Investments

Lecture 4 Risk and Return

Introduction to investments

 Investing  Definition of an investment: The current commitment of dollars for a period of time in order to derive future payments that will compensate the investor for: • 1. The time value of money • 2. The expected rate of inflation • 3. The risk associated with the investment

Introduction to investments

 Expected returns versus required returns  The expected return on a security is the return an investor expects to receive over some future time period.  The expected return is estimated by determining an expected future price and income for a security and measuring the return in a manner similar to the actual return calculation.

Characteristics of Probability Distributions

1) Mean: most likely value 2) Variance or standard deviation 3) Skewness * If a distribution is approximately normal, the distribution is described by characteristics 1 and 2

Measuring Mean: Scenario or Subjective Returns

Subjective returns E

(

r

) = S

s p

(

s

)

r

(

s

)

p(s) = probability of a state r(s) = return if a state occurs 1 to s states

Numerical Example: Subjective or Scenario Distributions State 1 2 3 4 5 Prob. of State .1

.2

.4

.2

.1

r in State -.05

.05

.15

.25

.35

E(r) = (.1)(-.05) + (.2)(.05)...+ (.1)(.35) E(r) = .15

Introduction to investments

 Expected returns versus required returns  We can also measure the expected return by estimating several potential outcomes and assigning a probability to each outcome. The expected return in this case would be equal to:   P i *E(r) i

Example #1

 Assume we have estimated that the price of General Motors will be $90 one year from now and GM will pay $2.00 in dividends during that time. If the current price of GM is $85 what is the expected annual return for GM?

Example #2

 Assume we have estimated GM’s expected return over four equally likely states of nature.      E(r i ) 5% 8% 12% 25% P i .25

.25

.25

.25

Example #3

 You have invested $100,000. Over the next year, you determine that there is a 60% chance that your money could equal $150,000 and a 40% chance that you could lose money with the ending wealth value being equal to $80,000. What is the expected return on the investment?

Introduction to investments

  The required rate of return  Recall that an investment is the current commitment of dollars for a period of time in order to derive future payments that will compensate the investor for: • a. The time the funds are committed; • b. The expected rate of inflation; and • c. The uncertainty of future payments.

These three factors determine an investor’s required rate of return

Introduction to investments

 Differences between the required rate of return and the expected rate of return  The expected return is merely the return that we expect from our investment in the future.  The development of these expectations can be based upon historical returns, fundamental analysis, market analysis, etc.  Thus, the expected return and the required return do not have to be the same.

Introduction to investments

 Comparing the required rate of return and the expected rate of return  a. E(r) > RR investment is undervalued, buy   b. E(r) = RR investment is fairly valued, indifferent c. E(r) < RR investment is overvalued, sell • In an efficient market, E(r) = RR on average • Choosing securities that have E(r) > RR is essence of any active investment technique.

Introduction to investments

 Actual return calculations  1. Percentage return on an investment =  ((P t -P t-1 ) + income t )/P t-1



Example #4

Example: We buy a Widget for $15 and sell it in a year for $18. The stock pays a dividend of 1.00 during the year. What is our percentage return on the stock?

Inflation-Adjusted Returns

    Nominal returns are returns that measure the dollar returns to a stock No mention of the change in purchasing power TR IA = ((1 + TR) / (1 + IF)) –1 Example – Jane bought and then sold a stock that returned 12% last year. The inflation rate was 3%. What is her return?

(1.12)/(1.03) – 1 = 8.74% Why is it important to find IA returns?

Introduction to investments

 Investment gains come in two forms  a. Capital gains (or losses)  b. Income

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 - 1 = 12.68%

Introduction to investments

 What if we don’t have the beginning and ending values of the investment?

 In this case the rate of return over the life of the investment is:  Rate of return for entire time period = [(1 + R 1 )*(1 + R 2 )*(1 + R 3 )*...*(1 + R n )]-1   R i = return in intermediate time period i, n = the number of intermediate time periods.

