Corporate Finance II

Week 1: Risk and Returns

Dr. Hui Zheng

Weekly Reading Schedule

Week Reading of Chapters 1 7 2 8 3 20, 26 4 21 5 22 6 17, 18 7 N/A 8 18, 24 9 3, 23

10 16 11 19 12 15 13 N/A

Outline of Today’s Lecture

1 Introduction – a century of market returns

2 The Problem

3 Measuring Returns : Arithmetic & Geometric Average

4 Measuring Risk : Variance (Standard Deviation)

5 A Recap of Variance and Covariance

6 How Securities Affect Portfolios

7 Risk and Return in Portfolios

8 The Relationship Between Risk and Return

$1

$10

$100

$1,000

$10,000

$100,000

1900

1910

1920

1930

1940

1950

1960

1970

1980

1990

2000

Start of Year

Dol

lars

(log

scal

e)

Common StockUS Govt BondsT-Bills

14,276

24171

2008

The Value of an Investment of $1 in Year 1900

$1

$10

$100

$1,000

1900

1909

1919

1929

1939

1949

1959

1969

1979

1989

1999

Start of Year

Dol

lars

(log

scal

e)

EquitiesBondsBills

581

9.85

2.87

2008

Real Returns

The Value of an Investment of $1 in 1900

-60.0

-40.0

-20.0

0.0

20.0

40.0

60.0

80.0

Rates of Return 1900-2008

Source: Ibbotson Associates Year

Per

cent

age

Ret

urn

Stock Market Index Returns

Measuring Risk

1 24

11 11

21

17

24

13

32

0

4

8

12

16

20

24

-50

to -4

0

-40

to -3

0

-30

to -2

0

-20

to -1

0

-10

to 0

0 to

10

10 to

20

20 to

30

30 to

40

40 to

50

50 to

60

Return %

# of Years

Histogram of Annual Stock Market Returns

(1900-2008)

Australian Data

Average rates of Return for Government Bonds, and Ordinary Shares for Australia 1882 - 1987

Portfolio Average Annual Rate of Return (Nominal) %/yr

Average Annual Rate of Return

(Real) %/yr

Average Risk Premium ( over bonds)

%/yr 10-Year Bonds 5.21

Shares 13.06 9.56 7.85

Source: R. R. Officer, in Ball et. al. 1989.

Volatility in Australian Shares

73 75 77 79 81 83 85 87 89 91 93 95

-30

-20

-10

0

10

20

30

40

50

6070

Total Return Percent

73 75 77 79 81 83 85 87 89 91 93 95

Year

Total returns, price changes plus dividends, for the Australian stockmarket. (Source:Datastream Australian Market Index - Total Returns)

• We have securities that entitle you to future cashflows: What is the value of these securities? Or, why does a security have “value”?

• Companies “issue” or sell securities

Company value is the value of the securities it has issued

• Individuals “acquire” or buy securities Individual wealth is the value of the securities they have bought/

invested in

The Problem

The above is a generalization . . . Companies and individuals can buy and sell securities

Note For ALL securities.

Price =PV [Expected future cashflows], = E[futurecashflows]

discount factor

Different models (usually some equation(s)) make different assumptions about:

1 The future cashflows, e.g. magnitude and timing 2 The probability associated with each future cashflow. 3 The present value calculation - the discount rate.

Recall Capital Budgeting in Corp I

The Problem

Prices affect the values of companies and individuals.

From prices we move easily to returns:

r1 = P1 − P0

P0 P1 = Price at time 1 P0 = Price at time 0

Say you own 100 shares worth $10 each, then Wealth = $1000

The Problem

Note Notation: E (r ) → Expected return. Assume that P0 is known; that is, it’s expected value is P0 .

Main point: When we “model” expected returns, we implicitly model expected prices. expected prices → expected wealth, hence our interest. In this course we will deal with both prices and returns.

Measuring Returns

Arithmetic Ave: 1 For historical returns: Add and divide by n . 2 For Future returns: Multiply by the probability and then add.

→ This suggests that 1/n is a proxy for the probability of a return

Geometric Ave: Multiply then raise to power of 1/n

Measuring Returns

Example A company is worth $100 on Monday. After some thought (analysis) you decide that on Wednesday:

Possible Wed. Values Probability $90 $110 $130

1 What is the arithmetic expected return? 2 What is the geometric expected return?

Which to use?

