1. The Trade-off between Risk and Return
Professor Thomson
Fin 3013

2. 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.

3. 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.

4. 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.

5. 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

6. 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?

7. 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?

8. 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

9. 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?

10. 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?

11. 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) =

12. Arithmetic Average Return
Add the individual hpr’s and divide by the number of years

13. 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%

14. IRR from a Constant Investment
P/YR=1 t CF -1 1 2 3
3.3166
Press  IRR = 5.10%

15. 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

16. 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

17. 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

18. 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

24. 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

25. 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

26. 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

27. 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.

28. 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

29. The Normal Distribution

30. 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.

31. 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

32. 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?

33. 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

34. 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.

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

36. 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

37. 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.

38. 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.

39. 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

41. 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

42. 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

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