Risk, Return and Utility - Characterising Risky Investments

We begin our study of portfolio theory by introducing the measures we will use for the risk of a portfolio (or individual asset) and the reward we believe we will receive for bearing that risk. We will then use these measures to develop a utility function that will serve as our way of unambiguously ranking competing risky investments. With this utility function at our disposal, we can then formally compute optimal portfolios. We will address this problem from the simplified setting of a single risky portfolio (or asset) and a single risk-free asset.

1. Gambling vs Speculating

Before we formally define our risk and reward measures, we first wish to understand different attitudes that investors can take to risk they face. To illustrate this discussion, we will consider the difference between a speculator and a gambler. Gamblers accept risk because they enjoy the risk itself. Consider the patron of a casino. At no stage can they truly believe that they have a better than equal chance of winning (the casino games are set up to ensure this) yet these patrons will continue to bet (invest) their wealth. The reason they do this is that the risk of the gamble provides happiness (what we will later call utility) to the gambler. Gamblers do not expect to be rewarded financially for bearing risk, the risk itself is their compensation. Later, we will call this kind of behavior irrational.

Speculators on the other hand do not behave in this way. In fact, a true speculator would not gamble at a casino. Speculators will accept risk if they believe that they will be financially compensated for it. This means that a speculator could still engage in what might be traditionally considered gambling. For example, a speculator could bet on horse races (or other sporting events). If they have a strong enough understanding of the horses, the track, its conditions, and a myriad of other factors, they may be able to determine when they can place bets and expect to make money on average. This would occur when the odds posted by book makers do not reflect the probabilities that the speculator believes to apply. We say that investors act rationally because they only accept risk if they expect their reward to be sufficiently high to overcome the risk of the bet (investment).

If speculators only place bets when they have an edge (probabilities in their favor), we then ask if speculators can gamble against one another? The answer to this question is yes and it is due to disagreement. If two speculators have different sets of beliefs, they may both believe they have an edge against the other speculator. The reason for this is that it is extremely difficult to ascertain any true probabilities in financial markets. This means that there is a lot of room for disagreement between investors which can lead to speculators.

To illustrate this point, consider a bet that two speculators make about whether a stock price will go up or down. If the stock goes up, agent $A$ must pay \$1 to agent $B$ and the opposite happens if the stock price goes down. If both agents believed the probability that the stock goes up/down is 50\%, then neither agent would take this bet on. This is because the expected payoff for each agent, $\mathbb{E}[P_{i}]$, $i \in {A, B}$, is given by

\begin{align*} \mathbb{E}[P_{A}] = p_{up}\times -1 + p_{down}\times1 = \tfrac{1}{2}(1 - 1) = 0\\ \mathbb{E}[P_{B}] = p_{up}\times 1 + p_{down}\times-1 = \tfrac{1}{2}(1 - 1) = 0 \end{align*}

and hence these agents do not receive compensation (in the form of a positive expected payoff) for taking on this bet. The condition where investors all agree on the probabilities of potential outcomes is called homogeneous expectations. This assumption is a key aspect to asset pricing theories such as the CAPM. If these agents disagree about the probabilities, they may still act rationally and take this gamble on. For example, say agent $A$ believes that $p_{down} > 0.5$ and agent $B$ believes that $p_{up} > 0.5$. Then each agents expected payoff is greater than 0 and they may take the gamble on as each expected to gain from the risk. This situation, where investors disagree about probabilities, is called heterogeneous expectations.

2. Utility and Risk Aversion

We said in the previous section that speculators may take a risky gamble (investment) on if the expected payoff is positive. However, they are not guaranteed to take such an investment on. They will only engage in this investment if the expected payoff is sufficiently large to overcome their aversion to risk. To quantify how an investor is going to behave, we need to quantify the measure of risk, reward and aversion to risk.

We assume that investors are only concerned with two aspects of the distribution of returns. They are concerned with its central tendency, the expected return. They are also concerned with how wide (symmetrically) the distribution is, which is measured by its standard deviation (equivalently variance). This is the investors measure of risk. No other aspects of the assets return distribution is considered by the investor. While this is not a perfect description of reality, for many purposes these assumptions are sufficient to capture the main ideas of how an investor should think about optimally constructing a portfolio. Given the investor is only concerned with the expected return and its variance, this is mathematically equivalent to the investor believing that returns are Normally distributed as this distribution is governed by only these two parameters. Alternatively, we could say that investors know that returns are not Normally distributed, but they do not care about higher order moments (skewness, kurtosis, etc.). Both scenarios are equivalent in terms of the technical treatment.