Example #5

 Suppose that we know that Widget’s stock earned 10%, 5%, -5%, 6%, and 12% over the last five years. What would the rate of return on Widget’s stock have been if we had held the stock over the last five years?

Introduction to investments

 Finding the return for subperiods of the entire life of the investment  Sometimes you will want to take a rate of return and split it into an equivalent compounded return for a smaller time period.   1  End value Beg value Beg value   1 /

k

 1

n

(   1

i

( 1 

HPY i

)) 1 /

k

 1

Example #6:

 From example #5, suppose that we wanted to find the annual rate of return that, when compounded over five years, will yield the total rate of return over the five year life of the investment.

• Thus, if we earned 5.43% compounded each year for five years, we would have earned the same return, 30.27%, as we earned on Widget’s stock over five years.

Example #7:

 We bought Widget for $40 a share a year ago, the company recently paid a dividend of $3 a share, and the stock is currently selling for $45. Suppose that we plan to sell the stock and want to find the weekly rate of return that, when compounded will yield the total rate of return over last year for Widget’s stock.

• Thus, if we had earned .35% per week compounded over the last year we would have earned the same rate of return (20%) as Widget’s stock.

Introduction to investments

 Arithmetic and geometric average returns

AM

  

i n

  1

HPY i

 

n GM

  

i n

  1 ( 1 

HPY i

)   1

n

 1

Example #8

Week 1 2 Beg. Value 1000 2000 End. Value 2000 1000 Return 100% -50%  The question of which average return measure is superior (geometric or arithmetic) becomes a question of whether or not the holding period of the investment is important.

Risk

   Uncertainty about future returns Suppose you can invest money now in an index fund with each share costing $100. You have a one year time horizon. You expect the dividend over the year to be $5.

 Dividend yield is ??

At year-end, your best guess is that the shares will be selling for $108.

 Capital gain is ??

 Holding period return is ?? (assuming dividend is paid at year end)

Risk

 However, there is considerable uncertainty about the price.

 Why? What might affect the price of the fund?

  Can we quantify our uncertainty about the price of the fund? - or more specifically about the state of the stock market by assigning probabilities to a variety of scenarios that we believe may occur E(r) =  P(s)*R(s) (see Example #2)

Measuring Variance or Dispersion of Returns

Subjective or Scenario

Variance

= S

s p

(

s

) [

r s

-

E

(

r

)]

2 Standard deviation = [variance] 1/2 Using Our Example: Var =[(.1)(-.05-.15) 2 +(.2)(.05- .15) 2 ...+ .1(.35-.15) 2 ] Var= .01199

S.D.= [ .01199] 1/2 = .1095

Risk

 The standard deviation can be used to measure risk. There are 2 different measures.

 1. The standard deviation of historical returns  

i n

  1 

HPY i n

  1

HPY

 2

Risk

 2. The standard deviation of expected returns   

P

(

s

)[

R

(

s

) 

E

(

r

)] 2  

n s

 1 The use of the correct standard deviation formula is very important and depends on whether returns are expected or observed.

Find the standard deviation from Ex. #2.

Example #9

 Find the standard deviation of the expected return from the investment described in example #3.

Actual Risk

 Standard deviation based on historical data is as follows  

t n

  1 (

r t



r

) 2

n

 1 .

Risk/Return Terminology

           Expected return Actual return Required return Risk-free rate Excess return Risk premium Risk aversion Real return Nominal return Business risk Financial risk

Equity Risk Premium

 Additional return available to stockholders over risk-free return   ERP = ((1 + TR CS ) / (1 + RF)) – 1 Importance of ERP  Consistency of ERP   Default premium for bonds Term premium for bonds

Portfolio Risk/Return

 Weighted average of the expected returns of the individual securities in the portfolio

E

(

r p

)  

i n

 1

w i E

(

r i

)  Measure of risk for a portfolio is the portfolio standard deviation  It is not a weighted average of the standard deviations of the individual securities in the portfolio.

(  ) 

i n

  1 [ R i E(R i )] 2 P i