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Investments II
Ken Choie
http://dasan.sejong.ac.kr/~kchoie/
Syllabus
1. Portfolio Theory:
Mean-Variance analysis (Markowitz Portfolio Theory)
A Quadratic Utility function & the Efficient Frontier
The Capital Market Line (Risk-free asset & the Market Portfolio)
The Capital Asset Pricing Model (Mean-Beta analysis)
The Capital Asset Pricing Model vs. the Arbitrage Pricing Model
The Black-Scholes Option Pricing Model
2. The Asset Allocation Decision:
3. The Asset Price Cycle: the Dynamic Analysis
The market psychology: the reflexivity theory of George Soros
The financial instability hypothesis of Hyman Minsky
Review of Asset Valuation:
How much is it worth to you? (e.g., estimate the value of the security)
Compare your valuation with the prevailing market price to decide if you want to buy/keep it.
The Theory of Valuation (The bottom-up approach ):
the expected cash flows (quality of profit potential),
your required rate of return
identify undervalued (overvalued) stocks to buy (sell).
Hard Assets:
Commodity
Real Estate
Financial Assets:
Fixed-Income Securities
Equity
Derivatives
Valuation of financial investments:
Bonds: promised cash flow
Preferred stock: stated dividend for an infinite period
Common stock:
the discounted cash flow methods
Dividends, Free cash flow to equity, Residual Income
the discount rate (the cost of equity)
the relative valuation method
P/Earnings, P/ Cash Flow, P/ Book Value, P/ Sales
The asset allocation Decision (the top-down approach):
Economy/market:
analysis of alternative economies (countries) and security markets (bonds, stocks,
cash)
Industry:
Identify promising and vulnerable industries
Individual stock:
Identify promising stocks and see if they are undervalued
Recapitulation of the Theory of Probability:
1. A particular outcome within random outcomes and its probability:
a certain outcome
Probability = the number of possible outcome
 To compute the number of possible outcome, study combination and permutation;
 Independent probability
 Conditional probability
2. The probability distribution of a random variable: the range of outcome of a random
variable and their associate probabilities.
A random variable:
 A certain outcome may take on a particular value within a range of outcomes;
 Which particular value the random variable may take on has an associated probability;
 The relationship between the value of the random variable and its associated probability
is called the probability distribution (or the probability distribution function)
The probability distribution of a random variable X:
A measure of its central tendency:
E (X) =
∑R f (X) ∗ X
=
: prob. weighted average
∑ P (Xi) ∗ Xi
A measure of the dispersion of outcomes:
Var (X) =
E [ Xi – E(Xi)]2
= ∑R f (X) ∗ [ Xi – E(Xi)]2
: prob. weighted dispersion
= ∑ P (Xi) ∗ [ Xi – E(Xi)]2
A measure of the relationship between the mean and the variance of X:
The coefficient of variance:
The Sharpe’s ratio:
σ
μ
E(X)− Rf
σ
3. The joint probability distribution of two random variables:
The relationship between two random variables, X and Y:
The mean:
E (X)
= ∑R X f(X)
= μx
The variance:
Var (X)
= E [ (X − μx )2 ]
= ∑R(X − μx )2 f(X)
= σ2 x
The mean:
E (Y)
= ∑R Y f(Y)
= μy
The variance:
Var (Y)
= E [ (Y − μy )2 ]
= ∑R(Y − μy )2 f(Y)
= σ2 y
The joint probability distribution function, f (X,Y):
The 3-D representation of the relationship;
The covariance:
= E [ (X – μx ) (Y − μy )]
Cov (X,Y)
= ∑R [(X − μx ) (Y - μy )] f (X,Y)
= σxy
The correlation coefficient:
ρxy =
Cov (X,Y)
σx σy
4. A linear combination of random variables is also a random variable.
Let Z be a linear combination of two random variables X and Y.
then Z has its own probability distribution function.
E (Z) =
E(aX + bY)
= a E(X) + b E(Y)
Var (Z) =
Var(aX + bY) =a2 Var(X) + 2ab Cov(X,Y) + b2 Var(Y)
5. Regression:
A deterministic linear relationship between two variables
Yi
=
a + b * Xi
∆𝑌
Where, b = ∆𝑋
A statistical linear relationship between two variables
= > a statistical linear relationship between two economic variables.
Yi
=
α + 𝛽 * Xi
E (Yi ) = α + β * Xi
+ ϵi
where, Yi ~ N [ E (Yi ) , σ2 ]
Visualize the statistical relationship: the slope and Y-axis intercept
(𝑋 − 𝑋̅)(𝑌 − 𝑌̅)
𝛽̂ = 𝑖 (𝑋 − 𝑋̅𝑖 )
𝑖
Yi
=
α + 𝛽 * Xi
+ ϵi
The estimator for β that minimizes the variance of [ Yi − E (Yi )] is:
β̂ =
Cov (Xi , Yi )
Var (Xi )
The estimator for the Y-axis intercept is:
̅ - β̂ ∗ X
̅)
̂ = (Y
α
Hence, the estimate of Yi is ̂
Yi :
Yi
=
α + 𝛽 * Xi
+ ϵi
̅ - β̂ ∗ ̅
= (Y
X) + β̂ * Xi
̂
Y
the estimate of σ2 is σ
̂2 :
𝜎2
=
σ2
̂
=
1
n
1
n
∑ [ Yi − 𝐸(𝑌i )] 2
∑ [ Yi − ̂
Yi ] 2
1. Portfolio Theory:
The modern portfolio theory (i.e., the mean-covariance analysis) assumes normally distributed
returns and a quadratic utility function.
Mean-Variance analysis (Markowitz Portfolio Theory): (Chap 50)
Uses expected rate of portfolio return as a measure of reward, variability (uncertainty) of rate of
portfolio return as a measure of risk. (Assume that the pdf is known)
n
E (Rp)
=

