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FINANCIAL ECONOMICS I
RETURN AND RISK CONCEPTS
A GLOSSARY OF BASIC CONCEPTS
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RETURN
Return is gain or loss over time, usually expressed as
an annual percentage of initial investment
Holding period rate of return, 𝐻𝑃𝑅 =𝑃1−𝑃0+𝐶
𝑃0
Effective annual rate (EAR) of return is useful for
comparing investments with different horizons
❖ (1 + 𝐸𝐴𝑅)𝑇= 1 + 𝐻𝑃𝑅, where 𝑇 is a fraction of 1 year
Annual percentage rate (APR) of return is the rate of
return for each period times number of periods in a year
❖ 𝐴𝑃𝑅 = 𝑚 × 𝐻𝑃𝑅 (Note: 𝑚 = 1/𝑇)3
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CONTINUOUS COMPOUNDING
Recall: 𝐴𝑃𝑅 = 𝑚 × 𝐻𝑃𝑅 =1
𝑇𝐻𝑃𝑅
This implies that 𝐻𝑃𝑅 = 𝑇 × 𝐴𝑃𝑅
So that (1 + 𝐸𝐴𝑅)𝑇= 1 + (𝑇 × 𝐴𝑃𝑅)
Alternatively 1 + 𝐸𝐴𝑅 = 1 + (𝑇 × 𝐴𝑃𝑅) 1/𝑇
When T → 0, we approach continuous compounding:
❖ 1 + 𝐸𝐴𝑅 = 𝑒𝑅𝐶𝐶, where 𝑒 ≈ 2.718282
❖ It follows that 1 + 𝐸𝐴𝑅 𝑇 = 𝑒𝑇×𝑅𝐶𝐶
𝑒𝑇×𝑅𝐶𝐶 is the total rate of return for a period, given the continuous compounding rate 𝑅𝐶𝐶 4
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RISK
Expected return, 𝐸 𝑅 = σ𝑖=1𝑛 𝑃𝑖𝑅𝑖
Standard deviation of returns, 𝑆𝐷 𝑅 = 𝑉𝑎𝑟(𝑅) 1/2
Variance of returns, 𝑉𝑎𝑟 𝑅 = σ𝑖=1𝑛 𝑃𝑖 𝑅𝑖 − 𝐸 𝑅 2
Expected holding period return can be broken down thus:
𝐸𝑥𝑝𝑒𝑐𝑡𝑒𝑑 𝑟𝑎𝑡𝑒 𝑜𝑓 𝑟𝑒𝑡𝑢𝑟𝑛 =𝑅𝑖𝑠𝑘 𝑓𝑟𝑒𝑒𝑟𝑎𝑡𝑒
+𝑅𝑖𝑠𝑘
𝑃𝑟𝑒𝑚𝑖𝑢𝑚
Therefore, risk premium (or excess return) is the
holding period return minus risk-free rate of return 5
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MARKOWITZ PORTFOLIO OPTIMIZATION
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RISK AND RETURN OF A PORTFOLIO
The expected return on a portfolio is simply the
weighted average of the expected returns on assets
comprising the portfolio, with the weights being
the relative proportions of each asset in the
portfolio:
𝐸 𝑅𝑃 =
𝑖=1
𝑛
𝑤𝑖𝐸 𝑅𝑖 = 𝑤1𝐸 𝑅1 +𝑤2𝐸 𝑅2 + …+𝑤𝑛𝐸 𝑅𝑛
Unlike the expected return on a portfolio, the
standard deviation of returns on a portfolio is not
merely a weighted average of risks on the assets
comprising the portfolio 7
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RISK AND RETURN OF A PORTFOLIO
Rather, the extent of association among returns on the assets comprising the portfolio matters ad must be incorporated in the computation of portfolio’s total risk
Therefore, portfolio risk (standard deviation) is computed as
𝑆𝐷 𝑅𝑃 =
𝑗=1
𝑛
𝑘=1
𝑛
𝑤𝑗𝑤𝑘𝐶𝑜𝑣 𝑅𝑗 , 𝑅𝑘
The covariance is obtained as
𝐶𝑜𝑣 𝑅𝑗 , 𝑅𝑘 =
𝑖=1
𝑛
𝑃𝑖 𝑅𝑖𝑗 − 𝐸 𝑅𝑗 𝑅𝑖𝑘 − 𝐸 𝑅𝑘
Or
𝐶𝑜𝑣 𝑅𝑗 , 𝑅𝑘 = 𝑆𝐷 𝑅𝑗 𝑆𝐷 𝑅𝑘 𝐶𝑜𝑟𝑟 𝑅𝑗 , 𝑅𝑘
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RISK AND RETURN OF A PORTFOLIO
Substituting the covariance formula into the portfolio standard deviation formula gives
𝑆𝐷 𝑅𝑃 =
𝑗=1
𝑛
𝑘=1
𝑛
𝑤𝑗𝑤𝑘𝑆𝐷 𝑅𝑗 𝑆𝐷 𝑅𝑘 𝐶𝑜𝑟𝑟 𝑅𝑗 , 𝑅𝑘
Suppose that 𝑛 = 2. the portfolio risk is computed as
