**Unformatted text preview: **Chapter 21
Modern Portfolio Theory
&
Chapter 22
Equilibrium Asset Pricing 1 21.2 Basic Mean-Variance Portfolio Theory
"MODERN PORTFOLIO THEORY"
(aka "Mean-Variance Portfolio Theory", or “Markowitz Portfolio
Theory” – Either way: “MPT” for short)
¾ DEVELOPED IN 1950s (by MARKOWITZ, SHARPE, LINTNER)
(Won Nobel Prize in Economics in 1990.)
¾ WIDELY USED AMONG PROFESSIONAL INVESTORS
¾ FUNDAMENTAL DISCIPLINE OF PORTFOLIO-LEVEL
INVESTMENT STRATEGIC DECISION MAKING. 2 See Chapter Appendix
I. REVIEW OF STATISTICS ABOUT PERIODIC TOTAL RETURNS:
(Note: these are all “time-series” statistics: measured across time, not across assets within a
single point in time.)
"1st Moment" Across Time (measures “central tendency”):
“MEAN”, used to measure:
)
Expected Performance ("ex ante", usually arithmetic mean: used in portf ana.)
)
Achieved Performance ("ex post", usually geometric mean) "2nd Moments" Across Time (measure characteristics of the deviation around the central
tendancy). They include…
1) "STANDARD DEVIATION" (aka "volatility"), which measures:
)
Square root of variance of returns across time.
)
"Total Risk" (of exposure to asset if investor not diversified)
2) "COVARIANCE", which measures "Co-Movement", aka:
)
"Systematic Risk" (component of total risk which cannot be "diversified away")
)
Covariance with investor’s portfolio measures asset contribution to portfolio total
risk.
3) "CROSS-CORRELATION" (just “correlation” for short). Based on contemporaneous
covariance between two assets or asset classes. Measures how two assets "move together":
)
important for Portfolio Analysis.
4) "AUTOCORRELATION" (or “serial correlation”: Correlation with itself across time), which
reflects the nature of the "Informational Efficiency" in the Asset Market; e.g.:
)
Zero
Î "Efficient" Market (prices quickly reflect full information; returns
lack predictability) Î Like securities markets
(approximately).
)
Positive
Î "Sluggish" (inertia, inefficient) Market (prices only gradually
incorporate new info.) Î Like private real estate
markets.
)
Negative
Î "Noisy" Mkt (excessive s.r. volatility, price "overreactions")
Î Like securities markets (to some extent). 3 "Picture" of 1st and 2nd Moments . . . 30 Actual Asset Value
1st & 2nd Moment of Return 25 L.R. Trend Asset Value
1st Moment Only of Return Asset Value 20 15 10 5 0
1977 2002 2027 2052 2077 2102 2127 2152 2177 2202 2227 Year Figure by MIT OCW. First Moment is "Trend“. Second Moment is "Deviation" around trend.
Food for Thought Question:
IF THE TWO LINES ABOVE WERE TWO DIFFERENT ASSETS, WHICH
WOULD YOU PREFER TO INVEST IN, OTHER THINGS BEING EQUAL? . . . 4 Historical statistics, annual periodic total returns:
Stocks, Bonds, Real Estate, 1970-2003…
S&P500 LTG Bonds Private Real
Estate Mean (arith) 12.7% 9.7% 9.9% Std.Deviation 17.5% 11.8% 9.0% 100% 27.2% 16.6% 100% -21.0% 1st Moments Correlations:
S&P500
LTG Bonds
Priv. Real Estate 2nd Moments 100% PORTFOLIO THEORY IS A WAY TO CONSIDER BOTH THE 1ST &
2ND MOMENTS (& INTEGRATE THE TWO) IN INVESTMENT
ANALYSIS. 5 21.2.1. Investor Preferences & Dominant Portfolios
SUPPOSE WE DRAW A 2-DIMENSIONAL SPACE WITH RISK (2ND-MOMENT)
ON HORIZONTAL AXIS AND EXPECTED RETURN (1ST MOMENT) ON
VERTICAL AXIS.
