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• supply curve:
– upward sloping
– higher int rate -> more supply of funds
CHAPTER 5
• demand curve:
– downward sloping
– higher int rate -> less demand of funds
RISK AND RETURN
=> equilibrium interest rate
• government
– shift S and D curves through fiscal (demand side) and
monetary (supply side) policies.
e.g. Increase in deficit D curve up  int rate up
e.g. expansionary mon. policy S up int rate down
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I. Basic concepts and issues
(ii)
Nominal interest rate:
considering Inflation rate
A. Interest Rate
• Inflation rate
= rate of change in Consumer Price Index
-
Basic rate:
interest rate on debt securities with no
possibility of getting default
it = (CPIt - CPIt-1)/CPIt-1
• Note:
The nominal interest rate is often regarded as
the minimum rate required by investors to
earn on a security. (risk-free rate)
(i) Real interest rate
(ii)Nominal interest rate
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(i)
-
Real interest rate
Real Rate vs. Nominal Rate
Fisher effect:
interest rate in a world with no inflation
• Approximation:
nominal rate = real rate + exp. inflation rate
R
=
r +
i
determined by:
(a)the supply of funds (e.g. personal deposit)
(b)the demand of funds (e.g. corp borrow $ )
(c)government fiscal & monetary policies.
-
• Exact:
(1+R)
=
(1+r ) x (1+i )
=> As long as the inflation rate > 0
the nominal rate (R) > the real rate (r)
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e.g. nominal rate (R) = 9%
expected inflation rate (i) = 6%
real rate (r) =?
• P1- P0 :
capital gains
(P1- P0 )/ P0 : capital gains yield
• D1:
D1/ P0 :
dividend
dividend yield
• HPR = (P1 – P0) / P0
+ D1 / P0
= [Capital gains yield] + [dividend yield]
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e.g.
You bought common stock of XYZ at $20 per
share at the beginning of 2021.
At the end of 2021, the price per share of
XYZ becomes $25 and dividend per share for
the year is $2.
B. Return
(i) Holding Period Return (HPR)
• Holding period:
the time interval of holding a security.
• Beginning wealth (Wo):
$investment at the beg of the period
• Ending wealth (W1):
$amount received from holding the security
during the holding period.
 HPR =
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Single Period Rate of Return for Stocks
(ii) Comparing HPRs of different
lengths of holding period
P1 = Ending price
P0 = Beginning price
D1 = Dividend during period one
• beginning wealth (W0)
= purchase price of the stocks ( Po )
• ending wealth (W1)
= P1 + D1
usually using annual HPR “annualized return”
converting with the assumption of reinvestment at the same
rate of return.
(i) “arithmetic averaging”
- HP=1/n yr: ra = r * n
- HP= n yrs: ra = (r1 + r2 + r3 + ... rn) / n
(ii) “geometric averaging” ~ considering compounding
- HP=1/n yr: rg = (1+r)n - 1
- HP= n yrs: rg = {[(1+r1) (1+r2) .... (1+rn)]}1/n - 1
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Arithmetic
e.g. A stock earns 8% on a semi-annual period.
(average without compounding)
Arithmetic averaging
The equivalent annual HPR
=
Geometric (average with compounding)
Geometric averaging
The equivalent annual HPR
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e.g. A bond earns 3% during a 90-day period.
Measuring Ex-Post (Past) Returns
Arithmetic averaging
Q: When should you use the GAR and when should you use
the AAR?
A: When you are evaluating PAST RESULTS (ex-post):
 Use the AAR (average without compounding) if you ARE
NOT reinvesting any cash flows received before the end of
each period.
 Use the GAR (average with compounding) if you
ARE reinvesting any cash flows received before the end of
each period.
The equivalent annual HPR
Geometric averaging
The equivalent annual HPR
5-17
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The following lists the stock price during
the past 5 years:
year
price
return
2017
$100
2018
75
2019
50
2020
75
2021
100
C. Risk
- uncertainty of cash flows generated from
holding the asset.
e.g.
A machine is expected to generate additional
$50,000 per year for the next 5 years.
 this series of CFs is subject to
uncertainty.
Find out the average annual return earned
on the stock?
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e.g. You bought a junk bond issued by XYZ
corp for $950, which promises to pay you
12% coupon rate per year for 10 years and
par value of $1000 at maturity.
D. To characterize risky returns:
Since the future HPR of a security is
uncertain (risky), one may set up a
scenario about the future HPR.
However, XYZ may not be able to pay the
interest and principle as promised.
Two measures regarding HPR are
particularly of our concern:
(1) expected return
(2) variance of return
the CF's generated from holding this
security is subject to uncertainty.
 The bond is a "risky" investment.
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Characteristics of Probability
Distributions
e.g. You bought AAPL's common stock at
$170 per share. You plan to sell them at the
end of the year. What is the return you'll
get?
