Essentials of Investments: Chapter 5 - Risk and Return Past and Prologue

Essentials of Investments: Chapter 5 - Risk and Return Past and Prologue includes Rates of Return, Returns Using Arithmetic and Geometric Averaging, Dollar Weighted Returns, Dollar Weighted Average Using Text, Quoting Conventions.

Chapter 5

Risk and Return: Past and Prologue

Rates of Return: Single Period

Rates of Return: Single Period Example

HPR = ( 24 - 20 + 1 )/ ( 20) = 25%

Data from Text Example p 154

Assets(Beg.) 1.0 1.2 2.0 .8

TA (Before Net Flows 1.1 1.5 1.6 1.0 Net Flows 0.1 0.5 (0.8) 0.0 End Assets 1.2 2.0 .8 1.0

Returns Using Arithmetic and Geometric Averaging

Dollar Weighted Returns

Internal Rate of Return (IRR) - the discount rate that results present value

of the future cash flows being equal to the investment amount

• Considers changes in investment

• Initial Investment is an outflow

• Ending value is considered as an inflow

• Additional investment is a negative flow

Dollar Weighted Average Using Text Example

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%

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

rr Symmetric distribution

Normal Distribution

rr Negative Positive

Skewed Distribution: Large Negative Returns Possible

Median

rr Negative Positive

Skewed Distribution: Large Positive Returns Possible

Median

Subjective returns

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

1 to s states

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

Numerical Example: Subjective or

Standard deviation = [variance]1/2

Measuring Variance or Dispersion of Returns

Real vs Nominal Rates

Fisher effect: Approximation nominal rate = real rate + inflation premium

Annual Holding Period Returns From Figure 6.1 of Text

Annual Holding Period Risk Premiums and Real Returns

-• Possible to split investment funds between safe and risky assets

• Risk free asset: proxy; T-bills

• Risky asset: stock (or a portfolio)

Allocating Capital Between Risky &

Risk-Free Assets

Allocating Capital Between Risky &

Risk-Free Assets (cont.)

• Issues

– Examine risk/ return tradeoff – Demonstrate how different degrees of risk aversion will affect allocations between

risky and risk free assets

E(rc) = yE(rp) + (1 - y)rf

E(rc) = yE(rp) + (1 - y)rf

For example, y = 75 E(rc) = 75(.15) + 25(.07)

E(rc) = 75(.15) + 25(.07)

= 13 or 13%

Expected Returns for Combinations

Capital Allocation Line

Borrow at the Risk-Free Rate and invest in stock (while not really possible, lets assume we can do it)

Using 50% Leverage

rc = (-.5) (.07) + (1.5) (.15) = 19

sc = (1.5) (.22) = 33 Note that we assume the T-bill is totally risk free

Capital Allocation Line

Slope: Reward to variability ratio:

ratio of risk premium to std dev.

Stock price and dividend history Year Beginning stock price Dividend Yield

2001 $100 $4

2002 110 $4

2003 90 $4

2004 95 $4

An investor buys three shares at the beginning of

2001, buys another 2 at the beginning of 2002, sells 1 share at the beginning of 2003, and sells all 4 remaining at the beginning of 2004

A What are the arithmetic and geometric average

• Using the historical risk premiums as your guide from the chart earlier, what is your estimate of the expected annual

HPR on the S&P500 stock portfolio if the current risk-free interest rate is

5.0% What does the risk premium represent?

For the period of 1926- 2004 the large cap stocks returned 10.0%, less t-

bills of 3.7% gives a risk premium of 6.3%.

– If the current risk free rate is 5.0%, then– E(r) = Risk free rate + risk premium

– E(r) = 5.0% + 6.3% = 11.3%

– The risk premium represents the additional return that is required to compensate you for the additional risk you are taking on to invest

• You manage a risky portfolio with an expected

return of 12% and a standard deviation of 25%

The T-bill rate is 4% Your client chooses to invest 70% of a portfolio in your fund and 30%

in a T-bill money market fund What is the expected return and standard deviation of your client’s portfolio?

– Clients Fund

E(r) (expected return) =.7 x 12% + 3 x 4% =

9.6%

σ (standard deviation) = 7 x 25 =

• Suppose your risky portfolio includes investments

in the following proportions What are the investment proportions in your clients portfolio

C What is the reward-to-variability ratio (s) of your risky

portfolio and your clients portfolio?

– Reward to Variability (risk premium / standard deviation)

– Fund = (12.0% – 4%) / 25 = .32 – Client = (9.6% – 4%) / 17.5 = 32

D Draw the CAL of your portfolio What is the slope of the CAL?

Slope of the CAL line

% Slope = 3704

17 P

14 Client

Standard Deviation 18.9 277

Maximizing Standard Deviation

Suppose the client in Problem 12 prefers to invest in your portfolio a proportion (y) that maximizes the expected return on the overall portfolio subject to the constraint that the overall portfolio’s standard

deviation will not exceed 20% What is the investment proportion? What is the expected return

on the portfolio?

Portfolio standard deviation 20% = (y) x 25%

Y = 20/25 = 80.0%

Mean return = (.80 x 12%) + (.20 x 4%) = 10.4%

Increasing Stock Volatility

• What do you think would happen to the expected return on stocks if investors perceived an increased volatility of stocks due to some recent event, i.e

Hurricane Katrina?

• Assuming no change in risk aversion, investors perceiving higher risk will demand a higher risk premium to hold the same portfolio they held before

If we assume the risk-free rate is unchanged, the increase in the risk premium would require a higher expected rate of return in the equity market

• A Do you understand rates of return?

• B Do you know how to calculate

return using scenario, probabilities,

and other key statistics used to

describe your portfolio?

• C Do you understand the

implications of using a risky and a free asset in a portfolio?