The way in which we determine which portfolio is best for an investor is via a utility function. A utility function essentially assigns a single number to a portfolio which measures how much happiness (or in economic parlance, how much utility) the portfolio provides. It is hence a function of both the portfolios and investors characteristics. While there are a variety of possible utility functions available, the one that has been used to develop portfolio theory is called quadratic utility. The quadratic utility function is given by

\begin{align} U = \mathbb{E}[r_{p}] - \tfrac{1}{2}A\sigma_{p}^{2} \label{utility function} \tag{2.1} \end{align}

The inputs are:

  • $\mathbb{E}[r_{p}]$ - the expected return on the portfolio (reward)
  • $\sigma_{p}^{2}$ - the variance of the portfolio (risk)
  • $A$ - the risk aversion of the investor

and the output is a number (scalar) $U$. Note that $U$ is unitless. This means that the number itself has no meaning, but we can compare utilities across portfolios. This means that we can use utilities to rank competing investments or, as we will see later, we can select portfolio weights to maximize the utility of a portfolio.

The risk aversion parameter $A$ captures how averse to risk the investor is. For rational investors, $A > 0$. These investors do not like risk and will only take it on if the expected reward is sufficiently high. Irrational investors have $A < 0$. Their utility associated with a portfolio increases as the risk increases. If $A = 0$, we say that the investor is risk neutral. Decisions they make are not based on risk at all. We can rank competing investments and calculate certainty equivalent via the utility function. A certainty equivalent tells us how much riskless return $r_{ce}$ you require to be indifferent between earning $r_{ce}$ and a risky investment.

Example 2.1.

Consider an investor with a risk aversion $A = 3$. What is the guaranteed rate of return (certainty equivalent) that is required to induce them away from a portfolio which has an expected return of 12\% and a standard deviation of 20\%?

First, compute the utility of the risky investment:

\begin{align*} U_{risky} = 0.12 - \tfrac{1}{2}\times 3\times 0.2^{2} = 0.06 \end{align*}

Now, set the utility of a risk-free investment equal to the utility just found:

\begin{align*} U_{risky} = 0.06 = \mathbb{E}[r_{riskless}] - \tfrac{1}{2}\times 3\times 0 = 0.06 \end{align*}

Hence, this investor is indifferent between the risky investment and a riskless return of 6\%. If the risk-free $r_{f} > 0.06$, then the investor will always select the risk-free investment.

3. Indifference Curves

Indifference curves are a graphical way of examining the utility function. They are defined as the set of points, $(\mathbb{E}[r_{p}], \sigma_{p})$ that produce a constant level of utility for a given value of the risk aversion $A$. This means that we can now see all portfolios for which this investor would be equally happy.

Python example 2.1. Indifference curve

In [22]:
import numpy as np
import matplotlib.pyplot as plt

# Set the parameters
A = 3                            # Risk aversion
U = 0.06                         # Utility
sigma = np.linspace(0,0.5,100)   # Generate values of sigma
Er = 0.5*A*sigma**2 + U          # Compute associated expected return

# Add in elements from the previous example
Er_eg = 0.12
sigma_eg = 0.2

# Plot
plt.plot(sigma, Er)
plt.plot(sigma_eg, Er_eg, 'ro')
plt.grid()
plt.xlabel('$\sigma$')
plt.ylabel('E[r]')
plt.show()

Examining the indifference curve from the above example, we can make some general statements about which portfolios are preferred. To see this, we split the plot at a given portfolio, $P$, identifying four quadrants (Q1, Q2, Q3, Q4)

All portfolios in Q1 are preferable to $P$ (lower risk and higher reward) while $P$ is preferable to all portfolios in Q4 (higher risk and lower reward). In quadrants Q2 and Q3, the preference is governed by the indifference curve (risk aversion) so we cannot make general statements. Nonetheless, what we have observe here gives rise to the mean-variance criterion which can be stated as:

Portfolio $A$ is preferred to portfolios $B$ ($A \succ B$) if

\begin{align} \mathbb{E}[r_{A}] &\geq \mathbb{E}[r_{B}] \label{MVC expected return} \tag{2.2}\\ \sigma_{A} &\leq \sigma_{B} \label{MVC sigma} \tag{2.3} \end{align}

with one of these inequalities being strict.