1) Mean : most likely value
2) Variance or standard deviation
3) Skewness
You don't know.
•If a distribution is approximately normal,
the distribution is described by
1) mean (expected return)
2) variance (risk)
As the following are subject to uncertainty:
(i) the price appreciation
(ii) the dividend payments
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< Measure of Risk >
Normal Distribution
 Variance
a measure of risk for a security when the
security is held alone.
s.d.
variance of a security's HPR is not a proper
measure of risk for the security when it is
held in a portfolio.
s.d.
r
Symmetric distribution
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Skewed Distribution: Large Negative
Returns Possible
D.1. Measuring Expected Return
E(r) =  ps
s
Median
Negative
r
x rs
ps = probability of state ‘s’ occurring
rs = return if state ‘s’ occurs
E(r) = expected rate of return
1 to s states
Positive
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e.g.
Skewed Distribution: Large Positive
Returns Possible
(s)
(Ps)
State
Probability of
(rs)
of Economy
State Occurring
XYZ
Boom
0.2
30%
Normal
0.6
10
-5
Recession
0.2
____________________________________________
Median
Negative
r
Positive
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Implication?
 is an incomplete
risk measure
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D.2 Measuring Variance or
Dispersion of Returns
Leptokurtosis
Var(r)
=  ps
s
[rs - E(r)]2
Standard deviation = [variance]1/2
5-27
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• Previous example:
E. Arbitrage and Market Equilibrium
2 =  Ps x (rs-E(r))2
Stand. dev. () = ²
1.
[Arbitrage]
Simultaneous
buying one security/portfolio and
selling another comparable
security/portfolio to yield risk-free
return.
2 =
=
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Example:
State
1
2
3
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5
e.g.
The same sweaters are traded at different
prices in store A vs store B.
 Arbitrage:
- Buy a sweater at Store A for $40 and
- sell it at Store B for $42.
=> You made $2 without risk.
but: "transaction cost"
Scenario Distributions
Prob.
of State
ri
.1
-5%
.2
5%
.4
15%
.2
25%
.1
35%
1.0
Pi x ri
Pi x( ri-E(r))2
E(r) = (.1)(-.05) + (.2)(.05)...+ (.1)(.35)
E(r) = .15
Var =[(.1)(-.05-.15)2+(.2)(.05- .15)2...+ .1(.35-.15)2]
Var= .01199
S.D.= [ .01199] 1/2 = .1095
e.g. Buy security XYZ in market A for $100
and sell it in market B for $102.
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e.g. Suppose you are able to duplicate the cash flows of
security S with a portfolio Q (composed of several
securities)
 the cash flows generated from S are the same as
those from Q under any condition.
Relation between E(r) and risk of a
security
In general,
the higher the risk
 the more E(r) required by investors
for compensation of the risk.
 i) Portfolio Q = stock A + stock B
ii) Security S
 If Price of Q  Price of security S
=> arbitrage opportunity
=> buy low + sell high to get "arbitrage profit"
(e.g. program trading - use stock index futures)
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e.g. Choice A: Sure $1,000
Choice B: 50% chance getting $2,000
50% chance getting $ 0
2. [Market Equilibrium]
- A market is in equilibrium if there is no
arbitrage opportunity.

- When there is any arbitrage opportunity:
=> for undervalued security:
buying pressure will drive up its price.
=> for overvalued security:
selling pressure will drive down its price.
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F.2. Risk-free Rate and Risk Premium
"Law of One Price"
Risk-free rate (rf):
the rate of return earned on a security with
no risk
empirical proxy: T-bill rate
- For all assets generating the same cash
flows under any conditions, they should be
priced the same.
Risk Premium :
the additional return earned on a risky
security due to its riskiness
[Def.] RPi = E(ri) – rf
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F. Investors' risk attitude and
risk premium
e.g. The risk-free rate currently is 2%.
The exp return on ABC = 14%
The exp return on XYZ = 18%
F.1. Investors’ risk attitude
i)
-
Risk Aversion:
Given the same expected return, investors prefer the asset with
less risk.
Investors require higher E(r) to compensate for additional risk.

ii) Risk Loving:
Given the same expected return, investors prefer the asset with
more risk.
iii) Risk Neutral:
As long as assets have the same expected return, investors are
indifferent among these choices.
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G. Mean-Variance Criterion
H. Certainty Equivalent
- In the M-V portfolio theory, it is assumed
that investors are only concerned with two
measures regarding their overall portfolio
return: E(rp) and Var(rp).
For every risky investment, there is a
value (C.E.), which is the return from a
risk-free investment, such that the
investor is indifferent between the risky
investment and the risk-free one.
