BMAN20072 Investment Analysis
Week 1-2: Risk and Capital Allocation
BKM Ch 5.3 – 5.4, Ch. 6.1 – 6.6
Overview
• How to design an optimal portfolio?
– Capital allocation
• How to optimally allocate capital between risk-free
and risky assets? [this week’s topic]
– Asset allocation
• How to optimally allocate budget across risky
assets? [next week’s topic]
• What is an optimal portfolio?
– Utility maximization
• Risk, Risk aversion, and Utility
1
Key terms
•
•
•
•
•
•
•
•
Expected return, Excess return, Risk premium
Measures of Risk
Utility function
Risk aversion
Mean-variance criterion
Indifference curve
Sharpe ratio (Reward-to-volatility ratio)
Capital Allocation Line (CAL)
2
Expected return, Excess return,
and Risk premium
• Expected return
𝐸 𝑟 = ෍ 𝑝 𝑠 𝑟(𝑠)
𝑠
• Excess return: 𝑅 = 𝑟 − 𝑟𝑓
• Risk premium = Expected excess return
𝐸 𝑅 = ෍ 𝑝 𝑠 𝑅(𝑠)
𝑠
Measures of Risk
• Variance
𝜎 2 = σ𝑠 𝑝(𝑠) 𝑟 𝑠 − 𝐸(𝑟)
• Standard deviation
𝜎=
2
𝜎2
• Risk
– Standard deviation of the excess return
Estimation of Return and Risk
If we treat each observation of return as an equally
likely scenario:
• Estimate of expected return = Arithmetic average
1
𝑛
𝐸 𝑟 = σ𝑠 𝑝 𝑠 𝑟 𝑠 = 𝑟ҧ = σ𝑛𝑠=1 𝑟(𝑠)
• Estimate of variance
𝜎ො 2
=
1
σ𝑛𝑠=1
𝑛−1
𝑟 𝑠 − 𝑟ҧ
• Estimate of standard deviation:
𝜎ො = 𝜎ො 2
2
Risk Aversion and Utility Values
• Utility function
1 2
𝑈 = 𝐸 𝑟 − 𝐴𝜎
2
•
•
•
•
•
U = utility
E(r) = expected return on the asset or portfolio
A = coefficient of risk aversion
𝜎 2 = variance of returns
½ = a scaling factor
6
Risk, Risk Aversion, and Utility
7
Risk Aversion and Utility Values
• Utility function
1 2
𝑈 = 𝐸 𝑟 − 𝐴𝜎
2
• Risk averse: A > 0
– Risk is “penalized”
• Risk-neutral: A = 0
– No penalty for risk
• Risk lover: A < 0
– Adjust the expected return upward
8
Trade-off between risk and return
Mean-variance (M-V) criterion:
Portfolio A dominates portfolio B if
𝐸 𝑟𝐴 ≥ 𝐸 𝑟𝐵 and 𝜎𝐴 ≤ 𝜎𝐵
9
Indifference Curve
10
Capital Allocation
• What is an optimal portfolio?
– A portfolio that maximize investor’s utility
• What is the optimum allocation of capital
between risk-free and risky assets?
– An allocation that maximizes investor’s utility
– Find the best from the feasible set of
allocation patterns
• How can we find such optimum allocation?
11
Portfolio of One Risky Asset and
a Risk-Free Asset
• Let y the proportion allocated to the risky
portfolio, P.
• Let 1-y the proportion allocated to the risk-free
asset, F.
•
•
•
•
Rate of return of P : 𝑟𝑃
Expected rate of return of P : 𝐸(𝑟𝑃 )
Standard deviation of P : 𝜎𝑃
Rate of return of F : 𝑟𝑓
• The risk premium on P : 𝐸 𝑟𝑃 − 𝑟𝑓
12
Portfolio of One Risky Asset and
a Risk-Free Asset
• The rate of return on the complete portfolio, C,
is
𝑟𝐶 = 𝑦𝑟𝑃 + 1 − 𝑦 𝑟𝑓
• The expected rate of return is
𝐸(𝑟𝐶 ) = 𝑦𝐸(𝑟𝑃 ) + 1 − 𝑦 𝑟𝑓
= 𝑟𝑓 + 𝑦 𝐸(𝑟𝑃 ) − 𝑟𝑓
• The standard deviation is
𝜎𝐶 = 𝑦𝜎𝑃
13
Portfolio of One Risky Asset and
a Risk-Free Asset
• Substitute 𝑦 = 𝜎𝐶 Τ𝜎𝑃 :
𝐸 𝑟𝐶
𝜎𝐶
= 𝑟𝑓 +
𝐸(𝑟𝑃 ) − 𝑟𝑓
𝜎𝑃
• 𝐸 𝑟𝐶 is a function of 𝜎𝐶 with intercept 𝑟𝑓 and
slope
𝐸(𝑟𝑃 ) − 𝑟𝑓
𝑆=
𝜎𝑃
• The slope is called the reward-to-volatility ratio
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or the Sharpe ratio.