3. Estimating Risk Aversion

To implement the ideas developed in the preceding sections, we need to be able to estimate the risk aversion of an investor. This is not something that can be done precisely as investors rarely behave in a consistent manner (there are many people who invest rationally in a professional setting yet visit casinos in their private time). However, we can get a sense of what an investors risk aversion is by studying decisions they make under uncertainty. Usually, these decisions need to be placed in a simplified context so that the investor (who may well be unsophisticated) can easily understand the risk/reward tradeoff and hence provide a useful response. A common example is to ask an investor how much they would pay to protect a large asset from destruction.

For example, imagine you have all your wealth invested in a property (real estate). Now, suppose that in any given year there is a probability $p$ that the property will be completely destroyed. In this case the return is -100\% (all wealth is lost). However, if the property isn't destroyed then the return is 0\% (we assume that the property hasn't increased in value. The figure below depicts this scenario.

We now compute the expected return and variance of returns for this scenario:

\begin{align*} \mathbb{E}[r] = p\times -1 + (1-p)\times 0 = -p\\ \textrm{Var}[r] = p(-1 + p)^{2} + (1-p)p^{2} = p(1-p) \end{align*}

Substitute into the utility function:

\begin{align*} U = -p - \tfrac{1}{2}Ap(1-p) \end{align*}

Now, say that for a cost of \$$v$ per year, per \$1 of asset, the property can be 100\% insured. Since $v$ is specified as a cost per dollar of asset value, it represents a proportion or return. This insurance will eliminate all risk and hence the utility of having the property and insurance will be equal to $-v$. This is our certainty equivalent. To determine our risk aversion, we set the utility of the uninsured property equal to the utility of the insured property:

\begin{align*} -v = -p - \tfrac{1}{2}Ap(1-p) \quad \Rightarrow \quad v = p\underbrace{[1 + \tfrac{1}{2}A(1-p)]}_{ELM} \end{align*}

We can now determine the investors risk aversion by asking how much they would pay, i.e. what $v$ is for them. If we know $v$, we can solve for $A$. For example, say the investor were risk neutral ($A = 0$). Then they would only be willing to pay the expected loss, $p$. If however they are more risk averse, they would be willing to pay more. We can examine the expected loss multiplier ($ELM$) to see what multiple of the expected loss they are willing to pay. By studying insurance premiums, we can hence gain an understanding of how risk averse investors tend to be. Typical values for $A$ lie in the range of 2-4. Later, we will examine how to estimate risk aversion of the representative investor in the CAPM.

Python example 3.1. Expected loss multiplier

In [26]:
import numpy as np
import matplotlib.pyplot as plt

# Specify the parameters
p1 = 0.001
p2 = 0.2
A = np.linspace(0,8,50)

# Compute the ELM
ELM1 = 1 + 0.5*A*(1-p1)
ELM2 = 1 + 0.5*A*(1-p2)

# Plot
plt.plot(A, ELM1, label='ELM1')
plt.plot(A, ELM2, label='ELM2')
plt.xlabel('$A$')
plt.ylabel('$ELM$')
plt.legend()
plt.grid()
plt.show()

4. Capital Allocation Across Assets

The construction of a portfolio means that we must choose how we allocate our wealth across a variety of assets. These could include asset classes such as low risk bonds (government bills/bonds), medium risk bonds (corporates) and higher risk equities (shares in companies). Identifying the broad asset classes that investors allocate their wealth to is called the asset allocation problem. Identifying the specific securities that the investors wish to invest in within each asset class is called the security selection problem. The theories that we will look at in this course will examine the asset allocation problem and the security selection problem as it applies to equities. Security selection within fixed income assets is a separate topic that is covered in fixed income courses.

It has been found that the asset allocation decision is responsible for the majority of returns that a portfolio generates. For instance, if the equity market is performing well, it is important that you are invested in equities more so than which specific stocks you have purchased (see Brinson, Hood and Beebower; 1986 for original findings).