Wi * E(Ri)
; weighted average of the prob. weighted averages
i 1
Var (Rp)
=

i
WiWj Cov (Ri, Rj)
j
=  Wi 2 
2
+
Where, Cov (Ri, Rj)

WiWj Cov (Ri, Rj)
j i
i
i
= E {[Ri- E(Ri)] [Rj – E(Rj)]}
=
Corr. coefficient =

i
ij

ij
j
=Cov (Ri, Rj)
[σ i *σ j]
1
If Wi = 𝑛 then
2
1 ̅̅̅̅̅
Var (Rp) = 𝑛  +
𝑛−1
𝑛
̅̅̅̅̅
𝐶𝑜𝑣
̅̅̅̅̅2 1−𝜌̅
= [ 𝑛 + 𝜌̅ ]
Where
̅̅̅̅̅̅
2

= average variance
̅̅̅̅̅
𝐶𝑜𝑣 = average covariance between tow stocks
𝜌̅ = average correlation coefficient
Objective:
minimize Var (Rp), given E (Rp):
reduce the portfolio variance through reducing the covariances.
Potential combinations of two assets.
The benefits of diversification: if Cov [R(i), R(j)] <0.
ERi
𝜌𝑖𝑗 = -1
𝜌𝑖𝑗 = 1
σi
A Quadratic Utility function & the Efficient Frontier:
A quadric utility function:
the diminishing marginal utility of wealth
In the mean-variance plain: first derivative >0, the second derivative >0
The optimal portfolio (in the mean-variance frame):
maximizes the investor’s utility.
The Efficient Frontier:
u4
ERi
u3
u3
u1
u3¹
Efficient frontier
u2¹
u1¹
0
i
The Capital Market Line
(Risk-free asset & the Market Portfolio): (Chap 51)
Risk-free asset:
the asset that has a zero variance also has a zero covariance with risky assets.
Market Portfolio:
the unique portfolio of risky assets on the efficient frontier.
Capital Market Line (CML):
combinations of the risky free asset and the Market Portfolio.
(The market portfolio is perfectly correlated with all other portfolio on the CML.)
E (Rp) =
w* Rf + (1- w) * E (Rm)
Var (Rp)=
(1 – w) * Var (Rm)
E (R i)
Capital market line
M
ER m
Efficient frontier
Rf
0
For individual assets within the market portfolio:
m

the relevant measure of the risk is its covariance with the Market Portfolio.
Rit =
a
Where
= Var (

2
i
b
=
i
Volatility
b
Mmt)
 
Total risk
b
+
i
2
i

Rmt +
i
, characteristic line
Cov (R i,R m )
, recall regression
Var (R m )
=
+
2
m
i
Var (
+


Var (Rit)
i
)
2
i
= systematic risk
+
unsystematic risk
=market/ economy
volatility
+
firm specific
volatility
The capital market line with borrowing at a rate higher than the risk-free rate will be flatter than
when the borrowing rate equals the risk free rate: the portfolio will not be as profitable as when
the borrowing rate equals the risk free rate.
The Capital Asset Pricing Model (Mean-Beta analysis):
The relationship b/w the beta (systemic risk) of a security and its expected rate of return:
[ E(Ri) – Rf] =

Where

i
E(Ri) = Rf +

i
i
[ E(Rm) – Rf]
=
; security market line
Cov (R i,R m )
, recall regression
Var (R m )
[ E(Rm) – Rf];
The expected rate of return in asset i is the sum of the risk-free rate and the risk premium
which is a function of the market risk premium.
The required rate of return of the security market, given a beta
(assuming homogeneous expectation on the expected rate of return)
vs. your own estimate of the expected rate of return.
E (Ri )
Security market line
ER m
Rf
0
 1
 eta
The Capital Asset Pricing Model vs. the Arbitrage Pricing Model:
The uni-variate pricing model (Sharpe):
E(Ri) = Rf

