FINS5513 Lecture 2A
Risk and Return and
Risk Aversion
Lecture Outline
❑
2.1 Constructing An Investment Portfolio
➢ Portfolio Basics
➢ Themes in Portfolio Management (Active/passive, traditional/alternative, growth/value)
❑
2.2 Measuring Return and Risk
➢ Return Measurement (HPR, APR, EAR)
➢ Measuring Expected Return (ex-ante and ex-post)
➢ Measuring Risk (ex-ante and ex-post)
➢ Sharpe Ratio
❑
2.3 Risk Aversion and Investor Preference
➢ Distribution of returns
➢ Risk aversion (mean-variance criterion)
➢ Preference and utility (utility functions)
➢ Indifference curves
2
2.1 Constructing An
Investment Portfolio
FINS5513
Portfolio Basics
FINS5513
4
What is a Portfolio?
❑
A combination of multiple assets and/or securities owned by an investor
➢ The aim of owning multiple assets is to achieve diversification
❑
We will be analysing assets/securities both individually as well as in the context of a portfolio
➢ Say an investor owns stocks A and B and wishes to add C
• In an isolated approach, an investor would look at the benefits and costs of C individually
• In a portfolio approach, an investor would compare the benefits and costs of portfolio A+B
to portfolio A+B+C
❑
Investors construct portfolios to achieve diversification and avoid the risks of investing all their
capital into a single asset
➢ As we will see, diversification allows investors to reduce risk without reducing the expected
rate of return on a portfolio
5
The Investment Process
❑
The process of portfolio construction can be undertaken top-down or bottom-up
❑
Top-down portfolio construction
➢ Asset allocation – choosing between broad asset classes and determining what
proportions of the portfolio should be invested in each asset class
➢ Top-down starts with asset allocation, then we decide which securities to hold in each asset
class
Video 2AV1: “Investopedia: Strategic Asset Allocation”
❑
Bottom-up portfolio construction
➢ Security selection – choosing which individual securities to hold within each asset class
➢ Construct portfolios from securities that are attractively priced with less concern for the
resulting asset allocation
➢ May lead to being overweight or underweight certain sectors or security types
Further Reading 2AR1: Various studies
on asset allocation vs security selection
6
Themes in Portfolio Management
FINS5513
7
Key Concepts
❑
Asset/fund managers are called the “buy-side” – they buy the services of sell-side firms such
as broker/dealers who sell securities and provide research/recommendations to the buy-side
➢ A number of important buy-side themes will be explored
❑
Efficient markets
➢ Efficient markets price securities quickly and accurately incorporating all relevant
information
➢ Are markets efficient in practice?
❑
Risk-return trade-off
➢ In efficient and competitive markets higher expected return should come with higher risk
➢ Does higher return always come with higher risk in practice?
❑
Active vs Passive management
8
Active vs Passive Management
❑
Active vs Passive management
➢ Active management – attempting to improve performance by identifying mispriced
securities (through security analysis and security selection)
• Active managers attempt to outperform a prescribed market benchmark such as the
S&P500 or the ASX200
➢ Passive management – holding diversified portfolios (little time spent on security selection)
• Passive managers attempt to track a prescribed market benchmark such as the S&P500
or the ASX200
➢ If markets are efficient, why bother with active management?
• If an investor were to believe in efficient markets, the only important decision is asset
allocation, not security selection
❑
There is a significant evidence that the majority of active fund managers underperform their
benchmarks, and overall returns from actively managed funds lag wider stock indices
Further Reading 2AR2: “SPIVA 2020 Scorecards”
Video 2AV2: “Interview with Andrew Innes S&P on active vs passive performance”
9
Other Key Themes
❑
Traditional vs Alternative
➢ Traditional – long-only, unleveraged funds focused on equity, fixed income and/or balanced
(multi-asset) asset classes
• Charge management fees based on FUM
➢ Alternative – hedge funds, private equity, venture capital - often leveraged
• Charge management fees based on FUM and performance fees (or “carried interest”)
❑
Growth vs Value
➢ Growth – focuses on early stage emerging companies whose growth is expected to
significantly outperform wider industry trends. Often follows momentum and trends
➢ Value – concentrates on stocks that appear to be trading for less than their intrinsic value.
Focuses on low P/E, low Price/Book, and high free cash flow stocks
➢ Traditionally, value stocks have provided higher returns than growth, however this trend
appears to have reversed post GFC
Further Reading 2AR3: “Where’s the value in value investing?”