Capital Allocation Line
Example: 𝐸 𝑟𝑃 = 15%, 𝜎𝑃 = 22%
𝑟𝑓 = 7%, 𝐸 𝑟𝑃 − 𝑟𝑓 = 8%
15
Utility Maximization
• Find an optimal allocation, 𝑦 ∗ , to maximize
investor’s utility.
• Solve the utility maximization problem:
1 2
max 𝑈 = 𝐸 𝑟𝐶 − 𝐴𝜎𝐶
𝑦
2
1 2 2
= 𝑟𝑓 + 𝑦 𝐸 𝑟𝑃 − 𝑟𝑓 − 𝐴𝑦 𝜎𝑃
2
• Optimal position:
𝐸 𝑟𝑃 − 𝑟𝑓
∗
𝑦 =
𝐴𝜎𝑃2
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Utility as a function of y (A=4)
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Calculations of Indifference
Curves
18
Indifference Curves for U = .09
and .05 with A = 2 and 4
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Expected Returns on Indifference
Curves and CAL (A = 4)
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Optimal Capital Allocation
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What is a typical value for A??
• A naive calculation from past data implies A =
2.94 :
𝐸 𝑟𝑀 − 𝑟𝑓
.081
∗
𝑦 =
=
= .656
2
2
𝐴 ×.2048
𝐴𝜎𝑀
𝐴=
.081
.656×.20482
= 2.94
Assumptions:
– 65.6% estimated from historical data.
– Portfolio M that mimics historical S&P500 risk
premium and risk: 8.1% and 20.48%.
• Several studies estimate in range of 2 to 4.
22
Key terms and formulas
• Expected return:
𝐸 𝑟 = σ𝑠 𝑝 𝑠 𝑟(𝑠)
• Excess return: 𝑅 = 𝑟 − 𝑟𝑓
• Risk premium = Expected excess return
𝐸 𝑅 = ෍ 𝑝 𝑠 𝑅(𝑠)
𝑠
• Variance:
𝜎 2 = σ𝑠 𝑝(𝑠) 𝑟 𝑠 − 𝐸(𝑟)
2
• Standard deviation: 𝜎 = 𝜎 2
23
Key terms and formulas
• Estimate of expected return = Arithmetic average
𝐸 𝑟 = σ𝑠 𝑝 𝑠 𝑟 𝑠 = 𝑟ҧ =
1 𝑛
σ 𝑟(𝑠)
𝑛 𝑠=1
• Estimate of variance
𝜎ො 2
=
1
σ𝑛𝑠=1
𝑛−1
𝑟 𝑠 − 𝑟ҧ
2
• Estimate of standard deviation:
𝜎ො =
𝜎ො 2
24
Key terms and formulas
• Utility function
•
•
•
•
1 2
𝑈 = 𝐸 𝑟 − 𝐴𝜎
2
Risk aversion
Mean-variance criterion
Indifference curve
Sharpe ratio (Reward-to-volatility ratio)
𝐸(𝑟𝑃 ) − 𝑟𝑓
𝑆=
𝜎𝑃
25
Key terms and formulas
• Capital Allocation Line (CAL)
𝜎𝐶
𝐸 𝑟𝐶 = 𝑟𝑓 +
𝐸(𝑟𝑃 ) − 𝑟𝑓
𝜎𝑃
= 𝑟𝑓 + 𝑆𝜎𝐶
• Optimal position of risky asset:
𝐸 𝑟𝑃 − 𝑟𝑓
∗
𝑦 =
𝐴𝜎𝑃2
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