- It is assumed that investors have following
characteristics:
a.) Non-satiation: the more the better
b.) Risk aversion
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Specifically:
- Suppose P and Q represent two portfolios
held by an investor:
E(rP)  E(rQ) , and
Var(rP)  Var(rQ)
 Investors will choose
If:
e.g. 2 Investments:
Choice A: Risk-free $1,000 return.
Choice B: 50% chance $2,000
50% chance $ 600
RA=$1,000 E(RB)=$1,300
If the investor is indifferent between A and B, then
 C.E. of B = $1,000.
The risk premium of B = E(RB)-RA=$300.
Each risky investment's C.E. depends upon the degree
of risk aversion of investors.
Given the same risky investment, the more risk averse
the investor is, the lower the C.E. of the risky
investment.
For risk averse investors, the C.E. return of a risky
investment is lower than its expected return.
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I.
mean-variance graph:
-
HISTORICAL RECORDS
• World Large stocks:
Assume that you are allowed to hold one
of the following portfolio alone:
• 24 developed countries, ~6000 stocks.
• U.S. large stocks:
• Standard and Poor's 500 largest cap.
• U.S. small stocks:
• Smallest 20% on NYSE, NASDAQ, and Amex.
• World bonds:
• Same countries as World Large stocks.
• U.S. Treasury bonds:
• Barclay's Long-Term Treasury Bond Index.
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Table 5.3: Historical Return and Risk
T-bills
Average
Risk premium
T-bonds
e.g. If  = 20% and  = 6%, then
Stocks
3.38
5.83
11.72
Na
2.45
8.34
approx. 68.26% chance that
<r<
approx. 95.44% chance that
<r<
approx. 99.74% chance that
<r<
3.12
11.59
20.05
e.g. If  = 20% and  = 11%, then
max
14.71
41.68
57.35
−0.02
−25.96
−44.04
approx. 68.26% chance that
<r<
min
approx. 95.44% chance that
<r<
approx. 99.74% chance that
<r<
Standard deviation
high volatility  high return
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Frequency distributions of annual HPRs
II. CALCULATION OF SECURITY
WIEGHTS IN A PORTFOLIO
A.
Portfolio return and risk:
A portfolio P containing N component securities:
rp = w1r1 + w2r2 + ... + wNrN
where wi:
r i:
rp:
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weight of sec i in the portf p
determined by $ amt not a
random variable
HPR of component sec i (r.v.)
HPR of portf p (r.v.)
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If a stock has an expected return of 
and a standard deviation of 
then
one may apply the normal distribution to
interpret the likelihood of the stock return.
Assume that:
B. Calculation of the weight of a security in a
portfolio:
•
r ~ N (, 2)
approx. 68.26% chance that
wi : fraction of $ invested in security i
relative to total $value of the portfolio.
=> wi = $ invested in security i
total $ of portfolio
- <r< +
approx. 95.44% chance that  - 2 < r <  + 2
•
approx. 99.74% chance that  - 3 < r <  + 3
•
Long position (purchase) security i
 wi > 0.
Short position (short sell) security i
 wi < 0.
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e.g. You invest
e.g. You have 10,000 to invest.
You short sell $2000 of stock A and
invest all the rest in stock B.
Q: What are the weights?
$2000 in IBM stocks and
$8000 in GE stocks
Then:

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B.2.Use Lintner's def. of short
sales:
e.g. Mutual Fund XYZ contains the
following stocks:
- Assume:
a) 100% margin requirement for short sales,
b) short sale proceeds cannot be used for further
investment
Stocks
PPS
N of shrs
A
$50
400
B
$40
250
C
$100
200
____________________________________
Total
 Thus:
note: still
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 |wi|=1
wi > 0 for long positions
wi < 0 for short positions
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e.g. You have $10,000 available for investment. You
sell short stock A for $2,000 and invest the rest in
stock B.
Q: What are the weights assigned for each security?
B.1. Use standard def of short sales
  wi = 1.
-
 Assuming that
a) no margin requirement in short sales,
b) short sale proceeds can be used for
further investment.
We will use standard definition for short sales (i.e.
wi =1) unless specified otherwise.
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III. PORTFOLIO
EXPECTED RETURN & RISK
Expected Value Operations
A. Portfolio return:
X and Y: random variables
a, b, and c: constant parameters
•
a) E(aX) =
2-asset portfolios.
A portfolio P containing two sec A and B:
rp = wArA + wBrB
b) E(X + b) =
c) E(X + Y ) =
d) E(aX + bY) =
 Given the values for rA and rB, based on
the weights you can get the portfolio
return.
e) E(aX + bY + c ) =
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e.g. A portfolio P contains A and B:
X and Y: random variables
a, b, and c: constant parameters
wA=0.5 and wB=0.5:
rp = 0.5rA + 0.5rB
If
State of
economy
Booming
Normal
Recession
Variance Operations
a) Var (aX) =
Prob.
