line (CML) is the Capital Allocation Line formed when the market portfolio is used as the risky portfolio.

What is the difference between the CAL and the CML?

CAL:
Can be formed using any risky portfolio and a risk-free asset.

CML:
Is a special CAL formed using the market portfolio and a risk-free asset.

How is the market portfolio approximated in practice?

The market portfolio is commonly approximated using:

• Broad stock market indices

• S&P 500

• FTSE All Share

• Global market indices

What is a passive investment strategy?

A passive investment strategy avoids direct security analysis and instead invests in a broad market portfolio or index.

Why might investors choose a passive strategy?

Passive strategies are inexpensive, well diversified and avoid the costs associated with active security selection.

What is the free-rider benefit?

Passive investors benefit from information incorporated into market prices by active investors without bearing the costs of generating that information.

What is the equation for the expected return of a complete portfolio?

E(rC) = yE(rP) + (1 − y)rf

Where:

E(rC) = Expected return of complete portfolio

E(rP) = Expected return of risky portfolio

rf = Risk-free rate

y = Weight invested in the risky portfolio

A risky portfolio has a standard deviation of 20%.

The weight invested in the risky portfolio is 0.7.

Calculate portfolio risk.

σC = yσP

σC = 0.7 × 20

σC = 14%

Why does the equation for complete portfolio risk not include the risk-free asset?

The risk-free asset has a standard deviation of zero and therefore contributes no risk to the portfolio.

What is the equation for the risk of a complete portfolio?

σC = yσP

Where:

σC = Risk of complete portfolio

σP = Risk of risky portfolio

y = Weight invested in risky portfolio

A portfolio has the following attributes:

Risky portfolio return = 10%

Risk-free rate = 2%

70% invested in risky portfolio

30% invested in risk-free asset

Calculate the expected return of the complete portfolio.

E(rC) = 0.7(10) + 0.3(2)

E(rC) = 7 + 0.6

E(rC) = 7.6%

What is the equation for the optimal allocation to risky assets?

y* = (E(rP) − rf) / (Aσ²P)

Where:

y* = Optimal allocation to risky assets

E(rP) = Expected return of risky portfolio

rf = Risk-free rate

A = Risk aversion coefficient

σ²P = Variance of the risky portfolio

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Calculate the optimal allocation to risky assets.

Expected return = 12%

Risk-free rate = 4%

Risk aversion coefficient = 4

Standard deviation = 20%

Variance = (0.20)²

Variance = 0.04

y* = (0.12 − 0.04)/(4 × 0.04)

y* = 0.08/0.16

y* = 0.50

Optimal allocation to risky assets = 50%