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**The Trade-off between Risk and Return**

Professor Thomson Fin 3013

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**A trade-off always arises between expected risk and expected return.**

Risk and Return The return earned on investments represents the marginal benefit of investing. Risk is one of the marginal costs of investing (the other is the pure time value of money). A trade-off always arises between expected risk and expected return.

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Risk and Return Valuing risky assets - a task fundamental to financial management Three-step procedure for valuing a risky asset 1. Determine the asset’s expected cash flows 2. Choose discount rate that reflects asset’s risk 3. Calculate present value (PV cash inflows - PV outflows) The three-step procedure is called discounted cash flow (DCF) analysis.

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Financial Return Total return: the total gain or loss experienced on an investment over a given period of time Components of the total return Income stream from the investment Capital gain or loss due to changes in asset prices Total return can be expressed either in dollar terms or in percentage terms.

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**Cash Flow Time Line Pt-1 cash payments Pt**

Time t Time t Pt = Price at time t (today) Capital Gain = Pt – Pt-1 = Price today – Price last period Dollar Return = Cash Payments + Capital Gain = Cash Payments + Pt – Pt-1

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Example 6.1 You purchased a stock last year for $25. It has paid $1 in dividends and is not worth $21. What is your Dollar Return?

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Example 6.2 You bought an 11% coupon bond one year ago for $ You can sell that bond today for $ What is your Dollar Return?

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**Holding Period Return (hpr) or Percentage Return**

This is the most common way to express the gains or losses over a period It is the $Return relative to the amount invested 1+hpr is often called the wealth relative

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Example 6.1 (Revised) You purchased a stock last year for $25. It has paid $1 in dividends and is not worth $21. What is your Dollar Return? What is your hpr?

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Example 6.2 Revised You bought an 11% coupon bond one year ago for $ You can sell that bond today for $ What is your Dollar Return? What is your hpr?

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**Measuring Wealth over Time**

Year hpr $1 Investment $1 Investment each year 1 15% 1*(1.15) = 1.15 Vt-1(1+it) = Vt 1*(1.15)=1.15Vt-Vt- Vt-1 (1+it) = Vt 2 -10% 1.15*(0.90) =1.035 (1+1.15)*(0.90) = 1.935 3 13% 1.035*1.13 = ( )*(1.13) =

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**Arithmetic Average Return**

Add the individual hpr’s and divide by the number of years

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**Geometric Rate of Return**

Multiply by the wealth relatives, raise to the 1/N power and subtract 1 Is the constant rate of wealth building over time that results in the observed future value By Financial Calculator: P/YR=1 I/YR(FV=1.1696, PV=-1, N=3) = 5.36%

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**IRR from a Constant Investment**

P/YR=1 t CF -1 1 2 3 3.3166 Press IRR = 5.10%

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**Value of $1 Invested in Equities, **

Treasury Bonds and Bills, Year $15,579 $148 $61 $22 10,000 100,000 1,000 100 10 1 Equities Bonds Bills Inflation

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**Geometric Return Calculation**

A $1 investment in Large Stocks (with dividends reinvested) was worth after 103 years. The geometric mean return can be computed as (P/Yr = 1) I/YR(FV=15579, PV=-1, N=103)=9.83% Stocks I/YR(FV=148, PV=-1, N=76)=4.97% Long US Bond I/YR(FV=61, PV=-1, N=76)=4.07% US Treasury Bills I/YR(FV=22, PV=-1, N=76)=3.05% Inflation

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**Geometric Real Rates of Return**

To compute the long run real rate of return one can divide the ending value of the investment by the ending value of the inflation figure to determine the purchasing power of the investment. Then compute the return using “Real Dollars” Real value of the Large Stocks at end of period is 15579/22 = I/YR(FV=708.14, PV=-1, N=103)=6.58% I/YR(FV=6.73, PV=-1, N=103)=1.87% Long US Bond I/YR(FV=2.73, PV=-1, N=103)=1.00% TBill

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**Arithmetic versus Geometric Returns (1900-2003)**

Stocks Bonds Bills Nominal Returns Arithmetic Avg 11.7 5.2 4.1 Nominal Returns Geometric 9.8 5.0 Real Returns Arithmetic Avg 8.5 2.3 1.1 Real Returns Geometric 6.6 1.9 1.0

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Take home message If you have a short holding period, stocks are very risky, but from a longer term perspective they have provided the best returns both recently and historically Investment is not about saving money for the future, its about earning money from the money you invested so that most of your portfolio is from the earning of that portfolio and not from your deposits into that fund

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**Percentage Returns on Bills, Bonds, and Stocks, 1900 - 2003**

