1. Risk and Return: Past and Prologue
Chapter 5
Risk and Return: Past and Prologue

2. Return over One Period: Holding Period Return (HPR)
HPR: Rate of return over a given investment period

3. Rates of Return: Single Period Example
Ending Price = 110
Beginning Price = 100
Dividend = 4

4. 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?

5. Return over Multiple Periods
Arithmetic Average: rA = (r1+r2)/2
Geometric Average: rG = [(1+r1)(1+r2)]1/2 – 1

6. 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

7. 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?

8. 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%

9. 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

10. 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%

11. 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

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

13. HPR - Expected Return

14. HPR - Risk Measure
Variance or standard deviation:

15. 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%

16. 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

17. Risk and return in the past
Year Ri(%)
Compute the mean and variance of this sample

18. 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

19. Figure 5.1 Frequency Distributions of Holding Period Returns

20. Figure 5.2 Rates of Return on Stocks, Bonds and Bills

21. 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)

22. 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.

23. 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 .

24. 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%

25. 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%

26. Complete Portfolio Risk Premium
In general:

27. 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

28. 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%

29. 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

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

31. 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

32. 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

33. 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

34. 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

35. Figure 5.5 Investment Opportunity Set with a Risk-Free Investment

36. 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.

37. Figure 5.6 Investment Opportunity Set with Differential Borrowing and Lending Rates

38. 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.

39. 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