We now take a simplified look at the asset allocation problem where we consider only two asset classes: a risk-free asset and a single risky portfolio. While this is a far simpler than what is done in a real-world setting, it provides a nice framework in which to introduce the main ideas. In later topics we will expand our analysis to consider many risky assets with a single risk-free asset.

4.1. The Risk-Free Asset

Before we conduct our analysis, we make a detour to discuss the risk-free asset. In our applications we will assume that there is an asset that delivers a guaranteed return of $r_{f}$ with no uncertainty (zero standard deviation). In reality, there is no such thing as an asset that is truly risk free. Typically, we use a short term, zero coupon government security (a Treasury Bill in the US for instance) as a proxy for the risk-free rate. However, it should be remembered that this return is guaranteed only if the government is guaranteed not to default on their payment and if the Bill is held until maturity. While large governments, like the US, are unlikely to default over a short time horizon, if the Bill isn't held until maturity, then you are subject to interest rate risk as the bond’s price will move due to fluctuations in the interest rate. In addition, there have been times in history where governments, even large and relatively stable ones, have presented a risk of default. For instance, in 1998 Russia defaulted on some of its debts. Also, the ability of the US government to borrow through its money market operations needs approval from Congress. This provides an element of political risk where disagreement between political parties can lead to the government being unable to borrow money to meet its financial obligations. This exact scenario occurred in 2011 and again in 2013. While the US government avoided defaulting on its obligations, it did partially shut down the government (some government facilities closed) until the debt ceiling could be raised. This led to some ratings agencies downgrading the US Federal governments credit ratings.

Nonetheless, the concept of a risk-free asset is extremely useful for our analysis and for finance in general. We often speak of the risk premium for instance which is the expected return above the risk-free rate. Additionally, the US (and many other large governments) has never defaulted on their obligations and hence default risk is still considered to be small. For this reason, we will continue to use the notion of a risk-free asset and proxy for it with short-term government securities.

4.2. The Allocation Problem

We now examine how to solve the allocation problem, that is determining what proportion of our wealth we should invest in the risky portfolio and what proportion we should invest in the risk-free asset. Let $x$ be the proportion of our wealth that we allocate to the risky portfolio $P$. Then the proportion we invest in the risk-free asset must be $1-x$ since the proportions must add up to 100\% (otherwise it would suggest that we invest more wealth than we have).

The expected return of this combined portfolio, $C$, is given by

\begin{align} \nonumber \mathbb{E}[r_{C}] &= (1-x)r_{f} + x\mathbb{E}[r_{P}]\\ &= r_{f} + x(\mathbb{E}[r_{P}] - r_{f}) \label{C exp return} \tag{4.1} \end{align}

and for the variance,

\begin{align*} \nonumber \mathrm{Var}[r_{C}] &= (1-x)^{2}\underbrace{\mathrm{Var}[r_{f}]}_{=0} + x^{2}\mathrm{Var}[r_{P}] + 2(1-x)x\underbrace{\mathrm{Cov}[r_{f},r_{P}]}_{=0}\\ &= x^{2}\sigma_{P}^{2} \quad \Leftrightarrow \quad \sigma_{C} = |x\sigma_{P}| \label{C vaiance} \tag{4.2} \end{align*}

Equations \eqref{C exp return} and \eqref{C vaiance} represent a straight line in $\sigma-\mathbb{E}[r]$ space which is called the Capital Allocation Line (CAL).

Python example 4.1. The CAL

In [27]:
import numpy as np
import matplotlib.pyplot as plt

# Specify the parameters
rf = 0.03
ErP = 0.15
sigmaP = 0.2

# Build the weights
x = np.linspace(-0.5,1.5,100)

# Compute the expected retruns and standard deviations
ErC = rf + x*(ErP - rf)
sigmaC = np.sqrt((x*sigmaP)**2)

# Plot
plt.plot(sigmaC, ErC)
plt.plot(sigmaP, ErP, 'ro')
plt.xlabel('$\sigma$')
plt.ylabel('E[r]')
plt.text(sigmaP,ErP-0.02,'P')
plt.grid()
plt.show()

