finc2012 week 1
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university of sydney finc2012 corporate finance 2 hw for week 1TRANSCRIPT
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