EC3050 Investment Analysis
Module A
Professor Patrick Honohan
http://www.tcd.ie/Economics/staff/phonohan
Valuing risky securities
Module A: Risk
•
Portfolio theory
•
Standard models (CAPM, APT)
•
Role of leverage
•
Are markets really efficient?
– Behavioural theories
– Irrational market pricing – bubbles
•
Options
Module B: Time (Dr. Denny)
Textbooks
Main text
Bodie, Zvi, Alex Kane and Alan J. Marcus: Investments (McGrawHill/Irwin) (7th Edition, 2007) (Earlier editions will do)
A good complementary approach:
Elton, Edwin J., Martin J. Gruber, Stephen J. Brown and William N.
Goetzmann: Modern Portfolio Theory (John Wiley & Sons) (7th
Edition, 2006).
Highly recommended supplementary reading:
Shiller, Robert J., Rational Exuberance (Princeton Univ Press) (“nd
Edition, 2005)
Topic 1: Portfolio theory and asset returns
•
•
•
•
•
•
•
•
Mean and variance
Diversification
Statistical models of return
Efficient frontier
Portfolio separation theorem for M-V investors
CAPM – market price of risk
Arbitrage, factor pricing
Performance measures
Bodie et al. Chaps 6-11(or Elton et al. Chaps 4-11)
Means and variances
Probabilities of these states occurring is Pr(s).
Payoff for asset i in state s is R(i,s)
The “expected value” or “mean” return for asset i is defined as:
S
E(i) =  i   Pr( s ) Ri ( s )
s 1
The variance of return is defined as
S
S
Var(i) =    Pr( s ) R ( s )     Pr( s )( Ri ( s )   i ) 2
2
i
s 1
2
i
2
i
s 1
The standard deviation σ is just the square root of the variance
…and covariances
The covariance of the returns for assets i and j is defined as:
S
Cov(i,j) =  ij   Pr( s)[( Ri ( s )   i )( R j ( s)   j )]
s 1
More about means and variances
• This does not necessarily exhaust the information about
future returns
• (For example the same mean and variance could be
associated with positive or negative skew).
• The mean of an average is the average of the mean
• This is not true for variances!
Three key probability distributions:
(i) Binomial distribution:
Toss n coins. Then the probability of getting exactly k heads is:
n!
Pr( k ) 
(1 / 2) k (1 / 2) n k
(n  k )! k!
That is because the number of combinations of k heads in n tosses is
n!
, where k! = k(k – 1)(k – 2)….3.2.1
C n,k 
(n  k )! k!
For repeated trials of a two-outcome (binomial) process with probability p
n!
Pr( k ) 
p k (1  p) n k
(n  k )! k!
Consider an asset for which the payoff for a head is 1, 0 otherwise
This has mean μ = np
And
Variance σ2 = np(1 – p)
(ii) Poisson distribution:
Let the number of time periods be very large, and suppose p is rather small
.
Then
Pr( k )  e

k
k!
where λ = np
Mean and variance are both λ
(iii) Normal (Gaussian) distribution
This is a continuous distribution:
outcome can be any real number
(not just integers)
Probability is given by this complicated formula
1
 2
e [ x   /  ]
2
/2
Guess what the mean and variance are!
The average of “any” distribution converges to this
The sum of any two normal distributions is Normal
So great for working with portfolios!
Coins and markets:
Similarities and differences
Choice 1: Choose between A and B
– A uses a single coin toss: pays 1 on H
– B uses two coins: pays 2 on HH, nothing otherwise
Choice 2: Choose between A and C
– C pays on next-day returns on EU and US market
index: pays 2 on EU and US both down, nothing
otherwise
• Lower variance means A is preferred to B
– but C may be preferred to A because the outcomes
(market movements) may be correlated with
remainder of player’s portfolio/prospects
– even though C has the same mean and variance as B
Lesson of coins and markets game
Mean and variance are sufficient information
in a coin-tossing game…
…but covariance kicks in more generally
i.e. covariance with the rest of investor’s
prospects
Diversification: Surprisingly little practised by US household investors
Morgan Kelly, 1995
Top 2 per cent of income distribution