In Python example 4.1. we allowed $x$, the allocation of wealth to $P$, to be greater 1. This means that $(1-x)$, the allocation to the risk-free asset, can be negative. A negative weight in the risk-free asset is called a short position and corresponds to borrowing money at an interest rate equal to the risk-free rate. We can then use this borrowed money to invest more than 100\% of our wealth into the risky portfolio $P$. We could of course allow also allow $x < 0$ (as we have in Python example 4.1.) and this corresponds to short selling $P$ and investing the proceeds in the risk-free asset. Shorting an asset is like borrowing money. You borrow the stock from another party, sell it in the market, then buy it back in the market at a later date to return the asset to the original owner. If the price of the asset falls you would make a positive return. In Python example 4.1. we can see that this would not be a wise strategy as we are selling an asset that is expected to return 15\% to purchase an asset with a return of 3\%. This means that the more of this that we do, the worse the expected return of our portfolio becomes.

4.2.1. Borrowing and Lending Rates

In the previous analysis we assumed that the rate at which investors can borrow and lend money is equal to the risk-free rate. However, in practice this is never the case. We always borrow money at a higher rate than we can lend it out (invest) at. This is precisely the method by which banks make their profits. What happens to our CAL in this more realistic scenario? The fundamental equations provided by \eqref{C exp return} and \eqref{C vaiance} do not change. However, we need to consider which inputs are appropriate for a given circumstance. Let there be two risk-free rates, $r_{f,I}$ and $r_{f,B}$ where $r_{f,I} < r_{f,B}$. When investing ($(1-x) > 0$), we receive the rate $r_{f,I}$ but when borrowing ($(1-x) < 0$) we must pay the rate $r_{f,B}$. While this has no impact on the variance of the portfolio, it does impact the expected return,

\begin{align*} \mathbb{E}[r_{C}] = \left\{\begin{array}{ll} r_{f,I} + x(\mathbb{E}[r_{P}] - r_{f,I}) & \textrm{if } x < 1\\ r_{f,B} + x(\mathbb{E}[r_{P}] - r_{f,B}) & \textrm{if } x > 1\end{array}\right. \end{align*}

This change in the expected return, based on which rate we are using, kinks the CAL.

Python example 4.4. The kinked CAL

In [28]:
import numpy as np
import matplotlib.pyplot as plt

# Specify the risks and returns
rfI = 0.03
rfB = 0.09
ErP = 0.15
sigmaP = 0.2

# Build the CAL for different values of x
n = 100                                  # Number of points to use
x = np.linspace(0,1.5,n)                 # Construct x
ErC = np.zeros(n)                        # Preallocate the array for E[rC]
for i in range(n):                       # Loop through the elements of x
    if x[i]<1:
        ErC[i] = rfI + x[i]*(ErP - rfI)  # Expected return if x<1
    else:
        ErC[i] = rfB + x[i]*(ErP - rfB)  # Expected return if x>1
sigmaC = np.sqrt((x*sigmaP)**2)          # Compute the standard deviation

# Plot
plt.plot(sigmaC, ErC)
plt.plot(sigmaP, ErP, 'ro')
plt.xlabel('$\sigma$')
plt.ylabel('E[r]')
plt.text(sigmaP,ErP-0.02,'P')
plt.grid()
plt.show()

4.2.2. The Sharpe Ratio

One of the most important characteristics of the CAL is the slope of the line. Recall that the slope is given by the rise divided by the run which in this case is

\begin{align} S = \frac{\mathbb{E}[r_{C}] - r_{f}}{\sigma_{C}} \label{sharpe ratio} \tag{4.3} \end{align}

This is also called the Sharpe ratio (named after its inventor, William Sharpe) and represents the reward earned per unit of portfolio risk. We can see that when the borrowing and lending rate are equal, all portfolios have the same Sharpe ratio. However, when the borrowing and investing rates are not the same, the Sharpe ratio can change as we shift from investing to borrowing.

4.2.3. Risk Aversion and Asset Allocation

We are now able to determine the optimal allocation of wealth between a risk-free asset and a single risky portfolio. The fundamental idea is that the optimal portfolio is that which maximizes the investors utility. To better see this concept, the example below plots utility of an investor with $A = 4$ as a function of $x$, their allocation to the risky portfolio $P$.

