Risk and Return: Past and Prologue Risk Aversion and Capital Allocation to Risk Assets

HPR: Rate of return over a given investment period

Ending Price = 110 Beginning Price = 100 Dividend = 4

 What is the average return of your investment per period? t = 012 $100$50$100 r1r1 r2r2 r 1, r 2 : one-period HPR

◦ Arithmetic Average: r A = (r 1 +r 2 )/2 ◦ Geometric Average: r G = [(1+r 1 )(1+r 2 )] 1/2 – 1

 Arithmetic return: return earned in an average period over multiple period ◦ It is the simple average return. ◦ It ignores compounding effect ◦ It represents the return of a typical (average) period ◦ Provides a good forecast of future expected return  Geometric return ◦ Average compound return per period ◦ Takes into account compounding effect ◦ Provides an actual performance per year of the investment over the full sample period ◦ Geometric returns <= arithmetic returns

Quarter HPR (.20).25 What are the arithmetic and geometric return of this mutual fund?

Arithmetic r a = (r 1 + r 2 + r r n ) / n r a = ( ) / 4 =.10 or 10% Geometric r g = {[(1+r 1 ) (1+r 2 ).... (1+r n )]} 1/n - 1 r g = {[(1.1) (1.25) (.8) (1.25)]} 1/4 - 1 = (1.5150) 1/4 -1 =.0829 = 8.29%

 Invest $1 into 2 investments: one gives 10% per year compounded annually, the other gives 10% compounded semi-annually. Which one gives higher return

APR = annual percentage rate (periods in year) X (rate for period) EAR = effective annual rate ( 1+ rate for period) Periods per yr - 1 Example: monthly return of 1% APR = 1% X 12 = 12% EAR = (1.01) = 12.68%

 Risk in finance: uncertainty related to outcomes of an investment ◦ The higher uncertainty, the riskier the investment. ◦ How to measure risk and return in the future  Probability distribution: list of all possible outcomes and probability associated with each outcome, and sum of all prob. = 1.  For any distribution, the 2 most important characteristics ◦ Mean ◦ Standard deviation

r or E(r) s.d.

Variance or standard deviation:

Suppose your expectations regarding the stock market are as follows: State of the economyScenario(s)Probability(p(s))HPR Boom10.344% Normal Growth20.414% Recession % Compute the mean and standard deviation of the HPR on stocks. E( r ) = 0.3* *14+0.3*(-16)=14% Sigma^2=0.3*(44-14)^2+0.4*(14-14)^2 +0.3*(-16-14)^2=540 Sigma=23.24%

Two variables with the same mean. What do we know about their dispersion?

Data in the n-point time series are treated as realization of a particular scenario each with equal probability 1/n

 YearRi(%)  Compute the mean and variance of this sample

Geom. Arith.Stan.Risk SeriesMean% Mean%Dev.% Premium World Stk US Lg Stk US Sm Stk Wor Bonds LT Treas T-Bills Inflation

Figure 5.2 Rates of Return on Stocks, Bonds and Bills

Figure 5.8 Standard Deviations of Real Equity and Bond Returns Around the World,

 A large enough sample drawn from a normal distribution looks like a bell-shaped curve. Probability Return on large company common stocks 99.74% – 3  – 48.3% – 2  – 28.1% – 1  – 7.9% % + 1  32.5% + 2  52.7% + 3  72.9% The probability that a yearly return will fall within 20.2 percent of the mean of 12.2 percent will be approximately 2/ % 95.44%

 The 20.14% standard deviation we found for large stock returns from 1926 through 2005 can now be interpreted in the following way: if stock returns are approximately normally distributed, the probability that a yearly return will fall within percent of the mean of 12.2% will be approximately 2/3.

Risk aversion: higher risk requires higher return, risk averse investors are rational investors Risk-free rate: the rate you can earn by leaving the money in risk-free assets such as T-bills. Risk premium (=Risky return –Risk-free return) It is the reward for investor for taking risk involved in investing risky asset rather than risk-free asset. The Risk Premium is the added return (over and above the risk- free rate) resulting from bearing risk.

 Historically, stock is riskier than bond, bond is riskier than bill  Return of stock > bond > bill  More risk averse, put more money on bond  Less risk averse, put more money on stock  This decision is asset allocation  John Bogle, chairman of the Vanguard Group of Investment Companies ◦ “The most fundamental decision of investing is the allocation of your assets: how much should you own in stock, how much in bond, how much in cash reserves. That decision accounts for an astonishing 94% difference in total returns achieved by institutionally managed pension funds.... There is no reason to believe that the same relationship does not hold true for individual investors.”

The complete portfolio is composed of: The risk-free asset: Risk can be reduced by allocating more to the risk-free asset The risky portfolio: Composition of risky portfolio does not change This is called Two-Fund Separation Theorem. The proportions depend on your risk aversion.

The Risky Asset Example Total portfolio value = $300,000 Risk-free value = 90,000 Risky (Vanguard & Fidelity) = 210,000 Vanguard (V) = 54% Fidelity (F) = 46%

The Risky Asset Example Continued Vanguard 113,400/300,000 = Fidelity 96,600/300,000 = Portfolio P 210,000/300,000 = Risk-Free Assets F 90,000/300,000 = Portfolio C 300,000/300,000 = 1.000

The Risk-Free Asset  Only the government can issue default- free bonds ◦ Guaranteed real rate only if the duration of the bond is identical to the investor’s desire holding period  T-bills viewed as the risk-free asset ◦ Less sensitive to interest rate fluctuations

 It’s possible to split investment funds between safe and risky assets.  Risk free asset: proxy; T-bills  Risky asset: stock (or a portfolio) Portfolios of One Risky Asset and a Risk- Free Asset