Video 2AV3: “Howard Marks’s thoughts on value vs growth investing”
10
Recent Trends
❑
Increase in passive investing due to lower cost and underperformance of active managers
❑
Increased variety and specialisation in ETFs eg thematic and factor based ETFs
❑
Increased use of high frequency trading and other quant methods using advanced statistical
and programming based techniques
➢ Attempt to take advantage of very short-term anomalies in the market
❑
Wider use of new data sources such as: social media; imagery and sensor data (customer
tracking, carpark monitoring, weather conditions etc); management psychological studies etc
to guide investing
❑
Robo-advisors and other algorithm-driven financial planning digital platforms (with no or little
human involvement)
11
2.2 Measuring Return and
Risk
FINS5513
Return Measurement
FINS5513
13
Holding Period Return
❑
Investor returns from holding an asset come from two basic sources:
➢ Income received periodically such as interest (debt security) and dividends (equity security)
➢ Capital gains (or losses) from the price of the asset increasing (decreasing)
❑
Holding Period Return (HPR) is the return on an asset during the period it is held
➢ The holding period ends when the asset is sold or matures/expires (for finite life assets)
Capital gain
component
𝐻𝑃𝑅 =
𝑃𝑇 − 𝑃0 + 𝐼𝑇
𝑃0
Income
component
𝑃0 = Price at the beginning of period T
𝑃𝑇 = Price at the end of period T
𝐼𝑇 = Total income received over the holding period T (eg interest, coupons, dividends)
❑
Example 2A1: You buy a share for $75 and sell it 9 months later for $84. It paid a div of $2.25:
➢
P0 = 75
PT = 84
IT = 2.25
𝐻𝑃𝑅 =
84 −75+2.25
75
= 0.15 = 𝟏𝟓%
14
APR and EAR
❑
❑
The HPR gives the total return over the holding period without regard to the time period
We can annualise the HPR in two ways:
➢ Assuming simple interest – we call this the Annualised Percentage Rate (APR)
➢ Assuming compound interest – we call this the Effective Annual Rate (EAR)
❑
Assume: T = holding period expressed in years (eg T=2 for 2-year hold; T=0.25 for 3 months;
T=5.5 for 5 years and 6 months)
❑
Annualized percentage rate (APR)
𝐴𝑃𝑅 =
❑
𝐻𝑃𝑅
𝑇
Effective Annual Rate (EAR)
𝐸𝐴𝑅 = 1 + 𝐻𝑃𝑅
1/T
-1
15
APR and EAR
❑
❑
The EAR accounts for compounding interest, not just simple interest (ie “interest on interest”)
➢ If compounding is annual: EAR = APR (at year end)
➢ If compounding is more frequent then annual: EAR > APR (at year end)
The relationship between EAR and APR is given by:
𝐴𝑃𝑅 =
1+𝐸𝐴𝑅
𝑇
𝑇
−1
❑
Yields are quoted as APRs for short-term bills/bonds (often called the bond equivalent yield)
❑
EAR is used to compare returns on investments with different time horizons
❑
Example 2A2: You invest $10,000 in a fund. With income reinvestment, your investment is
worth $16,000 after 4 years. Calculate the HPR and EAR. From your EAR calculate the APR
HPR =
➢
➢
EAR =
(1.6)1/4 –
16,000 −10,000+0
10,000
= 0.60 = 60%
1 = .1247 = 12.47% p.a. and APR =
1.1247
4
4
−1
= 15% p.a. = 0.60 / 4
Excel 2AE1: “2A - HPR, APR and EAR Calculations”
16
Other Return Calculations
❑
Gross vs net return – with regard to fund manager returns, refers to whether the fund’s return
is before fees and charges (gross return) or after fees and charges (net return)
❑