Python example 4.5. Utility and allocation

In [29]:
import numpy as np
import matplotlib.pyplot as plt

# Specify constants and build array for x
A = 4
n = 100
x = np.linspace(0,1,n)
sigmaP = 0.25
ErP = 0.15
rf = 0.07

# Compute utility
U = rf + x*(ErP - rf) - 0.5*A*(x*sigmaP)**2

# Plot
plt.plot(x,U)
plt.xlabel('$x$')
plt.ylabel('$U$')
plt.grid()
plt.show()

We can see from Python example 4.5. that there is a unique allocation at which the utility of an investor with $A=4$ is maximized. Our aim is now to determine a formula for the optimal allocation $x^{*}$.

The problem to solve is

\begin{align} \nonumber \max_{x}U &= \mathbb{E}[r_{C}] - \tfrac{1}{2}A\sigma_{C}^{2}\\ &= r_{f} + x\left(\mathbb{E}[r_{P}] - r_{f}\right) - \tfrac{1}{2}Ax^{2}\sigma_{P}^{2} \label{U max problem} \tag{4.4} \end{align}

To solve \eqref{U max problem} we need to differentiate with respect to $x$, set equal to 0 and solve for $x$.

\begin{align} \nonumber \frac{dU}{dx} &= \mathbb{E}[r_{P}] - r_{f} - Ax\sigma_{P}^{2} = 0\\ \Rightarrow x^{*} &= \frac{\mathbb{E}[r_{P}] - r_{f}}{A\sigma_{P}^{2}} \label{opt soln} \tag{4.5} \end{align}

Equation \eqref{opt soln} is the optimal allocation formula. Note its behavior with respect to $A$. As $A$ increases, $x$ decreases. This is because as investors become more risk averse, they seek to place more of their wealth in the risk-free asset. Th opposite is also true in that as $A$ decreases the allocation to the risky asset, $x$, increases as this investor seeks more return due to their higher tolerance for risk. Also note the two extreme cases. For a completely risk averse investor, $A \to \infty$ and in this case $x^{*} \to 0$. This investor will avoid all risk at all costs and hence only allocates wealth to the risk-free asset. Alternatively, consider the risk neutral investor, $A\to0$. In this case $x^{*} \to \infty$ if $\mathbb{E}[r_{P}] - r_{f}>0$ and $x^{*} \to -\infty$ if $\mathbb{E}[r_{P}] - r_{f}<0$. This investor does not consider risk at all and hence will only concern themselves with expected returns to make their investment decisions. This means that they will short sell as much as possible of the asset with the lower expected return and use this to fund as large a long position as possible in the asset with the higher expected return. In the absence of any limits, these positions are, theoretically, infinite.

The analysis we have just performed can also be done with indifference curves introduce in Section 3. To illustrate, we plot indifference curves with increasing levels of utility.

Python example 4.6. Increasing indifference curves

In [30]:
import numpy as np
import matplotlib.pyplot as plt

# Set the parameters
A = 3                              # Risk aversion
U1 = 0.06                          # Utility
U2 = 0.09                          # Utility
sigma = np.linspace(0,0.5,100)     # Generate values of sigma
Er1 = 0.5*A*sigma**2 + U1          # Compute associated expected return
Er2 = 0.5*A*sigma**2 + U2          # Compute associated expected return

# Plot
plt.plot(sigma, Er1, label='U1=0.06')
plt.plot(sigma, Er2, label='U2=0.09')
plt.grid()
plt.xlabel('$\sigma$')
plt.ylabel('E[r]')
plt.legend()
plt.show()

The main point to take from the plot in Python example 4.6. is that the indifference curves move up as the level of utility increases. This is because we are considering portfolios with higher levels of expected return for the same levels of risk. Note also that the space in which we have plotted the indifference curves is the same as that used to plot the CAL, $\sigma-\mathbb{E}[r]$ space. Hence, by plotting the indifference curves and CAL together, we can use the way in which they intersect to identify the optimal portfolio.

