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FIN437Vicentiu Covrig
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Portfolio managementPortfolio management Optimum asset allocationOptimum asset allocation
(see chapter 7 Bodie, Kane and Marcus)(see chapter 7 Bodie, Kane and Marcus)
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How Finance is organizedHow Finance is organized
Corporate finance
Investments
International Finance Financial Derivatives
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Risk and ReturnRisk and Return
The investment process consists of two broad tasks:
• security and market analysis
• portfolio management
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Risk and ReturnRisk and Return
Investors are concerned with both:
Expected return: comes from a valuation model
Risk
As an investor you want to maximize the returns for a given level of risk.
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Return: Calculating the expected return Return: Calculating the expected return for each alternativefor each alternative
occurs outcomeeach y that probabilit p
outcomeeach for return expected k
outcomes ofnumber n where
kP....kP k
return of rate expected k
nn11
^
^
Outcome Prob. of outcome Return in 1(recession) .1 -15%2 (normal growth) .6 15%
3 (boom) .3 25%
k^ =expected rate of return = (.1)(-15) + (.6)(15) +(.3)(25)=15%
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What is investment risk?What is investment risk? Investment risk is related to the probability of earning a low or negative
actual return. The greater the chance of lower than expected or negative returns, the riskier
the investment.
Expected Rate of Return
Rate ofReturn (%)100150-70
Firm X
Firm Y
Firm X (red) has a lower distribution of returns than firm Y (purple) though both have the same average return. We say that firm X’s returns are less variable/volatile (lower standard deviation ) and thus X is a less risky investment than Y
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Selected Realized Returns, Selected Realized Returns, 1926 – 20011926 – 2001
Average Standard Return Deviation
Small-company stocks 17.3% 33.2%Large-company stocks 12.7 20.2L-T corporate bonds 6.1 8.6L-T government bonds 5.7 9.4U.S. Treasury bills 3.9 3.2
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Investor attitude towards risk:Investor attitude towards risk:Does it matter?Does it matter?
Risk aversion – assumes investors dislike risk and require higher rates of return to encourage them to hold riskier securities.
Some individuals are risk lovers, meaning that they purchase/ invest in instruments with negative expected rate of return
Ex:
Risk premium – the difference between the return on a risky asset and less risky asset, which serves as compensation for investors to hold riskier securities
Very often risk premium refers to the difference between the return on a risky asset and risk-free rate (ex. a treasury bond)
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Loss AversionLoss Aversion
First decision: Choose between
Choice 1: sure gain of $ 85,000 Choice 2: 85% chance of receiving $100,000 and 15% chance of receiving nothing
Second decision: Choose between
Choice 1: sure loss of $ 85,000 Choice 2: 85% chance of losing $100,000 and 15% chance of losing nothing
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Behavioral Finance vs Standard FinanceBehavioral Finance vs Standard Finance
Behavioral finance considers how various psychological traits affect investors
Behavioral finance recognizes that the standard finance model of rational behavior can be true within specific boundaries but argues that this model is incomplete since it does not consider the individual behavior.
Currently there is no unified theory of behavioral finance, thus the emphasis has been on identifying investment anomalies that can beexplained by various psychological traits.
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Top Down Asset AllocationTop Down Asset Allocation
1. Capital Allocation decision: the choice of the proportion of the overall portfolio to place in risk-free assets versus risky assets.
2. Asset Allocation decision: the distribution of risky investments across broad asset classes such as bonds, small stocks, large stocks, real estate etc.
3. Security Selection decision: the choice of which particular securities to hold within each asset class.
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Expected Portfolio Rate of ReturnExpected Portfolio Rate of Return
- Weighted average of expected returns (Ri) for the individual investments in the portfolio
- Percentages invested in each asset (wi) serve as the weights
E(Rport) = wi Ri
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Portfolio Risk (two assets only)Portfolio Risk (two assets only)
When two risky assets with variances 12 and 2
2, respectively, are combined into a portfolio with portfolio weights w1 and w2, respectively, the portfolio variance is given by:
p2
= w121
2 + w222
2 + 2W1W2 Cov(r1r2)
Cov(r1r2) = Covariance of returns for Security 1 and Security 2
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Correlation between the returns of two securitiesCorrelation between the returns of two securities
Correlation, : a measure of the strength of the linear relationship between two variables
21
21 ),cov(
RR
-1.0 < < +1.0 If= +1.0, securities 1 and 2 are perfectly positively
correlated If= -1.0, 1 and 2 are perfectly negatively correlated If= 0, 1 and 2 are not correlated
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Efficient Diversification Efficient Diversification
Let’s consider a portfolio invested 50% in an equity mutual fund and 50% in a bond fund.