𝑤1𝑤1𝑆𝐷 𝑅1 𝑆𝐷 𝑅1 𝐶𝑜𝑟𝑟 𝑅1, 𝑅1 𝑤1𝑤2𝑆𝐷 𝑅1 𝑆𝐷 𝑅2 𝐶𝑜𝑟𝑟 𝑅1, 𝑅2𝑤2𝑤1𝑆𝐷 𝑅1 𝑆𝐷 𝑅2 𝐶𝑜𝑟𝑟 𝑅2, 𝑅1 𝑤2𝑤2𝑆𝐷 𝑅2 𝑆𝐷 𝑅2 𝐶𝑜𝑟𝑟 𝑅2, 𝑅2
𝑤12𝑉𝑎𝑟 𝑅1 𝑤1𝑤2𝑆𝐷 𝑅1 𝑆𝐷 𝑅2 𝐶𝑜𝑟𝑟 𝑅1, 𝑅2
𝑤2𝑤1𝑆𝐷 𝑅2 𝑆𝐷 𝑅1 𝐶𝑜𝑟𝑟 𝑅2, 𝑅1 𝑤22𝑉𝑎𝑟 𝑅2
𝑆𝐷 𝑅𝑃 = 𝑤12𝑉𝑎𝑟 𝑅1 +𝑤2
2𝑉𝑎𝑟 𝑅2 + 2𝑤1𝑤2𝑆𝐷 𝑅1 𝑆𝐷 𝑅2 𝐶𝑜𝑟𝑟 𝑅1, 𝑅29
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RISK AVERSION AND UTILITY VALUES
Investors are risk averse if they demand positive risk
premium for a given quantity of risk
Investment opportunities (assets) are attractive if
they have low risk and high expected returns
If risk increases with return, investors must trade off
risk for return
The trade-off is achieved by assigning utility to the
assets based on their risk and expected return
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RISK AVERSION AND UTILITY VALUES
The following utility function has gained wide usage in evaluating investment opportunities:
𝑈 = 𝐸 𝑅 − ൗ1 2𝐴𝜎2
where 𝑈 is the utility value, 𝐴 is investor’s risk aversion index; Τ1 2 is a scaling convention
More risk averse investors have large values of 𝐴 which penalizes risky investments more severely
The utility score can be interpreted as the certainty equivalent rate of return
(the rate of return that a risk-free asset must earn to be as competitive as the risky portfolio in question)
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MEAN-VARIANCE CRITERION
Mean-variance dominance criterion states that asset A
dominates asset B if
𝐸 𝑅𝐴 ≥ 𝐸 𝑅𝐵
and
𝑉𝐴𝑅 𝑅𝐴 ≤ 𝑉𝑎𝑟 𝑅𝐵
At least one equality must be strict to rule out
indifference
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INDIFFERENCE CURVES
Indifference curves are expected return-standard
deviation trade-off function (yield constant utility)
Consider portfolios P and Q in the figure below
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I
IV
II
III
𝐸 𝑅𝑃
𝜎𝑃
Indifference curve
P
Q
Utility increases
Uti
lity
in
crease
s
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PORTFOLIOS OF ONE RISKY ASSET AND
ONE RISK-FREE ASSET
Assume an investor holds the proportion 𝑦 of her wealth
in a risky portfolio, P, and (1 − 𝑦) in a risk-free asset*, F
Thus, the complete portfolio’s riskiness will vary with 𝑦
Let’s define terms as follows:
𝑟𝑃 – the rate of return on the risky portfolio, 𝑃
𝐸 𝑟𝑃 – the expected return on risky portfolio, 𝑃
𝜎𝑃 – the standard deviation of risky portfolio, 𝑃
𝑟𝑓 – return on risk-free asset, 𝐹14
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PORTFOLIOS OF ONE RISKY ASSET AND
ONE RISK-FREE ASSET CONT’D
Thus, the expected return, E 𝑟𝐶 , on the complete
portfolio, 𝐶, is
E 𝑟𝐶 = 𝑦𝐸 𝑟𝑃 + (1 − 𝑦)𝑟𝑓
= 𝑟𝑓 + 𝑦 𝐸 𝑟𝑃 − 𝑟𝑓 … … … …(1)
Because asset 𝐹 is risk-free, the standard deviation of
the returns on portfolio 𝐶 is simply:
𝜎𝐶 = 𝑦𝜎𝑃 … … … … … … … … (2)
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PORTFOLIOS OF ONE RISKY ASSET AND
ONE RISK-FREE ASSET CONT’D
Solving for 𝑦 in eq. (2), substituting the result