Difference between average return of stocks and bills = 7.6% Difference between average return of stocks and bonds = 6.5% Risk premium: the difference in returns offered by a risky asset relative to the risk-free return available

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**Why are Treasury Bills considered risk free?**

If the government default on Treasury Bills, your last concern will be the money you might have earned on the TB When you buy a Treasury Bill, you purchase it at a discount and redeem it at par, so you know when you buy it, what your return will be If you buy a stock, you don’t know what you will sell it for, or what dividends it will pay; thus, it is risky The yield on Treasury Bills, is generally taken to be the risk free return

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**Distribution of Historical Stock Returns, 1900 - 2003**

< to to to 0 to to to 30 to to >50 -20 -10 10 20 30 40 50 Percent return in a given year Probability distribution for future stock returns is unknown. We can approximate the unknown distribution by assuming a normal distribution.

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**Variability of Stock Returns**

Normal distribution can be described by its mean and its variance. A Normal Distribution is symmetric around the mean Variance (2) - the expected value of squared deviations from the mean Units of variance (%-squared) - hard to interpret, so calculate standard deviation, a measure of volatility equal to square root of 2

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**The Normal Distribution**

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**Volatility of Asset Returns**

Asset classes with greater volatility pay higher average returns. Average return on stocks is more than double the average return on bonds, but stocks are 2.5 times more volatile.

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**Average Returns and St. Dev. for Asset Classes, 1900-2003**

Stocks Bills Bonds Standard Deviation (%) Investors who want higher returns have to take more risk The incremental reward from accepting more risk seems constant

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**Average Return and St. Dev. for Individual Securities, 1994-2003**

Average risk for all stocks in this period was 60% For various asset classes, a trade-off arises between risk and return. Does the trade-off appear to hold for all individual securities?

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**Average Return and St. Dev. for Individual Securities, 1994-2003**

Wal-Mart Anheuser-Busch American Airlines Archer Daniels Midland Standard Deviation (%) No obvious pattern here

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Diversification Most individual stock prices show higher volatility than the price volatility of portfolio of all common stocks. How can the standard deviation for individual stocks be higher than the standard deviation of the portfolio? Diversification: investing in many different assets reduces the volatility of the portfolio. The ups and downs of individual stocks partially cancel each other out.

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The standard deviation of the portfolio is lower than the standard deviation of either Coke or Wendy’s

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**The Impact of Additional Assets on the Risk of a Portfolio**

Number of Stocks Systematic Risk Portfolio of 11 stocks AMD Unsystematic Risk AMD + American Airlines AMD + American Airlines + Wal-Mart Portfolio Standard Deviation

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**Systematic and Unsystematic Risk**

Diversification reduces portfolio volatility, but only up to a point. Portfolio of all stocks still has a volatility of 21%. Systematic risk: the volatility of the portfolio that cannot be eliminated through diversification. Unsystematic risk: the proportion of risk of individual assets that can be eliminated through diversification, for example, by buying mutual funds. Because this risk can be eliminated, there is no reward for holding unsystematic risk What really matters is systematic risk….how a group of assets move together.

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**Systematic and Unsystematic Risk**

Anheuser Busch stock had higher average returns than Archer-Daniels-Midland stock, with smaller volatility. American Airlines had much smaller average returns than Wal-Mart, with similar volatility. The tradeoff between standard deviation and average returns that holds for asset classes does not hold for individual stocks. Because investors can eliminate unsystematic risk through diversification, market rewards only systematic risk. Standard deviation contains both systematic and unsystematic risk.

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**Risk and Return Investment performance is measured by total return.**

Trade-off between risk and return for assets: historically, stocks had higher returns and volatility than bonds and bills. One measure of risk: standard deviation (volatility) Unsystematic and (systematic) risk: risk that can (cannot) be eliminated through diversification, respectively

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**Total dollar return = income + capital gain / loss**

Dollar Returns Total dollar return = income + capital gain / loss Terrell bought 100 shares of Micro-Orb stock for $25 A year later: Dividend = $1/share Sold for $30/share Dollar return = (100 shares) x ($1 + $5) = $600 Owen bought 50 shares of Garcia Inc. stock for $15 A year later: No dividends paid Sold for $25/share Dollar return = (150 shares) x ($15) = $500

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Percentage Returns Terrell’s dollar return exceeded Owen’s by $100. Can we say that Terrell was better off? No, because Terrell and Owen’s initial investments were different: Terrell spent $2,500 in initial investment, while Owen spent $750. Percentage return: total dollar return divided by the initial investment

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Percentage Returns In percentage terms, Owen’s investment performed better than Terrell’s did.

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