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**Risk and Return: Past and Prologue**

Chapter 5 Risk and Return: Past and Prologue

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**Return over One Period: Holding Period Return (HPR)**

HPR: Rate of return over a given investment period

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**Rates of Return: Single Period Example**

Ending Price = 110 Beginning Price = 100 Dividend = 4

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**Return over Multiple Periods**

$100 $50 $100 r1 r2 t = 0 1 2 r1, r2: one-period HPR What is the average return of your investment per period?

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**Return over Multiple Periods**

Arithmetic Average: rA = (r1+r2)/2 Geometric Average: rG = [(1+r1)(1+r2)]1/2 – 1

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**Return over Multiple Periods**

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

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**Data from Table 5.1 Quarter 1 2 3 4 HPR .10 .25 (.20) .25**

What are the arithmetic and geometric return of this mutual fund?

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**Returns Using Arithmetic and Geometric Averaging**

ra = (r1 + r2 + r rn) / n ra = ( ) / 4 = .10 or 10% Geometric rg = {[(1+r1) (1+r2) .... (1+rn)]} 1/n - 1 rg = {[(1.1) (1.25) (.8) (1.25)]} 1/4 - 1 = (1.5150) 1/4 -1 = = 8.29%

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

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**Quoting Conventions 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%

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Risk and return 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

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Return distribution s.d. s.d. r or E(r)

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HPR - Expected Return

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HPR - Risk Measure Variance or standard deviation:

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Problem 4, Chapter 5(p.154) Suppose your expectations regarding the stock market are as follows: State of the economy Scenario(s) Probability(p(s)) HPR Boom % Normal Growth % 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%

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**Historical Mean and Variance**

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

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**Risk and return in the past**

Year Ri(%) Compute the mean and variance of this sample

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**Annual Holding Period Returns From Table 5.3 of Text**

Historical Returns: Arith. Stan Risk Series Mean% Dev.% Premium World Stk US Lg Stk US Sm Stk Wor Bonds LT Treas T-Bills Inflation

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**Figure 5.1 Frequency Distributions of Holding Period Returns**

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**Figure 5.2 Rates of Return on Stocks, Bonds and Bills**

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**Risk aversion and Risk Premium**

Risk aversion: higher risk requires higher return, risk averse investors are rational investors Risk-free rate Risk premium (=Risky return –Risk-free return)

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Asset Allocation 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 Asset allocation decision accounts for 94% of difference in return of portfolio managers.

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**Portfolios: Basic Asset Allocation**

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. Basic decomposition into safe asset and market portfolio .

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**Complete Portfolio Expected Return**

Example: Let the expected return on the risky portfolio, E(rP), be 15%, the return on the risk-free asset, rf, 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% What is the return on the portfolio if all of the funds are invested in the risky portfolio? 15% 8%

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**Complete Portfolio Expected Return**

Example: Let the expected return on the risky portfolio, E(rP), be 15%, the return on the risk-free asset, rf, 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%

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**Complete Portfolio Risk Premium**

In general:

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**Portfolio Standard Deviation**

where sc - standard deviation of the complete portfolio sP - standard deviation of the risky portfolio srf - standard deviation of the risk-free rate y - weight of the complete portfolio invested in the risky asset

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**Portfolio Standard Deviation**

Example: Let the standard deviation on the risky portfolio, sP, 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%

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**Capital Allocation Line**

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

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**Capital Allocation Line**

Fig. 5.5 Risk Tolerance and Asset Allocation: More risk averse - closer to point F Less risk averse - closer to P

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

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

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Borrowing 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 = 1.5 1-y = -0.5 Negative sign means short position. Instead of earning risk-free rate as before, now have to pay risk-free rate

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**Borrowing at 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

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**Figure 5.5 Investment Opportunity Set with a Risk-Free Investment**

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**Borrowing with rate > risk-free rate**

Example: Let the expected return on the risky portfolio, E(rP), be 15%, the return on the risk-free asset, rf, be 7%, the borrowing rate, rB, be 9% and the standard deviation on the risky portfolio, sP, 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 rf.

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**Figure 5.6 Investment Opportunity Set with Differential Borrowing and Lending Rates**

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**Risk aversion and allocation**

More risk-averse, the complete portfolio C is close to F Less risk averse, C is close to P Why we should invest in stock in practice even if we don’t like risk at all A portfolio of 100% T-bond and a portfolio of 83% T-bond, 17% blue chip stocks have the same amount of risk, but the mix portfolio gives higher return Typical portfolio: 60% stock, 40% bond. To reduce risk, we can increase the proportion of bonds However, a more effective way is 75% stock, 25% bill, give the same return but less risk In short term, stock can be bad In long term, stocks outperform other investments.

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**Summary 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

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