( Τ𝑦 = 𝜎𝐶 𝜎𝑃) in equation (1) and rearranging:
E 𝑟𝐶 = 𝑟𝑓 + 𝜎𝐶𝐸 𝑟𝑃 − 𝑟𝑓
𝜎𝑃… … … … … …(3)
Equation (3) is called the capital allocation line (CAL*)
The expected excess return on portfolio 𝐶, per unit of
additional risk is the slope of the CAL, also called the
reward-to-volatility ratio, (or the Sharpe ratio):
𝑆 =𝐸 𝑟𝑃 − 𝑟𝑓
𝜎𝑃
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PORTFOLIOS OF ONE RISKY ASSET AND
ONE RISK-FREE ASSET CONT’D
Lending and Borrowing
Consider the graph of equation (3):
17
P𝐸 𝑟𝑃
𝜎𝑃
CAL
𝑆 =𝐸 𝑟𝑃 − 𝑟𝑓
𝜎𝑃
𝑟𝑓
C
T
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PORTFOLIOS OF ONE RISKY ASSET AND
ONE RISK-FREE ASSET CONT’D
Suppose P is the investor’s ‘desired’ risky portfolio
Thus, she can invest in 𝑃 and also invest (lend) in the
risk-free asset, 𝐹, to form their optimal portfolio, (e.g. 𝐶)
Investors can also borrow at the risk-free rate, if
possible, to form a portfolio to the right of 𝑃 (e.g. 𝑇)
However, since the investor has borrowed, the
proportion of her wealth in portfolio 𝑃, 𝑦 > 1
❖ Accordingly, (1 − 𝑦)must be negative
Clearly, portfolio 𝑇 must be riskier than portfolio 𝐶18
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PORTFOLIOS OF ONE RISKY ASSET AND
ONE RISK-FREE ASSET CONT’D
In practice, private (nongovernmental) investors
usually borrow at above the risk-free rate (say, 𝑟𝑓𝐵)
Private investors borrow through margin accounts
Thus, their capital allocation line will be kinked at P:
19
𝑅𝑓𝐵
𝑅𝑓
P𝐸 𝑅𝑃
𝜎𝑃
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RISK TOLERANCE AND ASSET ALLOCATION
The CAL gives provides investors with the set of all feasible capital allocation choices (opportunity set)
We can determine the portfolio (combination of risky and
risk-free asset) at which investors maximize utility thus:
𝑀𝑎𝑥 𝑈 = 𝐸 𝑅𝐶 − ൗ1 2𝐴𝜎𝐶2 = 𝑟𝑓 + 𝑦 𝐸 𝑟𝑃 − 𝑟𝑓 − Τ1 2𝐴𝑦2𝜎𝑃
2
Differentiating 𝑈 w.r.t. 𝑦, setting the derivative equal to 0 and solving for 𝑦 gives
𝑦∗ = Τ𝐸 𝑟𝑃 − 𝑟𝑓 𝐴𝜎𝑃2
Thus, the optimal portfolio is inversely proportional to risk and risk tolerance coefficient but directly proportional to the risk premium
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RISK TOLERANCE AND ASSET ALLOCATION
Theoretically, individuals choose portfolios from the
opportunity set based on their degrees of risk aversion
(less risk-averse investors hold less of the risk-free
asset etc.)
The degree of risk aversion is reflected in the slope of
each investor’s indifference curve
The slope of the indifference curve is steeper the higher
is the investor’s risk aversion coefficient, A
See illustration next slide and in Excel 21
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RISK TOLERANCE AND ASSET ALLOCATION
– INDIFFERENCE CURVES
220.000
0.100
0.200
0.300
0.400
0.500
0.600
0.700
0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50
A = 4
A = 2
U = 0.15
U = 0.05