0.5
0.3
0.2
rA
20%
10%
-10%
rB
10%
5%
0%
rp
b) Var (X + a) =
c) Var (X + Y)
=
d) Var (aX + bY)
=
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Covariance / Correlation coefficient
• If rA and rB are not certain or yet realized,
then they are random variables:
-
Given E(rA) and E(rB)
 can get E(rp).
-
Given Var(rA), Var(rB), and Cov (rA,rB)
 can get Var(rp).
X and Y: random variables
Cov(X, Y) =  Pk x (Xk – E(X))(Yk-E(Y))
.If Cov(X,Y) > 0 => X & Y tend to move in the
same direction
.If Cov(X,Y) < 0 => X & Y tend to move in the
opposite directions
.If Cov(X,Y) = 0 => X & Y are not linearly related
Make use of fundamental statistical
operations to find E(rp) and Var(rp).
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Correlation coefficient
• N-asset case:
• Correlation coefficient between X & Y
-
Cov(X,Y)
 (X,Y) = ------------------X Y
A portfolio P is composed of N different
assets:
rp =w1 r1 + w2  r2 +...+ wN  rN
X Y
X,Y = ------------------X Y
• -1    +1
E(rp ) =E [ w1r1 + w2  r2 +...+ wN  rN]
• Correlation has the same sign as covariance
E(rp)
Cov(rA,rB)
 (rA,rB) = --------------------A B
=>
=w1  E(r1) + w2  E(r2) +….+ wNE(rN)
Cov(rA,rB) = A,B x A x B
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e.g.
e.g.
State of
economy
Booming
Normal
Recession
Prob. rA
0.5
20%
0.3
10%
0.2
-10%
rB
10%
5%
0%
P x (rA-E(rA))(rB-E(rB))
E(rA)=10% E(rB)=20%
wA
1.2
1
0.8
0.5
0.2
0
-0.2
-1
E(rA) =
E(rB) =
Cov (rA, rB) =  Pk x (rA,k – E(rA))(rB,k-E(rB))
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wB
-0.2
0
0.2
0.5
0.8
1
1.2
2
E(rp)
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C. the variance of a portfolio return.
B. Expected Return of a Portfolio.
• Two-asset case:
A portfolio P contains stocks A and B only:
• Two-asset case:
rp =
wArA + wBrB
where wA + wB = 1
Var(rp)
= Var(wArA + wBrB)
= wA2Var(rA) + wB2 Var(rB) + 2wAwB cov(rA,rB)
 E (rp) = E (wArA + wBrB)

E(rp) =
Or:
p2 = wA2 A2 + wB2 B2 + 2 wA wB AB
wAE(rA) + wBE(rB)
Note:
cov(rA,rB)  AB = Ax B x A,B
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e.g. A = 8% B = 15% A,B=0.2
What is the risk of your portfolio, if
a) You invest $800 in A & $200 in B?
b) You invest $300 in A & $700 in B?
e.g. Portfolio P consists of A and B:
If
wA=0.5 and wB=0.5:
rp = 0.5rA + 0.5rB
State of
economy
Booming
Normal
Recession

Prob.
0.5
0.3
0.2
rA
20%
10%
-10%
rB
10%
5%
0%
rp
rB
0%
5%
10%
rp
In this case,
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wA
wB
1.2
1
0.8
0.5
0.3
0.2
0
-0.2
-0.2
0
0.2
0.5
0.7
0.8
1
1.2
Var(rp)
P
e.g. Portfolio P consists of A and B:
If
wA=0.5 and wB=0.5:
rp = 0.5rA + 0.5rB
State of
economy
Booming
Normal
Recession
Prob.
0.5
0.3
0.2
rA
20%
10%
-10%
In this case,

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e.g. A=10%, B=10%
wA =0.5, wB=0.5
P = ? assuming different correlations
a) if A,B = 0
* Three-asset case:
3 securities: 1, 2, and 3, each with standard deviation of 1 ,2,
and 3
P2
b) if AB = 0.6
= w1212 + w2222 + w3232
+ 2 w1w212
+ 2 w1w313
+ 2 w2w323
The equation above can be re-written as:
P2 =
w1212
+ w1w212
+ w1w313
+ w2w112
+ w2222
+ w2w323
+ w3w131
+ w3w223
+ w3232
c) if AB = -0.6
= i=1,3 wi2i2 + i=1,3 j=1,3 wiwjij
j i
• Note:
1.) Cov(X,X) = Var(X)
2.) Cov(X,Y) = Cov(Y,X)
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*N-asset case:
P2 = w1212
+ w2w121
+….
+ wNw1N1
+ w1w212 +… + w1wN1N
+ w2222 +… + w2wN2N
+ wNw2N2 +… + wN2N2
P2 =i=1,N wi2i2 + i=1,N j=1,N wiwjij
j i
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