A RISK-AVERSE INVESTOR MIGHT HAVE A UTILITY (PREFERENCE)
SURFACE INDICATED BY CONTOUR LINES LIKE THESE (investor is indifferent
along a given contour line): RETURN P Q RISK THE CONTOUR LINES ARE STEEPLY RISING AS THE RISK-AVERSE
INVESTOR WANTS MUCH MORE RETURN TO COMPENSATE FOR A LITTLE
MORE RISK.
6 A MORE AGGRESSIVE INVESTOR MIGHT HAVE A UTILITY
(PREFERENCE) SURFACE INDICATED BY CONTOUR LINES LIKE THESE. RETURN P Q RISK THE SHALLOW CONTOUR LINES INDICATE THE INVESTOR DOES NOT
NEED MUCH ADDITIONAL RETURN TO COMPENSATE FOR MORE RISK. BUT BOTH INVESTORS WOULD AGREE THEY PREFER POINTS TO
THE "NORTH" AND "WEST" IN THE RISK/RETURN SPACE. THEY
BOTH PREFER POINT "P" TO POINT "Q".
7 FOR ANY TWO PORTFOLIOS "P" AND "Q" SUCH THAT:
EXPECTED RETURN "P" ≥ EXPECTED RETURN "Q"
AND (SIMULTANEOUSLY): RISK "P" ≤ RISK "Q"
IT IS SAID THAT: “Q” IS DOMINATED BY “P”.
THIS IS INDEPENDENT OF RISK PREFERENCES.
Î BOTH CONSERVATIVE AND AGGRESSIVE INVESTORS WOULD
AGREE ABOUT THIS.
IN ESSENCE, PORTFOLIO THEORY IS ABOUT HOW TO AVOID INVESTING
IN DOMINATED PORTFOLIOS.
DOMINATES
"Q" RETURN P
DOMINATES
"Q" Q
DOMINATED
BY
"Q" RISK
PORTFOLIO THEORY TRIES TO MOVE INVESTORS
FROM POINTS LIKE "Q" TO POINTS LIKE "P". 8 Section 21.2.2…
III. PORTFOLIO THEORY AND DIVERSIFICATION...
"PORTFOLIOS" ARE "COMBINATIONS OF ASSETS".
PORTFOLIO THEORY FOR (or from) YOUR GRANDMOTHER:
“DON’T PUT ALL YOUR EGGS IN ONE BASKET!”
WHAT MORE THAN THIS CAN WE SAY? . . .
(e.g., How many “eggs” should we put in which “baskets”.) In other words,
GIVEN YOUR OVERALL INVESTABLE WEALTH, PORTFOLIO THEORY TELLS YOU HOW
MUCH YOU SHOULD INVEST IN DIFFERENT TYPES OF ASSETS. FOR EXAMPLE:
WHAT % SHOULD YOU PUT IN REAL ESTATE?
WHAT % SHOULD YOU PUT IN STOCKS?
TO BEGIN TO RIGOROUSLY ANSWER THIS QUESTION, CONSIDER... 9 AT THE HEART OF PORTFOLIO THEORY ARE TWO BASIC
MATHEMATICAL FACTS:
1) PORTFOLIO RETURN IS A LINEAR FUNCTION OF THE ASSET
N
WEIGHTS: r P = ∑ wn r n
n=1 IN PARTICULAR, THE PORTFOLIO EXPECTED RETURN IS A
WEIGHTED AVERAGE OF THE EXPECTED RETURNS TO THE
INDIVIDUAL ASSETS. E.G., WITH TWO ASSETS ("i" & "j"):
rp = ωri + (1-ω)rj
WHERE ωi IS THE SHARE OF PORTFOLIO TOTAL VALUE INVESTED
IN ASSET i. e.g., If Asset A has E[rA]=5% and Asset B has E[rB]=10%, then a
50/50 Portfolio (50% A + 50% B) will have E[rP]=7.5%.
10 THE 2ND FACT:
2) PORTFOLIO VOLATILITY IS A NON-LINEAR FUNCTION OF THE
N N
ASSET WEIGHTS: VAR P = ∑ ∑ wi w j COVij
I =1 J =1 SUCH THAT THE PORTFOLIO VOLATILITY IS LESS THAN A
WEIGHTED AVERAGE OF THE VOLATILITIES OF THE
INDIVIDUAL ASSETS. E.G., WITH TWO ASSETS:
sP = √[ ω²(si)² + (1-ω)²(sj)² + 2ω(1-ω)sisjCij ]
≤ ωsi + (1-ω)sj
WHERE si IS THE RISK (MEASURED BY STD.DEV.) OF ASSET i. e.g., If Asset A has StdDev[rA]=5% and Asset B has
StdDev[rB]=10%, then a 50/50 Portfolio (50% A + 50% B) will
have StdDev[rP] < 7.5% (conceivably even < 5%).