More desirable portfolios
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CAPITAL ALLOCATION USING INDIFFERENCE
CURVES
Assume 𝜎𝑃 = 15%; 𝐸 𝑅𝑃 = 35%; and 𝑅𝑓 = 3%
230.000
0.100
0.200
0.300
0.400
0.500
0.600
0.700
0.800
0.900
0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.16 0.18 0.20
CAL
Optimal portfolio, C
Risky
portfolio, P
𝑅𝑓
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PASSIVE (RISKY) PORTFOLIOS
Portfolio P above is formed by the investor after researching each asset’s promised returns and risk
An alternative is to invest in a passive portfolio (which do not require research and regular rebalancing)
Passive portfolios are available in some of Africa’s stock markets [e.g. South Africa’s index funds and/or ETFs (Satrix, Stanlib etc.)]
Advantages of passive portfolios
1. Lower cost
2. Free-rider benefit
A capital allocation line representing a strategy with a passive risky portfolio is called the capital market line (CML) 24
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DIVERSIFICATION
The risk of an asset can be split into two components:
1. Systematic risk: the general component attributable to broad macroeconomic factors (e.g. GDP)
2. Non-systematic or diversifiable risk: related to factors specific to the asset issuer (e.g. managerial changes)
Non-systematic risk can be reduced (possibly eliminated) by investing in several non-related assets
❖ This strategy is called diversification 25
Systematic risk
Non-systematic risk
No. of securities
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PORTFOLIOS OF TWO RISKY ASSETS
Assume an investor holds a portfolio of two assets, 1 and
2 with expected returns 𝐸 𝑅1 and 𝐸 𝑅2 respectively
The proportion of the investor’s wealth in asset 1 is 𝑤1and the proportion in asset 2 is 𝑤2 (where 𝑤2 = 1 − 𝑤1)
The expected return on the two-asset portfolio is
𝐸 𝑟𝑃 = 𝑤1𝐸 𝑟1 +𝑤2𝐸 𝑟2
The variance of the two-asset portfolio is
𝑉𝑎𝑟 𝑅𝑃 = 𝑤12𝑉𝑎𝑟 𝑅1 +𝑤2
2𝑉𝑎𝑟 𝑅2 + 2𝑤1𝑤2𝐶𝑜𝑣 𝑅1, 𝑅226
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PORTFOLIOS OF TWO RISKY ASSETS CONT’D
The covariance of returns can be expressed as
𝐶𝑜𝑣 𝑅1, 𝑅2 = 𝑆𝐷 𝑅1 𝑆𝐷 𝑅2 𝐶𝑜𝑟𝑟(𝑅1, 𝑅2)
Thus, the two-asset portfolio variance is
𝜎𝑃2 = 𝑤1
2𝜎12 +𝑤2
2𝜎22 + 2𝑤1𝑤2𝜎1𝜎2𝜌12
Note:
1. Portfolio risk is usually less than the weighted average of risks of individual assets comprising it
2. The upper bound portfolio risk is realized when the correlation coefficient is +1
3. The lower bound is realized when the correlation coefficient is: -1 27
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MINIMUM VARIANCE PORTFOLIO
The combination of assets 1 and 2 that gives a portfolio
with the minimum variance can be obtained by
differentiating the variance equation with respect to 𝑤1,
setting the result equal to zero, then solving for 𝑤1∗:
𝑑𝑉𝑎𝑟 𝑅𝑃𝑑𝑤1
= 2𝑤1𝜎𝑋2 − 2𝜎𝑌
2 + 2𝑤1𝜎𝑌2 + 2𝑟𝑋,𝑌𝜎𝑋𝜎𝑌 − 4𝑤1𝑟𝑋,𝑌𝜎𝑋𝜎𝑌 = 0
