Chapter 5.pptx risk and return

8/14/2019 Chapter 5.pptx risk and return

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Risk and Return: Past andPrologue

Chapter 5


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Real and Nominal Rates of Interest

Nominal interest rate:Growth rate of your money

Real interest rate:

Growth rate of your purchasing power

(1 + rnom)= (1 + rreal) x (1 + i)

rnom rreal +i


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Real and Nominal Rates of Interest

Fisher Equation:

rnom= rreal+ E(i)


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Rates of Return

Holding-period return (HPR):

Example: Suppose an investor wants to invest in astock-index fund, which sells for $100 per sharetoday. Suppose also that this fund is expected to

pay a $4 cash dividend and sell for $110 one yearlater. What is the expected HPR? Capital gainsyield? Dividend yield?

t

ttt

P

DPPHPR 11


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Rates of Return

Measuring returns over multiple periods:

Arithmetic average:

Geometric average: Single per-period return that givesthe same cumulative performance as the sequence ofactual returns.

Dollar weighted return: The IRR of an investment.

n

ssr

n 1)(

1

1)1)...(1)(1( /121 n

nrrrg


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Rates of Return

Example: Consider a fund that starts with $1 millionunder management at the beginning of the year. Thisfund receives both additional funds to invest andredemptions from existing shareholders as given below.

Find the arithmetic and geometric averages, and thedollar-weighted return.

1st

Quarter

2nd

Quarter

3rd

Quarter

4th

Quarter

AUM at the start of the

quarter

1.0 1.2 2.0 0.8

HPR (%) 10.0 25.0 (20.0) 25.0

Net Inflow ($ million) 0.1 0.5 (0.8) 0.0

AUM at the end of the

quarter

1.2 2.0 0.8 1.0


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xpec e e urns an an arDeviation

Scenario analysis: What HPRs are possible and howlikely are they?

Expected return:

Variance (Var):

Standard Deviation (Std):

( ) ( ) ( )s

E r p s r s

2

2 ( ) ( ) ( )s

p s r s E r

2STD


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Expected Returns and StandardDeviation

Example: Suppose that, over the next year, there arefour possible scenarios with their probabilities andHPRs. Find the expected return and the standarddeviation of returns.

Scenario Probability HPR (%)

SevereRecession

0.05 -37

MildRecession

0.25 -11

NormalGrowth

0.40 14

Boom 0.30 30


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The Normal Distribution


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The Normal Distribution

Suppose that riis normally distributed with expectedreturn E(ri) and standard deviation i.

Then, the standardized return: =()

is normally

distributed with a mean of zero and a standard deviation of1. Therefore, is a standard normal variable.


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Risk

Value at Risk (VaR):Attempts to answer the followingquestion:How many dollars can I expect to lose on my portfolio in a

given time period at a given level of probability?

The typical probability used is 5%. We need to know what HPR corresponds to a 5% probability.

If returns are normally distributed then we can use a standard normaltable or Excel to determine how many standard deviations below themean represents a 5% probability:

From Excel: =Norminv (0.05,0,1) = -1.64485 standard deviations From the standard deviation we can find the

corresponding

level of the portfolio return:

VaR = E[r] + -1.64485


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Risk

Example: A $500,000 stock portfolio has an annualexpected return of 12% and a standard deviationof 35%. What is the portfolio VaR at a 5%probability level?


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Using Time Series of Return

tt rn

r 1

2)(1

1)(

ttt

rrn

rVar

)(trVarSTD


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Risk Premium and Risk Aversion

Risk-free rate: Risk premium:

Speculation vs. Gamble

Excess return:

Risk Aversion: Risk averse investors reject investment portfolios that are

fair games or worse

These investors are willing to consider only risk-free or

speculative prospects with positive risk premiums Intuitively one would rank those portfolios as more attractive

with higher expected returns


15/25


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Utility Function

Where

U= utility

E ( r ) = expected return on the asset or portfolio

A= coefficient of risk aversion

2= variance of returns

Types of investors:

1. Risk Averse2. Risk Neutral

3. Risk lover (risk seeking)

21

( ) 2U E r A


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Utility Values of Possible Portfolios for an Investorwith Risk Aversion, A = 4


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The Trade-off Between Risk and Returns of aPotential Investment Portfolio, P


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The Indifference Curve


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Indifference Curves

Given A, there is one indifference curve for each level of utility.

Curve 1

Curve 2

Increasing

Utility

E(r)


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Mean-Variance Criterion

A dominates B if E(rA) E(rB) and AB, withat least one strict inequality.

Goal of Risk-Averse Investors: Either

(i) Maximize expected return for a given level ofrisk (), or(ii) Minimize risk () for a given expected return.


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Mean-Variance Dominance

Expected Return

Variance or Standard Deviation

B

A

RiskyC

D

RF

A dominates B

A dominates D

C dominates D


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Asset Allocation Across Risky and Risk-freePortfolios

Asset Allocation: Portfolio choice among broadinvestment classes.

John Bogle of the Vanguard Group of InvestmentCompanies:

The most fundamental decision of investing is theallocation of your assets: how much should you hold instock? how much in bonds? how much in cash reserves.

That decision can account for an astonishing 94% of thedifferences in total returns achieved by institutionallymanaged pension funds.


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Asset Allocation Across Risky and Risk-freePortfolios

Example: Assume that the total market value of aninvestors portfolio is $300,000. Of that $90,000 isinvested in shares of Ready Asset money market fund(a risk-free asset). The remaining $210,000 is in risky

securities, say, $113,400 in shares of Vanguards S&P500 Index Fund and $96,000 in shares of FidelitysInvestment Grade Bond Fund.


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Portfolio Expected Return and Risk

Example: Suppose that, in the previous example,E(rp) = 15%, p= 22%, and rf= 7%. Draw the CapitalAllocation Line and calculate the Sharpe ratio.

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Chapter 5.pptx risk and return

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