+
i
[ E(Rm) – Rf]
A Multi-factor asset pricing model (Ross):
E(Ri) = Rf
j  ij
+
[ E(Rg) – Rf]
The Black-Scholes Option Pricing Model:
The weighted average (given the pdf) of the PV of the European call option value on the
expiration date:
C(S, X, Rf, t,  2 ) =



f(S)
*
-rt
e
Where f(S) ~ lognormal
*
(S - X) ds
2. The Asset Allocation Decision: (Chap 49)
The process of managing portfolio starts with developing an investment policy (plan) statement
that reflects the investor’s goal, constraints and risk tolerance.
Investment Policy Statement:
A road map that guides the investment process:


it articulates investor’s realistic goals and
sets the standards (a benchmark portfolio) for evaluating portfolio performance.
Investment Objectives:
capital preservation vs. capital appreciation;
current income vs. total return (capital gains + reinvestment of current income)
how much risk is right for you?
Investment Constraints:
Liquidity needs,
investment time horizon,
tax & legal factors,
unique preferences
Given that,
 managing investment portfolio consists of managing risk and managing returns,

as investment needs change over a person’s life cycle, do does the optimal asset
allocation between income (bonds) and capita gains (stocks),
accumulation phase, consolidation phase, spending phase, gifting phase.
The portfolio management process:
investment policy statement
- > constructing a portfolio
- > evaluate portfolio
- > monitor investor’s changing needs
- > modify the portfolio
Asset allocation distributes the investment fund among different asset classes and countries
consistent with the investment policy statement in a constantly changing economic environment.
The importance of asset allocation:
How one manages the risk and return of one’s investment fund is reflected in the asset
allocation decision.
The asset allocation decision determines most of the portfolio’s returns over time.
About 90% of a single investment fund;
About 40% of the variation in fund returns across all funds
3. The Asset Price Cycle: the Dynamic Analysis
The Price Cycle theory (George Soros):
The reflexivity theory says that greed (fear) will reinforce the cycle of price rise (fall)
The Financial Instability hypothesis (Hyman Minsky):
Liquidity is the result of the appetite of borrowers (i.e., leveraged investors) to underwrite risk
and the appetite of (unleveraged) savers to provide leverage to borrowers (i.e., leveraged
investors) who want to underwrite risk.
There are three types of borrowers:
Hedge unit:
Income from the asset purchased with a loan is sufficient to pay the interest and
to amortize the loan
Speculative unit:
Income from the asset purchased with a loan is sufficient to pay the interest only
but not enough to amortize the loan
Ponzi / Madoff unit:
Income from the asset purchased with a loan is insufficient for amortizing the
loan or even for paying all the interest.
As asset prices are rising at a steady pace for a long period, borrowers walk the path from being
hedge units to speculative units and, finally, to Ponzi/ Madoff units in the self-full-filling
prophecy (the reflexivity process). As the risk appetite of borrowers (i.e., the leveraged investors)
increases and they would go after increasingly marginal opportunities, and the real interest rate
(the real Fed fund rate) would rise.
The cycles of economic activities and asset prices:
Ken Choie
An expanding economy brings profits to entrepreneurs who are the beneficiaries of either their
skills or luck.
Greater profit in the real sector spurs greater demand for assets.
Rising asset prices elicit greater demand for assets.
Greed is infused with optimism, confidence, animal spirits, etc
The leveraged-up speculators push asset prices to ever-higher levels. e.g., asset prices start
outpacing earnings.
The role of the central bank at this point is to pop the bubble early by raising the interest
rates; it is a politically unpopular thing for the central banker to do so.
People become aware of the unsustainably large gap between price and value.
Asset prices stop rising, which prompts a weaker demand for assets.
Fear and pessimism start supplanting greed and optimism.
Banks cut the credit lines to speculators; speculators get margin calls; speculators’ equity
evaporates.
Fear morphs into panic; the asset prices fall precipitously.
Will you buy my seat in the burning theater?
As the market value of collaterals collapses, creditors/banks suffer big losses; debtors are forced
to take austerity measures.
Demand for (and supply of) credit/ loan dries up.
The collapse in the financial sector drags down the real sector of the economy.
The main-street folks, the 99%, now would have to rescue the wall-street folks, the 1%,
and clean up grudgingly the mess the financial collapse left behind.
Expansionary fiscal policies and lower interest rate monetary policy ensue.
Eventually, the economy may come back.
History repeats itself: the cycle repeats itself all over again.
The instability of asset prices is a fact of life: suckers are born every day.