2𝑤1 𝜎𝑋2 + 𝜎𝑌
2 − 2𝑟𝑋,𝑌𝜎𝑋𝜎𝑌 + 2𝑟𝑋,𝑌𝜎𝑋𝜎𝑌 − 2𝜎𝑌2 = 0
Dividing through by 2 and solving for 𝑤1∗, we obtain
𝑤1∗ =
𝜎𝑌2− 𝑟𝑋,𝑌𝜎𝑋𝜎𝑌
𝜎𝑋2+ 𝜎𝑌
2− 2𝑟𝑋,𝑌𝜎𝑋𝜎𝑌
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PORTFOLIOS OF TWO RISKY ASSETS
CONT’D
Consider two assets 𝑋 and 𝑌 :
Suppose you can form ten portfolios with weights 100%,
90%, 80%, 70%, 60%, 50%, 40%, 30%, 20%, 10% and 0%
respectively in asset 𝑋
The expected returns and standard deviations can be
obtained under various correlation assumptions 29
Asset Expected return Standard deviation
X 10% 3.5%
Y 8.5% 2%
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PORTFOLIO OPPORTUNITY SET
30
8.4
8.6
8.8
9
9.2
9.4
9.6
9.8
10
10.2
0 0.5 1 1.5 2 2.5 3 3.5 4
Ex
pe
cte
d r
etu
rn
(%
)
Standard deviation of returns (%)
Lower bound Upper bound Port Opp Set
More desirable portfolios
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PORTFOLIO OPPORTUNITY SET
Notice that the benefits of diversification are greater
the lower the correlation coefficient
Negative correlations give deeper curvatures and give
more desirable portfolios
Notice, too, that perfect positive correlation is not
useful for diversification
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PORTFOLIO CHOICE
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EFFICIENT FRONTIER WITH 𝑵 RISKY ASSETS
Consider the portfolio opportunity set (minimum
variance frontier) of 𝑁 risky assets in Figure 1
B
𝐸(𝑅𝐷)
𝜎 𝜎𝐴,𝐷,𝐸
𝜎𝐴,𝐷,𝐸𝜎𝐴,𝐷,𝐸
D
C 𝐸(𝑅𝐶,𝐸)
E
𝐸(𝑅𝐴) A
∎
∎
∎
∎
∎
Figure 1: The opportunity set of risky portfolios
∎Y
∎X
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On Figure 1
The area enclosed by the curvature plus the boundary represents portfolios of risky assets
The opportunity set is clearly an infinite set
Point B is the minimum variance portfolio
Portfolios along the frontier 𝐵𝑌 will be preferred by investors to portfolios below the frontier
Accordingly, the frontier 𝐵𝑌 is known as the efficient frontier or efficient set
The efficient frontier is the market determined investors’ marginal rate of transformation (MRT) between risk and return
❖ It is the set of portfolios that offers the highest possible expected return for any given level of risk or entails the lowest risk for any given level of expected return 34
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Because all portfolios on the frontier 𝐵𝑌 are efficient, rational investors can choose any one of them
❖ Each individual’s exact portfolio choice is informed by the individual’s subjective marginal rate of substitution (MRS) between risk and return
❖ The MRS is determined by investors’ degree of risk aversion (slope of the investor’s indifference curves)
Indifference curves are maps of the individual’s risk-return trade-off that yield the same total utility
The individual’s total utility is maximized when his/her MRS exactly equals MRT