Î This is the beauty of Diversification. It is at the core of Portfolio Theory. It
is perhaps the only place in economics where you get a “free lunch”: In this
case, less risk without necessarily reducing your expected return!
11 The diversification effect is greater the less correlated are the assets…
Stocks & bonds (+30% correlation): Each dot is one year's returns.
Stock & Bond Ann. Returns, 1970-2003:
+30% Correlation
50% 40% Bond Returns 30% 20% 10% 0%
-30% -20% -10% 0% 10% 20% 30% 40% 50% -10% -20% Stock Returns 12 The diversification effect is greater the less correlated are the assets…
Stocks & real estate (+17% correlation): Each dot is one year's returns.
Real Est. & Stock Ann. Returns, 1970-2003:
+17% Correlation
30%
25%
20% R.E. Returns 15%
10%
5%
0%
-30% -20% -10% 0% 10% 20% 30% 40% 50% -5%
-10%
-15% Stock Returns 13 The diversification effect is greater the less correlated are the assets…
Bonds & real estate (-21% correlation): Each dot is one year's returns.
Real Est. & Bond Ann. Returns, 1970-2003:
-21% Correlation
30%
25% Real Estate Returns 20%
15%
10%
5%
0%
-20% -10% 0% 10% 20% 30% 40% 50% -5%
-10%
-15% Bond Returns 14 For example, a portfolio of 50% bonds & 50% stocks would not have provided much
volatility reduction during 1981-98, though over the longer 1970-2003 period it would
have reduced the half&half portfolio to just bond volatility:
Annual Periodic Total Returns, Long-Term Bonds and Stocks, 1970-2003
50% 40% 30% 20% 10% 0% -10% -20% Stocks Returns:
Mean
Std.Dev. Bonds 2002 2000 1998 1996 1994 1992 1990 1988 1986 1984 1982 1980 1978 1976 1974 1972 1970 -30% Bonds/Stocks Bonds Stocks Half&Half 9.7% 12.7% 11.2% 11.8% 17.5% 11.8% 15 Here the portfolio of 50% bonds & 50% real estate would have provided a more
consistent diversification during 1970-2003, with less volatility than either asset class
alone even though a very similar return:
Annual Periodic Total Returns, Long-Term Bonds and Real Estate, 1970-2003
50% 40% 30% 20% 10% 0% -10% -20% Real Estate Returns:
Mean
Std.Dev. Bonds 2002 2000 1998 1996 1994 1992 1990 1988 1986 1984 1982 1980 1978 1976 1974 1972 1970 -30% Bonds/Real Estate Bonds R.Estate Half&Half 9.7% 9.9% 9.8% 11.8% 9.0% 6.6% 16 This “Diversification Effect” is greater, the lower is the correlation among the
assets in the portfolio.
NUMERICAL EXAMPLE . . .
SUPPOSE REAL ESTATE HAS:
EXPECTED RETURN
= 8%
RISK (STD.DEV)
= 10% SUPPOSE STOCKS HAVE:
EXPECTED RETURN
= 12%
RISK (STD.DEV)
= 15% THEN A PORTFOLIO WITH ω SHARE IN REAL ESTATE & (1-ω) SHARE IN STOCKS WILL
RESULT IN THESE RISK/RETURN COMBINATIONS, DEPENDING ON THE CORRELATION
BETWEEN THE REAL ESTATE AND STOCK RETURNS:
C = 100%
C = 25%
C = 0%
C = -50%
rP
sP
rP
sP
rP
sP
rP
sP
ω
0%
12.0% 15.0% 12.0% 15.0% 12.0% 15.0% 12.0% 15.0%
25%
11.0% 13.8% 11.0% 12.1% 11.0% 11.5% 11.0% 10.2%
50%
10.0% 12.5% 10.0% 10.0% 10.0%
9.0% 10.0%
6.6%
75%
9.0% 11.3%
9.0%
9.2%
9.0%
8.4%
9.0%
6.5%
100%
8.0% 10.0%
8.0% 10.0%
8.0% 10.0%
8.0% 10.0%
where:
C = Correlation Coefficient between Stocks & Real Estate.