Real vs nominal return – refers to whether the return is adjusted for inflation (real) or
unadjusted for inflation (nominal)
❑
After-tax vs pre-tax return – refers to whether taxes have been deducted from the return
(after-tax) or the return is before taxes (pre-tax)
❑
Unleveraged vs leveraged return – where the investor borrows money, unleveraged returns
are calculated before deduction of interest expenses while leveraged returns are calculated
after deduction of interest expenses
❑
Absolute vs relative return – refers to the raw return of a fund (absolute) or the return
compared to the fund’s benchmark index return (relative)
17
Measuring Expected Return
FINS5513
18
Expected Return: Ex-Ante
❑
The reward from an investment is its return
➢ Since returns are generally uncertain (or “stochastic”) we deal with expected returns
❑
On a forward-looking basis under uncertainty, we form return expectations. Ex-ante analysis
is expectations-based analysis before an event
➢ Ex-ante analysis attempts to place probabilities on possible future scenarios
❑
Expected return 𝐸(𝑟) on an ex-ante basis is given by:
𝐸 𝑟 = σ𝑠 𝑝 𝑠 × 𝑟(𝑠)
p(s) = Probability of a scenario
r(s) = Return if a scenario (s) occurs
❑
r(s) can be thought of as the expected return if a particular scenario occurs
19
Expected Return: Ex-Ante
❑
Example 2A3: After extensive simulations, Quant Fund has determined that the distribution
of returns for Walmart (WMT) in different probability weighted economic future scenarios is
given by:
Economic Scenario
➢
Scenario Probability
Scenario Return
Boom
0.25
38.0%
Growth
0.50
14.0%
Flat
0.20
-7.5%
Recession
0.05
-32.0%
The expected return 𝐸(𝑟) is the probability weighted return:
𝐸(𝑟) = (.25)(.38) + (.50)(.14) + (.20)(−.075) + (0.05)(−.32)
𝐸(𝑟) = .1340 or 13.40%
Excel 2AE2: “2A – Calculating Ex-Ante & Ex-Post ER, Var & SD”
20
Expected Return: Ex-Post
❑
Estimating expected returns by projecting future scenarios can have a high level of
forecasting error
❑
Therefore, expected return is often estimated using the average (or mean) historical
(backward-looking or ex-post) sample rates of return, denoted 𝒓ത by using the formula:
𝑟ҧ =
1
𝑛
σ𝑛𝑡=1 𝑟𝑡
𝑟𝑡 = Return at time t
❑
Example 2A4: Quant Fund analysed 10-year historical returns for WMT as shown:
WMT Returns
➢
2020
2019
2018
2017
2016
2015
23.3%
30.2%
-3.4%
46.5%
16.0%
-26.6%
2014
2013
2012
2011
11.9% 18.2% 17.0% 13.9%
The expected return can be estimated from the historical average return 𝑟ҧ :
𝑟ҧ = (.233 + .302 −.034 + .465 + .16 − .266 + .119 + .182 + .17 + .139) / 10
𝑟ҧ = .147 or 14.70%
21
Measuring Risk
FINS5513
22
What is Risk?
❑
We seek to maximise return because return maximises wealth
➢ However, we seek return in a world of uncertainty
➢ Under uncertainty, we face risk
❑
In finance, risk refers to the possibility that realised outcomes differ (better or worse) from
expectation
➢ We seek not to avoid risk, but to incorporate it appropriately into decision making
❑
❑
Think of return as the “reward” and risk as the “cost” of that reward
So, how do we measure risk?
➢ In a quantitative sense, risk is a measure of the volatility of our returns
➢ So, how do we measure volatility?
Video 2AV4: RB: “Why is risk - measured by volatility - a problem for fund managers?”