Measuring Returns

The Arithmetic average is frequently used as another term for the expected value. Returning to our example:

Possible Wed. Return Probability −10%

+10% +30%

1 The arithmetic mean or “expected” return

= 10%

Now assume the question is: You started with $100 and invested for 3 years. The amount you got for: Year 1 was down by 10%, Year 2 was up by 10% and Year 3 was up by 30%. What is the average return you actually earned over the three year investment?

Measuring Returns

2 The geometric “version” of the “expected” return calculation:

nonsense! Note the negative term. 3 The compound rate of return:

The compound rate of return → geometric mean/average.

1.287

Measuring Returns

Which to use when? Whichever describes the changes in wealth!

Use arithmetic mean when estimating costs of capital from historical returns. Use arithmetic mean when estimating the opportunity cost of capital (i.e. what is the opportunity cost of capital p.a.). Use geometric mean when estimating compounding changes in wealth (i.e. how much you will get by investing $1 from the start).

Measuring Risk

If we square the difference and call this the “variance” or σ2

(3)

For example, you flip a coin. If head you get $10, if tail you get $-10. What is the variance of this bet?

Hence we often standardize and use standard deviation = σ = 10%. Advantage → risk and return measures are in comparable units.

A natural concept of risk is “a deviation from what is expected”

$

Measuring Risk

Covariance is a measure of “co-movement”. The expression for the covariance between x and y is covar(x , y) or σx,y

If x achieves below average returns at the same time when y achieves above average returns Covariance is negative.

If x achieves above average returns at the same time when y achieves above average returns Covariance is positive.

If x achieves below average returns at the same time when y achieves below average returns. Covariance is positive.

A Recap of Variance and Covariance

⇒ covar(1, 2) ⇒ covariance between security 1 and security 2

where

A Recap of Variance and Covariance

2. Calculate as the weighted sum of the individual security variances and covariances:

where N is the number of securities. are %’s invested in each security.

is the covariance between security and security .

Portfolio variance: can be calculated in two ways:

1. Take the portfolio as ONE security, and then you can calculate its average return and variance following what we did just now; Or

Which one is better?

Example

Suppose you invest 60% of your portfolio in Exxon Mobil and 40% in Coca Cola. The expected dollar return on your Exxon Mobil stock is 10% and on Coca Cola is 15%. The expected return on your portfolio is:

%12)1540(.)1060(. ReturnExpected =×+×=

2222

22

211221

2112212221

21

)3.27()40(.σx3.272.181

60.40.σσρxxCola-Coca

3.272.181 60.40.σσρxx

)2.18()60(.σxMobil-Exxon

Cola-CocaMobil-Exxon

×=×××

×=×××

×=×=

Example

Suppose you invest 60% of your portfolio in Exxon Mobil and 40% in Coca Cola. The expected dollar return on your Exxon Mobil stock is 10% and on Coca Cola is 15%. The standard deviation of their annualized daily returns are 18.2% and 27.3%, respectively. Assume a correlation coefficient of 1.0 and calculate the portfolio variance.

Example

Suppose you invest 60% of your portfolio in Exxon Mobil and 40% in Coca Cola. The expected dollar return on your Exxon Mobil stock is 10% and on Coca Cola is 15%. The standard deviation of their annualized daily returns are 18.2% and 27.3%, respectively. Assume a correlation coefficient of 1.0 and calculate the portfolio variance.

% 21.8 0.477 DeviationStandard

0.47718.2x27.3)2(.40x.60x ]x(27.3)[(.40)

]x(18.2)[(.60) Variance Portfolio22

22

==

=++

=

Portfolio Risk The shaded boxes contain variance terms; the remainder contain covariance terms.

1 2 3 4 5 6

N 1 2 3 4 5 6 N

STOCK

STOCK To calculate portfolio variance add up the boxes

Portfolio Risk Example Correlation Coefficient = .4 Stocks σ % of Portfolio Avg Return ABC Corp 28 60% 15% Big Corp 42 40% 21% Standard Deviation: = [(282)(.62) + (422)(.42) + 2(.4)(.6)(28)(42)(.4)]1/2

= 28.1 Return : r = (15%)(.60) + (21%)(.4) = 17.4%

Let’s Add stock New Corp to the portfolio

Portfolio Risk Example Correlation Coefficient = .3 Stocks σ % of Portfolio Avg Return Portfolio 28.1 50% 17.4% New Corp 30 50% 19% NEW Standard Deviation = Portfolio = 23.43 NEW Return = weighted avg = Portfolio = 18.20%

NOTE: Higher return & Lower risk How did we do that? DIVERSIFICATION