(This table was simply computed using the formulas noted previously.) 17 This “Diversification Effect” is greater, the lower is the correlation among the
assets in the portfolio.
Correlation = 100% Portf Exptd Return 12% 11% 1/4 RE 10% 1/2 RE 9% 3/4 RE 8%
9% 10% 11%
12%
13%
Portf Risk (STD) 14% 15% 14% 15% Correlation = 25% Portf Exptd Return 12% 11%
1/4 RE
10% 1/2 RE
3/4 RE 9% 8%
9% 10% 11% 12% 13% Portf Risk (STD) 18 IN ESSENCE,
PORTFOLIO THEORY ASSUMES:
YOUR OBJECTIVE FOR YOUR OVERALL WEALTH
PORTFOLIO IS:
Î MAXIMIZE EXPECTED FUTURE RETURN
Î MINIMIZE RISK IN THE FUTURE RETURN
GIVEN THIS BASIC ASSUMPTION, AND THE EFFECT OF
DIVERSIFICATION, WE ARRIVE AT THE FIRST MAJOR
RESULT OF PORTFOLIO THEORY. . . 19 To the investor, the risk that matters in an
investment is that investment's contribution to the
risk in the investor's overall portfolio, not the risk in
the investment by itself. This means that covariance
(correlation and variance) may be as important as
(or more important than) variance (or volatility) in
the investment alone.
(e.g., if the investor's portfolio is primarily in stocks &
bonds, and real estate has a low correlation with stocks
& bonds, then the volatility in real estate may not matter
much to the investor, because it will not contribute much
to the volatility in the investor's portfolio. Indeed, it may
allow a reduction in the portfolio’s risk.)
THIS IS A MAJOR SIGNPOST ON THE WAY TO FIGURING OUT
"HOW MANY EGGS" WE SHOULD PUT IN WHICH "BASKETS". 20 21.2.4 STEP 1: FINDING THE EFFICIENT FRONTIER
SUPPOSE WE HAVE THE FOLLOWING RISK & RETURN
EXPECTATIONS… Mean
STD
Corr
Stocks
Bonds
RE Stocks
10.00%
15.00% Bonds
6.00%
8.00% RE
7.00%
10.00% 100.00% 30.00%
100.00% 25.00%
15.00%
100.00% INVESTING IN ANY ONE OF THE THREE ASSET CLASSES WITHOUT
DIVERSIFICATION ALLOWS THE INVESTOR TO ACHIEVE ONLY ONE
OF THREE POSSIBLE RISK/RETURN POINTS… 21 INVESTING IN ANY ONE OF THE THREE ASSET CLASSES WITHOUT
DIVERSIFICATION ALLOWS THE INVESTOR TO ACHIEVE ONLY ONE OF
THE THREE POSSIBLE RISK/RETURN POINTS DEPICTED IN THE GRAPH
BELOW…
3 Assets: Stocks, Bonds, RE, No Diversification
12%
Stocks E(r) 10% 8% Real Est
6%
Bonds
4%
6% 8% 10% 12% 14% 16% Risk (Std.Dev)
Stocks Bonds Real Ests IN A RISK/RETURN CHART LIKE THIS, ONE WANTS TO BE ABLE TO GET AS
MANY RISK/RETURN COMBINATIONS AS POSSIBLE, AS FAR TO THE
“NORTH” AND “WEST” AS POSSIBLE.
22 ALLOWING PAIRWISE COMBINATIONS (AS WITH OUR PREVIOUS STOCKS
& REAL ESTATE EXAMPLE), INCREASES THE RISK/RETURN
POSSIBILITIES TO THESE…
3 Assets: Stocks, Bonds, RE, with pairwise combinations
12%
Stocks E(r) 10% 8%
Real Est
6%
Bonds 4%
6% 8% 10% 12% 14% 16% Risk (Std.Dev)
RE&Stocks Stks&Bonds RE&Bonds 23 FINALLY, IF WE ALLOW UNLIMITED DIVERSIFICATION AMONG ALL THREE
ASSET CLASSES, WE ENABLE AN INFINITE NUMBER OF COMBINATIONS,
THE “BEST” (I.E., MOST “NORTH” AND “WEST”) OF WHICH ARE SHOWN
BY THE OUTSIDE (ENVELOPING) CURVE.