Example: Let the expected return on the risky portfolio, E(r P ), be 15%, the return on the risk-free asset, r f, be 7%. What is the return on the complete portfolio if all of the funds are invested in the risk-free asset? What is the risk premium? 7% 0 What is the return on the portfolio if all of the funds are invested in the risky portfolio? 15% 8%

Example: Let the expected return on the risky portfolio, E(r P ), be 15%, the return on the risk-free asset, r f, be 7%. What is the return on the complete portfolio if 50% of the funds are invested in the risky portfolio and 50% in the risk-free asset? What is the risk premium? 0.5*15%+0.5*7%=11% 4%

In general:

where  c - standard deviation of the complete portfolio  P - standard deviation of the risky portfolio  rf - standard deviation of the risk-free rate y - weight of the complete portfolio invested in the risky asset

Example: Let the standard deviation on the risky portfolio,  P, be 22%. What is the standard deviation of the complete portfolio if 50% of the funds are invested in the risky portfolio and 50% in the risk-free asset? 22%*0.5=11%

We know that given a risky asset (p) and a risk-free asset, the expected return and standard deviation of any complete portfolio (c) satisfy the following relationship: Where y is the fraction of the portfolio invested in the risky asset

Risk Tolerance and Asset Allocation: More risk averse - closer to point F Less risk averse - closer to P

S is the increase in expected return per unit of additional standard deviation S is the reward-to-variability ratio or Sharpe Ratio

Example: Let the expected return on the risky portfolio, E(r P ), be 15%, the return on the risk-free asset, r f, be 7% and the standard deviation on the risky portfolio,  P, be 22%. What is the slope of the CAL for the complete portfolio? S = (15%-7%)/22% = 8/22

So far, we only consider 0<=y<=1, that means we use only our own money. Can y > 1? Borrow money or use leverage Example: budget = 300,000. Borrow additional 150,000 at the risk-free rate and invest all money into risky portfolio y = 450,000/150,000 = y = -0.5 Negative sign means short position. Instead of earning risk-free rate as before, now have to pay risk- free rate

The slope = 0.36 means the portfolio c is still in the CAL but on the right hand side of portfolio P

Example: Let the expected return on the risky portfolio, E(r P ), be 15%, the return on the risk-free asset, r f, be 7%, the borrowing rate, r B, be 9% and the standard deviation on the risky portfolio,  P, be 22%. Suppose the budget = 300,000. Borrow additional 150,000 at the borrowing rate and invest all money into risky portfolio What is the slope of the CAL for the complete portfolio for points where y > 1, y = 1.5; E(Rc) = 1.5(15) + (-0.5)*9 = 18% Slope = ( )/0.33 = 0.27 Note: For y  1, the slope is as indicated above if the lending rate is r f.

SPECIAL CASE OF CAL (I.e., P=MKT) The line provided by one-month T-bills and a broad index of common stocks (e.g. S&P500) Consequence of a passive investment strategy based on stocks and T-bills

48 E(r) E(R m ) = 12% r f = 3% 20% 0 M F  S=0.45 P1? P3? P2?

FIN 8330 Lecture 7 10/04/0749  Risk Preference ◦ Risk averse  Require compensation for taking risk ◦ Risk neutral  No requirement of risk premium ◦ Risk loving  Pay to take risk  Utility Values: A is risk aversion parameter

Table 6.5 Utility Levels for Various Positions in Risky Assets (y) for an Investor with Risk Aversion A = 4

Figure 6.6 Utility as a Function of Allocation to the Risky Asset, y

Table 6.1 Available Risky Portfolios (Risk-free Rate = 5%)

Utility Function Where U = utility E ( r ) = expected return on the asset or portfolio A = coefficient of risk aversion   = variance of returns

Table 6.2 Utility Scores of Alternative Portfolios for Investors with Varying Degree of Risk Aversion

55  Greater levels of risk aversion lead to larger proportions of the risk free rate.  Lower levels of risk aversion lead to larger proportions of the portfolio of risky assets.  Willingness to accept high levels of risk for high levels of returns would result in leveraged combinations.

56  Solve the maximization problem:  Two approaches: 1.Try different y 2.Use calculus:  Solution:

Figure 6.8 Finding the Optimal Complete Portfolio Using optimal solution If A = 4, rf = 7%, E(Rp) = 15%,

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Table 6.6 Spreadsheet Calculations of Indifference Curves

Figure 6.7 Indifference Curves for U =.05 and U =.09 with A = 2 and A = 4

Figure 6.8 Finding the Optimal Complete Portfolio Using Indifference Curves

Table 6.7 Expected Returns on Four Indifference Curves and the CAL

FIN 8330 Lecture 7 10/04/0766  If CAL is from 1-month T-bills and a broad index of common stocks, then CAL is also called Capital Market Line (CML)  Why passive strategy: (1) strategies are costly; (2) market is competitive.  Mutual fund separation theorem: capital should be invested in the (same optimal) risky portfolio and risk- free asset.

Passive Strategies: The Capital Market Line  Passive strategy involves a decision that avoids any direct or indirect security analysis  Supply and demand forces may make such a strategy a reasonable choice for many investors

Passive Strategies: The Capital Market Line Continued  A natural candidate for a passively held risky asset would be a well-diversified portfolio of common stocks  Because a passive strategy requires devoting no resources to acquiring information on any individual stock or group we must follow a “neutral” diversification strategy

Definition of Returns: HPR, APR and AER. Risk and expected return Shifting funds between the risky portfolio to the risk-free asset reduces risk Examples for determining the return on the risk-free asset Examples of the risky portfolio (asset) Capital allocation line (CAL) All combinations of the risky and risk-free asset Slope is the reward-to-variability ratio Risk aversion determines position on the capital allocation line