Equity fund Bond fundE(Return) 11% 7%Standard dev. 14.31% 8.16%Correlation -1
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Portfolo Risk and Return Combinations
5.0%
6.0%
7.0%
8.0%
9.0%
10.0%
11.0%
12.0%
0.0% 2.0% 4.0% 6.0% 8.0% 10.0% 12.0% 14.0% 16.0%
Portfolio Risk (standard deviation)Po
rtfol
io R
etur
n
% in stocks Risk Return0% 8.2% 7.0%5% 7.0% 7.2%10% 5.9% 7.4%15% 4.8% 7.6%20% 3.7% 7.8%25% 2.6% 8.0%30% 1.4% 8.2%35% 0.4% 8.4%40% 0.9% 8.6%45% 2.0% 8.8%
50.00% 3.08% 9.00%55% 4.2% 9.2%60% 5.3% 9.4%65% 6.4% 9.6%70% 7.6% 9.8%75% 8.7% 10.0%80% 9.8% 10.2%85% 10.9% 10.4%90% 12.1% 10.6%95% 13.2% 10.8%
100% 14.3% 11.0%
100% bonds
100% stocks
Note that some portfolios are “better” than others. They have higher returns for the same level of risk or less. We call this portfolios EFFICIENT.
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The Minimum-Variance FrontierThe Minimum-Variance Frontierof Risky Assetsof Risky Assets
E(r)
Efficientfrontier
Globalminimum
varianceportfolio Minimum
variancefrontier
Individualassets
St. Dev.
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Two-Security Portfolios with Various Two-Security Portfolios with Various Correlations Correlations
100% bonds
retu
rn
100% stocks
= 0.2
= 1.0
= -1.0
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The benefits of diversificationThe benefits of diversification
Come from the correlation between asset returns
The smaller the correlation, the greater the risk reduction potential greater the benefit of diversification
If= +1.0, no risk reduction is possible
Adding extra securities with lower corr/cov with the existing ones decreases the total risk of the portfolio
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Estimation IssuesEstimation Issues Results of portfolio analysis depend on accurate statistical inputs Estimates of
- Expected returns - Standard deviations- Correlation coefficients
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Portfolio Risk as a Function of the Number of Portfolio Risk as a Function of the Number of Stocks in the PortfolioStocks in the Portfolio
Nondiversifiable risk; Systematic Risk; Market Risk
Diversifiable Risk; Nonsystematic Risk; Firm Specific Risk; Unique Risk
n
Portfolio risk
Thus diversification can eliminate some, but not all of the risk of individual securities.
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Optimal Risky Portfolios and a Risk Free AssetOptimal Risky Portfolios and a Risk Free Asset
What if our risky securities are still confined to the previous securities but now we can also invest in a risk-free asset (e.g. T-bill)?
You have to decide how much to invest in risky securities and how much in the risk-free rate
You want the risky portfolio to be efficient
We use the Capital Allocation Line (CAL) to answer this question
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Capital Allocation LineCapital Allocation Line
E(rc) = yE(rp) + (1 - y)rf = rf + y[E(rp) - rf ]
c = y p
p
fp rErS
][)( fpp
cfc rErrrECAL
is the risk premium per unit of risk also called the reward-to-variability ratio
CAL shows all available risk-return combinations
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Optimal Risky Portfolios and a Risk Free AssetOptimal Risky Portfolios and a Risk Free Asset
Example:
1 year term deposit: rf = 3% f = 0
Bond fund: rb = 7% b = 8.19%
Equity fund: re = 11% e = 14.31%
(rb,re) = 0.3
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M
E(rp)
CAL (Globalminimum variance)
CAL (A)CAL (O)
O
A
rf
O M
A
G
O
M
p
Optimal Risky Portfolios and a Risk Free AssetOptimal Risky Portfolios and a Risk Free Asset
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Optimal Risky Portfolios and a Risk Free AssetOptimal Risky Portfolios and a Risk Free Asset
The CAL (O) corresponding to the tangency portfolio O provides the highest reward (risk premium) per unit of risk.
Why? Because it has the biggest slope.The efficient portfolio O is the optimum portfolio.
The coordinates of the optimum portfolio O are:ErO = 8.69% and O = 8.71%
In practice, you find the risk and return of the optimumportfolio using a computer program that looks for the portfoliowith the highest risk premium per unit of risk (S). (see your project)
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Optimal Risky Portfolios and a Risk Free AssetOptimal Risky Portfolios and a Risk Free Asset
The choice of weight a, how much to invest in the optimum risky portfolio, depends on your tolerance for risk and return requirement.
For example, in our case, the investor chooses to investa = 90% of his money in the optimum risky portfolio
And portfolio O consists of :wb = 57.8% in the bond fundwe = 42.2% in equity fund
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Optimal Risky Portfolios and a Risk Free AssetOptimal Risky Portfolios and a Risk Free Asset
The percentage of total portfolio invested in
bonds: a•wb = 0.9•0.578=0.52 or 52%
equity: a•we = 0.9•0. 422 =0.38 or 38%Total Portfolio Allocation
Risk FreeBondsEquity
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Optimal Risky Portfolios and a Risk Free AssetOptimal Risky Portfolios and a Risk Free Asset
Optimum risky portfolio: ErO = 8.69% O = 8.71%
Total portfolio :ErC = 0.1x3% + 0.9x8.69% = 8.12 % C = 0.9x8.71 = 7.84 %
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• Know the three steps of the top down asset allocation• Discuss the benefits of diversification. • Everything covered in these Recommended end-of chapter problems: 1,2,3 and 12
Learning objectivesLearning objectives