23
Risk: Ex-Ante
❑
Volatility is the sum of total (squared) deviations from our expectations
➢ This is known as the variance which on an ex-ante basis is given by:
𝜎 2 = σ𝑠 𝑝 𝑠 [𝑟 𝑠 − 𝐸 𝑟 ]2
➢
To return to original units (rather than squares), we use the Standard Deviation:
𝜎=
❑
𝜎2
Example 2A5: Determine the standard deviation for Quant Fund’s ex-ante analysis of WMT
➢ Step #1 – Derive 𝐸(𝑟) = 13.40%
➢ Step #2 – Take the actual return in each scenario and subtract 𝐸(𝑟)
➢ Step #3 – Square the difference
➢ Step #4 – Multiply each scenario’s squared differences by its probability and sum them:
𝜎 2 = .25(.38 −.134)2 + .50(.14 −.134)2 + .20(− .075 −.134)2 + .05(−.32 −.134)2 = 0.034
➢
Step #5 – Standard deviation is the square root of the variance: 𝜎 =
0.034 = 𝟏𝟖. 𝟒𝟗%
24
Risk: Ex-Post
❑
We can also use historical (ex-post) data to estimate the risk
❑
When conducting ex-post (backward-looking) analysis, each historical data point is
considered equally likely and therefore we do not probability weight them. However, we
divide by n – 1 (rather than n) to account for estimation error as 𝒓ത is only an estimation of 𝐸(𝑟)
❑
The unbiased standard deviation estimate 𝜎ො is given by:
𝜎ො =
❑
σ𝑛
𝑡=1 𝑟𝑡 − 𝑟ҧ
2
𝑛−1
Example 2A6: Determine the standard deviation for Quant Fund’s ex-post analysis of WMT
➢ 𝜎
ො 2 = [ (.233 - .147)2 + (.302 - .147)2 + (−.034 - .147)2 + (.465 - .147)2 + (.16 - .147)2 +
(−.266 - .147)2 + (.119 - .147)2 + (.182 - .147)2 + (.17 - .147)2 + (.139 - .147)2 ] / 9
= 0.3386 / 9 = 0.0376
➢
𝜎ො =
0.0376 = 𝟏𝟗. 𝟒𝟎%
Excel 2AE2: “2A – Calculating Ex-Ante & Ex-Post ER, Var & SD”
25
Sharpe Ratio
FINS5513
26
Reward to Volatility (Sharpe) Ratio
❑
Now that we have quantified return and risk individually, how do we relate the risk/reward
relationship in one measure?
➢ Divide return (the “reward”) by risk (the “cost”) and state it as a ratio
➢ This reward-to-risk ratio is often called the Sharpe ratio
❑
We often look at the “excess return” above the risk-free rate, rather than the total return
➢ This is because part of the return can be earned for no risk by investing in a risk-free asset
➢ On a forward looking basis we often refer to the expected excess return above the risk-free
rate for a risky asset as the Risk Premium
❑
The Sharpe Ratio is given by:
Sharpe ratio for security i : 𝑆𝑖 =
𝐸 𝑟𝑖 − 𝑟𝑓
𝜎𝑖
𝑟𝑓
= risk-free rate
𝐸 𝑟𝑖 − 𝑟𝑓 = Risk premium for security i
𝜎𝑖
= Standard deviation of excess returns for security i
27
The Importance of the Sharpe Ratio
❑
❑
The Sharpe ratio measures return per unit of risk. The higher the Sharpe ratio - the higher
the incremental return received per unit of risk
➢ In other words, the higher the Sharpe ratio the better (the more attractive the investment)
➢ As the Sharpe ratio is straight forward to calculate and easy to interpret, it is one of the most
widely used appraisal measures for assessing risk against reward
• However, within a portfolio context it does have limitations which we will explore later
Example 2A7: Determine the Sharpe ratio for both Quant Fund’s ex-ante and ex-post
analysis of WMT. Assume a risk-free rate of 3.0%
➢
Ex-ante: SWMT =
.1340 − .03
= 0.562
.1849
➢
Ex-post: 𝑆መ WMT =
.1470 − .03
= 0.603
.1940
Further Reading 2AR4: “The Sharpe Ratio Broke Investors’ Brains”
28
2.3 Risk Aversion and
Investor Preference
FINS5513
Distribution of Returns
FINS5513
30
Which Fund is Preferred?
❑
Consider two funds: All Weather (AW) and Traditional Portfolio (TP)
➢ AW and TP have the same expected return 𝐸(𝑟) of 10%, but which would you choose?
Probab ility
0.2
s AW =5%
AW A
0.15
0.1
TPB
0.05
s TP =10%
0
0
5
10
15
20
Retu rn (%)
E( r AW) = E (rTP) = 10%
❑
TP has a much wider dispersion of returns - as reflected in the higher 𝜎
31
Which Fund is Preferred?
❑
❑
❑
Assume both fund returns are normally distributed
Let’s say we are judging both funds by the probability they will make a negative return
➢ Probability AW will return less than 0%:
Prob(rAW < 0%) = N[(0% - 10%) / 5%]
= N(-2.0) = 2.3%
A negative return is 2 standard deviations from the mean which has a probability of 2.3%
➢ Probability TP will return less than 0%
Prob(rTP<0%) = N[(0% - 10%) / 10%]
= N(-1.0) = 15.9%
A negative return is 1 standard deviation from the mean which has a probability of 15.9%
TP Fund is riskier - risk averse investors would prefer AW
32
Normal Distributions
❑
❑
Investment management is simplified when returns (which are uncertain) are approximated
as a normal distribution:
➢ Normal distribution assumes returns are symmetric around the mean
➢ Under symmetric returns, standard deviation is an effective measure of risk
➢ Future return probabilities can be estimated using only mean and standard deviation
➢ Interdependence of returns between securities can be estimated by their correlations
If returns are not normally distributed, standard deviation is no longer a complete measure of
risk and we must also consider skewness and kurtosis
Further Reading
Skew
Kurtosis
2AR5: “How ‘Tail Risk’
changes over the
market cycle”
Video 2AV5:
“Nassim Taleb - What is
a "Black Swan?”