Python example 4.7. Indifference curves and the CAL

In [31]:
import numpy as np
import matplotlib.pyplot as plt

# Asset parameters
rf = 0.07
ErP = 0.15
sigmaP = 0.22

# Set the parameters for two indifferent curves
A = 4                            # Risk aversion
U1 = 0.06                        # Utility for indifference curve 1
U2 = 0.12                        # Utility for indifference curve 2
n = 100                          # Number of points to compute
x = np.linspace(0,1,n)           # Build allocation weight array
sigma = x*sigmaP                 # Compute portfolio sigma
Er1_U = 0.5*A*sigma**2 + U1      # Compute associated expected return for U1
Er2_U = 0.5*A*x*sigma**2 + U2    # Compute associated expected return for U2

# Compute the CAL
ErCAL = rf + x*(ErP - rf)

# Compute the optimal x, utility and expected return
x_star = (ErP - rf)/(A*sigmaP**2)
U_star = rf + x_star*(ErP - rf) - 0.5*A*(x_star*sigmaP)**2
Er_star_U = 0.5*A*sigma**2 + U_star

# Compute the optimal portfolio
sigma_star = x_star*sigmaP
Er_star = rf + x_star*(ErP - rf)

# Plot
plt.plot(sigma, Er1_U, '--', label='U1=0.06')
plt.plot(sigma, Er2_U, '--', label='U2=0.12')
plt.plot(sigma, Er_star_U, '--', label='U3=U*')
plt.plot(sigma, ErCAL, label='CAL')
plt.plot(sigma_star, Er_star, 'bo', label='C*')
plt.grid()
plt.xlabel('$\sigma$')
plt.ylabel('E[r]')
plt.legend()
plt.show()
print("The optimal utility is U* = % s" % round(U_star,4))

The optimal utility is U* = 0.0865

What we observe in Python example 4.7. is that the CAL and indifference curves can interact. Recall that the CAL represents feasible portfolios, namely those that can be constructed. We cannot build a portfolio with characteristics that are not included on the CAL. Hence, if the indifference curve never touches the CAL (as is the case with U2), then this indifference curve represents portfolios that we cannot construct. The point where the indifference curve intersects the CAL represents a feasible portfolio with the utility equal to the indifference curve. For instance, where U1 and the CAL intersect represents a portfolio that provides a utility level of 0.06. However, we can find a better portfolio by looking for one with increased utility which, graphically, corresponds to the indifference curve moving up in $\sigma-\mathbb{E}[r]$ space. Moving the indifference curve as high as possible while still intersecting with the CAL yields the indifference curve U3 which just touches (is tangential to) the CAL. Where these two lines touch represents the optimal portfolio, which has utility equal to U* = 0.0865.

5. Passive Strategies

While the allocation framework we have examined in this topic is very simple, it nonetheless represents a common method of investing called a passive strategy. The principle behind a passive strategy is to select a risky portfolio (or asset) that represents the market. Some examples are funds that track indices such as:

  • S\&P 500 (USA)
  • ASX 200 (Australia)
  • FTSE 100 (UK)
  • DAX (Germany)
  • Hang Seng (Hong Kong)
  • Hang Seng China Enterprises (China)

If such a fund is chosen as the risky portfolio $P$, and the risk-free asset is selected to be as government bond (or bill), then the CAL becomes the Capital Market Line (CML). The precise reason for this will be provided when we study the CAPM.

The main idea behind such strategies is that by splitting wealth between a risk-free asset and a well-diversified risky portfolio, risk can be easily controlled by simply shifting wealth between these two assets while still providing access to higher average returns through the equity index. While such a simple approach may appear naive at first glance, research has found that these passive investment strategies are very tough to beat in practice. The reasons for this include:

  • Reduced trading costs - the only trading that occurs is between two assets.
  • Little cost of obtaining information - passive managers (in contrast to active managers) do not need to analyze lots of (expensive) data to make their decisions
  • Added value via execution strategies - some passive managers make extra returns by exploiting inefficiencies in the market. One example is index rebalance dates. Some funds have a mandate that they must track the index as closely as possible while other funds can deviate slightly (these are usually called enhanced passive funds). When an index changes, the market is given prior warning. For example, Standard and Poor’s lets investors know of changes to the ASX200 about 5 days before the changes takes place. On the date the index changes take effect, passive investors who need to track the index as closely as possible will leave their trades to this date. Enhanced passive investors however can trade early and hence avoid the demand shocks created by the true passive investors. In this way, enhanced passive funds can add value through the way they execute their passive decisions.