fin437 vicentiu covrig 1 portfolio management optimum asset allocation optimum asset allocation (see...

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

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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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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



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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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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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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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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

fin437 vicentiu covrig 1 portfolio management optimum asset allocation optimum asset allocation (see...

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