33
Risk Aversion
FINS5513
34
Mean-Variance Criterion
❑
❑
Mean-variance analysis requires the mean and standard deviation of returns
➢ We graph expected return (y-axis) against standard deviation (x-axis)
The Mean-Variance Criterion states:
Portfolio A dominates portfolio B if:
E (rA )  E (rB )
and*
sA sB
* At least one inequality must be strict to rule out indifference
Expected Return
2
1
4
3
➢
2 dominates 1 - higher return
➢
2 dominates 3 - lower risk
➢
4 dominates 3 - higher return
Variance or Standard Deviation
35
Risk Aversion
❑
Modern Portfolio Theory rests on the assumption investors are risk averse:
➢ Risk averse investors follow the mean-variance criterion – for the same level of 𝐸(𝑟), they
will choose the asset with the lowest risk
➢ Risk neutral investors judge assets solely by their 𝐸 𝑟 and are indifferent to risk
➢ Risk seekers prefer higher levels of risk
❑
Historical market returns show there is a risk premium in the market (indicates risk aversion):
➢ Since 1926, U.S. risk-free assets (1-month T-bills) returned ~3.4% annually while risky
assets (US stocks) returned ~11.7% – resulting in a ~8.3% risk premium with 𝜎 = 20.4%
➢ Market takes additional risk only for a commensurate return – indicating risk aversion
❑
What happens when return increases with risk?
➢ Individual investors have different degrees of risk aversion
➢ It will depend on each investor’s individual risk-reward trade-off
➢ Hence the need to understand investor preference and utility
Further Reading 2AR6: “Australian Investor Study 2020” Figures 20, 22, 48, 49, 52-55
36
Preference and Utility
FINS5513
37
Preference and Utility
❑
Utility is a measure of satisfaction / welfare / happiness of an investor
❑
When an investor prefers Asset A over Asset B, we say that Asset A provides the investor with
greater utility
❑
In order to work with preferences mathematically, we use utility functions
➢ A utility function assigns a value to each outcome so that preferred outcomes get higher
utility values
❑
For simplicity, the fundamental assumption in finance is that utility is derived from wealth
➢ We assume that the more money an investor has, the better their ability to achieve
preferred outcomes
➢ Therefore, in finance:
• More is better – maximise return which maximises wealth
• More certainty is better – risk aversion
38
Logarithmic Wealth Utility Function
❑
A common specification of the wealth utility function is U(W) = ln(W)
The logarithmic expression results in a concave function
➢ The concavity indicates that the incremental utility we gain from increases in wealth is less
than the utility we lose from equivalent decreases in wealth
➢ The concavity captures risk aversion – risk averse investors would not take a 50/50 bet
4.5
Utility curve
4
3.5
Wealth level: average of A and B
Wealth level B: 41
3
Utility
❑
2.5
2
Average utility from A and B
1.5
1
0.5
Wealth level A: 1
0
1
11
21
31
41
Wealth
39
Risk-Reward Trade-off
❑
Choosing the preferred asset when one dominates is straight forward
❑
But what about where no asset dominates?
Fund
Expected Return
Risk Premium
(rf = 5%)
Risk
σ
Low-Risk
7.0%
2.0%
5.0%
Medium-Risk
9.0%
4.0%
10.0%
High-Risk
13.0%
8.0%
20.0%
❑
In the example, return increases but so does risk (all funds have the same Sharpe ratio)
❑
Each portfolio receives a utility score indicating the investor’s risk/return trade-off
❑
The portfolio with the highest utility score is preferred
40
Utility Function
❑
What is a reasonable method for determining a utility score?