Thus, the point of tangency between each individual’s highest indifference curve and the efficient frontier is the location of the individual’s optimal portfolio: the point of tangency therefore sets the individual’s subjective price of risk
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B
𝐸(𝑅𝐿)
𝜎 𝜎𝐿
𝜎𝐴,𝐷,𝐸𝜎𝐴,𝐷,𝐸
𝑶𝑯 𝐸(𝑅𝐻) ∎
∎
∎
Figure 2: Optimal portfolio choice
∎Y
𝑶𝑳
𝜎𝐻
𝜎𝐴,𝐷,𝐸𝜎𝐴,𝐷,𝐸
OPTIMAL PORTFOLIO CHOICE
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On Figure 2
The figure shows the optimal choices of two
investors with dissimilar degrees of risk aversion
Investor 𝐻, who has a relatively high degree of risk
aversion (steep indifference curve), chooses his
optimal portfolio on the lower segment of the
efficient frontier, portfolio 𝑂𝐻,
Investor 𝐿 with relatively low degree of risk
aversion chooses her optimal portfolio on the upper
part of the frontier, point 𝑂𝐿, where she bears more
risk but enjoys the prospect of higher returns
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RISK-FREE ASSETS AND PORTFOLIO CHOICE
Consider an investor who buys a risky portfolio in the proportion 𝑤1 of her wealth and invests the remaining proportion (1 − 𝑤1) in a risk-free asset
An asset is risk-free if it offers the same return possibility in all states of nature
The expected return on the resulting portfolio is
𝐸 𝑅𝑃 = 𝑤1𝐸 𝑅1 + 1 − 𝑤1 𝑅𝑓 (1)
where 𝐸 𝑅1 is the expected return on the risky-asset portfolio and 𝑅𝑓 is the return on the risk-free asset
The variance of the resulting portfolio is
𝑉𝑎𝑟 𝑅𝑃 = 𝑤12𝜎1
2 + 1 − 𝑤12𝜎𝑓
2 + 2𝑤1 1 − 𝑤1 𝑟1,𝑓𝜎1𝜎𝑓 = 𝑤12𝜎1
2
(2) 38
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From equation (2), the standard deviation of the portfolio is
𝑆𝐷 𝑅𝑃 = 𝑤1𝜎1 (3)
Equations (2) and (3) show that the risk-free asset does not contribute to the total risk of the resulting portfolio
The portfolio that results from the combination of a risky-asset portfolio and a risk-free asset has return characteristics that are linear in 𝐸(𝑅𝑃) and 𝜎𝑃 space
Proof of linearity is straightforward:
𝑑𝐸 𝑅𝑃
𝑑𝑤1= 𝐸 𝑅1 − 𝑅𝑓
𝑑𝑆𝐷 𝑅𝑃
𝑑𝑤1= 𝜎1
❖ Therefore, the slope of the function that describes the return and risk of the portfolio is:
𝑑𝐸 𝑅𝑃
𝑑 𝑆𝐷 𝑅𝑃=
Τ𝑑𝐸 𝑅𝑃 𝑑𝑤1
Τ𝑑 𝑆𝐷 𝑅𝑃 𝑑𝑤1=
𝐸 𝑅1 −𝑅𝑓
𝜎1(4) 39
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COMBINING THE RISK-FREE ASSET WITH
A RISKY PORTFOLIO
Suppose that the market is frictionless (no
transaction costs and other inefficiencies) so that
investors can lend unlimited amounts of money at
the risk-free rate, 𝑅𝑓
Since the return-risk combination that results is a
straight-line function (already proved), we can
draw a straight line from the risk-free rate to any
risky-asset portfolio on the efficient frontier
Points along the straight lines represent various
possible portfolio combinations – see Figure 340
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COMBINING THE RISK-FREE ASSET WITH