❑
As wealth is dependent on risk and return, we derive a utility function based on risk and return
❑
For investments, we assume a quadratic utility function:
U = E ( r ) − 1 As 2
2
U = Utility
A = Coefficient of risk aversion (a constant)
½ = A scaling factor
❑
For a risk-free asset, U = r, as r is a known constant and 𝜎2 = 0
➢ Therefore, what is the meaning of the utility score U for a risky investment?
• It is the risk-free rate which would result in an investor being indifferent between the riskfree asset and a risky investment with the same utility score – often called the certainty
equivalent return
41
Estimating Risk Aversion
❑
❑
❑
For each individual investor, the unique element in the utility function is the value of A
So how do we estimate an individual’s risk aversion coefficient?
➢ Often depends on life cycle and personality type
➢ Questionnaires
➢ Discussion with broker/advisor
➢ Observe how much people are willing to pay to avoid risk
➢ Observe individuals’ decisions when confronted with risk
• Would you take $100 for certain or flip of a coin for $200
Note that the higher the risk aversion coefficient A the more risk averse the investor:
➢
➢
Conservative investors have high risk aversion coefficients
Aggressive investors have low risk aversion coefficients
Further Reading 2AR7: “Wealth Management Risk Profile Questionnaire”
42
Example: Applying the Utility Function
❑
Example 2A8: Three investors are analysing the Low-Risk, Medium-Risk and High-Risk
funds from earlier. For an Aggressive investor, the risk aversion coefficient A = 2.0; Moderate
investor A = 3.5; Conservative investor A = 5.0. Which fund would each investor choose?
➢
Replace the risk aversion coefficients and fund return and risk into each utility function.
Then rank each fund based on its utility score:
Investor
❑
Risk
Aversion
A
Low Risk Fund
Utility Score
E(r) = 7% σ = 5%
Medium-Risk Fund
Utility Score
E(r) = 9% σ = 10%
High-Risk Fund
Utility Score
E(r) = 13% σ = 20%
Aggressive
2.0
.07 – ½ x 2.0 x .052 = .0675 .09 – ½ x 2.0 x .12 = .0800
.13 – ½ x 2.0 x .22 = .090
Moderate
3.5
.07 – ½ x 3.5 x .052 = .0656
.09 – ½ x 3.5 x .12 = .0725
.13 – ½ x 3.5 x .22 = .060
Conservative
5.0
.07 – ½ x 5.0 x .052 = .0638
.09 – ½ x 5.0 x .12 = .0650
.13 – ½ x 5.0 x .22 = .030
Aggressive investor chooses the High-Risk fund, the others choose the Medium-Risk fund
➢ Risk aversion doesn’t mean the investor doesn’t take risk – rather it means the investor puts
a higher price (return) on taking risk. For example: even the conservative investor does not
pick the Low-Risk fund
43
Indifference Curves
FINS5513
44
Indifference Curves
❑
We can illustrate our preferences through indifference curves
➢ Plotted in the risk-return (𝐸 𝑟 − 𝜎) space that connect points giving equal utility
➢ For example, to draw the indifference curve for U = 10%, choose all asset portfolio
combinations of E(r) and σ which yield a utility score of 10%
➢
➢
Graphical representation of the utility function. Called an Indifference “Curve” because the
utility function is a quadratic equation
Note that two indifference curves with different utility levels never intersect
U = 25%
❑
U = 20%
E(r)
U = 15%
U = E(r) – ½* A*σ2 = 10%
σ
Indifference curves for riskaverse (A > 0) investors are
upward sloping
45
How to Plot an Indifference Curve
❑
Each plot point on an indifference curve represents a risk and return combination which
provides the same utility score
❑
Example 2A9: For an Aggressive investor with risk aversion coefficient A=2, and a
Conservative investor with A=5, plot two indifference curves with U=0.03 and U=0.09
➢
➢
Given a specific value of A,
indifference curves above and to
the left offer higher utility than lower
curves and don’t intersect
More risk averse investors (A)
have steeper indifference curves
(higher E(r) for each increase in 𝜎)
A=5 is
steeper
than A=2
Certainty
equivalent
return.
Plot first.
Higher Utility
Excel 2AE3: “2A – Indifference Curves”
Video 2AV5: “How to plot a simple
Indifference Curve”
46
Next Lecture
❑
BKM Chapter 6 and 7
❑
2.4 Introduction to Modern Portfolio Theory (MPT)
❑
2.5 Optimal Risky Asset Portfolio Construction
47