A RISKY PORTFOLIO
B
𝜎
∎
∎
∎
Figure 3: Combining the risky-asset portfolio with the risk-free asset
∎Y
𝑅𝑓
M
C
∎X
A ∎
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1 2
COMBINING THE RISK-FREE ASSET WITH
A RISKY PORTFOLIO
Notice that portfolios along the line 𝑅𝑓𝑀 dominate all
the portfolios along the lines below it
Notice, too, that portfolios on the curvature 𝑀𝑋, originally on the efficient frontier, are also dominated by the portfolios on the line 𝑅𝑓𝑀 and are therefore no
longer efficient
Accordingly, when the risk-free asset with return, 𝑅𝑓, is
introduced, all rational investors will choose portfolios from the line segment 𝑅𝑓𝑀 and the curvature 𝑀𝑌
(That is, the new efficient frontier is 𝑅𝑓𝑀𝑌)
Each investor’s optimal portfolio will, again, be determined by their degrees of risk aversion
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COMBINING THE RISK-FREE ASSET WITH
A RISKY PORTFOLIO
𝜎
∎
Figure 4: Portfolio choice with lending at the risk-free rate
∎Y
𝑅𝑓
M
∎X
∎
∎
𝑶𝑳
𝑶𝑯
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QUESTIONS FOR SELF-STUDY
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SELF-STUDY QUESTION
You are presented with the following information
relating to the expected performance of the stock
of company J after one year
Required
1. Determine the expected return, the standard
deviation of returns and coefficient of variation
for the above stock 45
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Possible Return (GhC) 100 120 130 150 180 220
Probability 0.05 0.14 0.20 0.36 0.20 0.05
SELF-STUDY QUESTION
Suppose you have decided to invest in the above
stock, based on the values computed in (i).
However, you have read in the financial press
about the risk reduction benefits of diversification
and decided to combine stock J with other
securities.
Your Financial Analyst has provided you with the
following information relating to two stocks of
companies K and L, and an outstanding bond
issued by company M. Further the correlation
coefficients between the returns on pair-wise
combinations of the four securities are provided: 46
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SELF-STUDY QUESTION
Your maximum acceptable standard deviation on the resulting portfolio is GhC 130 and you have a total of GhC 1,500,000 to invest in the above securities. Your financial analyst has suggested that GhC 300,000 and GhC 500,000 should be invested in stock L and bond M, respectively
1. How much money must you invest in each of the remaining securities to attain your maximum risk target?
2. Compute the coefficient of variation of the resulting portfolio
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Security
Expected
Return (GhC)
Standard
Deviation (GhC)
Correlation With
J K L
K 250 50 0.50
L 80 10 0.10 -0.20
M 100 20 0.15 0.10 0.30