3 Assets with Diversification: The Efficient Frontier
12% E(r) 10% 8% 6% 4%
6% 8% 10% 12% 14% 16% Risk (Std.Dev)
Effic.Frontier RE&Stocks Stks&Bonds RE&Bonds THIS IS THE “EFFICIENT FRONTIER” IN THIS CASE (OF THREE
ASSET CLASSES). 24 IN PORTFOLIO THEORY THE “EFFICIENT FRONTIER”
CONSISTS OF ALL ASSET COMBINATIONS
(PORTFOLIOS) WHICH MAXIMIZE RETURN AND
MINIMIZE RISK.
THE EFFICIENT FRONTIER IS AS FAR “NORTH” AND
“WEST” AS YOU CAN POSSIBLY GET IN THE
RISK/RETURN GRAPH.
A PORTFOLIO IS SAID TO BE “EFFICIENT” (i.e.,
represents one point on the efficient frontier) IF IT HAS THE
MINIMUM POSSIBLE VOLATILITY FOR A GIVEN
EXPECTED RETURN, AND/OR THE MAXIMUM
EXPECTED RETURN FOR A GIVEN LEVEL OF
VOLATILITY.
(Terminology note: This is a different definition of "efficiency"
than the concept of informational efficiency applied to asset
25
markets and asset prices.) SUMMARY UP TO HERE:
DIVERSIFICATION AMONG RISKY ASSETS ALLOWS:
¾ GREATER EXPECTED RETURN TO BE OBTAINED
FOR ANY GIVEN RISK EXPOSURE, &/OR;
¾ LESS RISK TO BE INCURRED
FOR ANY GIVEN EXPECTED RETURN TARGET.
(This is called getting on the "efficient frontier".) PORTFOLIO THEORY ALLOWS US TO:
¾ QUANTIFY THIS EFFECT OF DIVERSIFICATION
¾ IDENTIFY THE "OPTIMAL" (BEST) MIXTURE OF RISKY
ASSETS 26 MATHEMATICALLY, THIS IS A "CONSTRAINED
OPTIMIZATION" PROBLEM ==> Algebraic solution using calculus ==> Numerical solution using computer and
"quadratic programming". Spreadsheets such as Excel
include "Solvers" that can find optimal portfolios this
way. 27 21.2.5. STEP 2: PICK A RETURN TARGET FOR YOUR OVERALL
WEALTH THAT REFLECTS YOUR RISK PREFERENCES...
E.G., ARE YOU HERE (7%)?...
Optimal portfolio (P) for a conservative investor: Target=7%
12%
max
risk/return
indifference
curve E(r) 10% Efficient
Frontier 8% P = 16%St, 48%Bd, 36%RE for 7% Target
6% 4%
6% 8% 10% 12% 14% 16% Risk (Std.Dev) 28 OR ARE YOU HERE (9%)?...
Optimal portfolio (P) for an aggressive investor: Target=9%
12%
max risk/return
indifference
curve E(r) 10% 8% P= 67%St, 0%Bd, 33%RE
for 9% Target Efficient
Frontier 6% 4%
6% 8% 10% 12% Risk (Std.Dev) 14% 16% 29 21.2.6
Major Implications of Portfolio Theory for Real Estate Investment
100% ASSET COMPOSITION OF THE EFFICIENT FRONTIER
(based on Exhibit 21-1a expectations)
RE Share 75% Bond Share
Stock
Share 50% 25% 0%
6.0% 7.0% 8.0% 9.0% 10.0% INVESTOR RETURN TARGET Core real estate assets typically make up a large share of efficient
(non-dominated) portfolios for conservative to moderate return
targets. 30 GENERAL QUALITATIVE RESULTS OF PORTFOLIO THEORY
1) THE OPTIMAL REAL ESTATE SHARE DEPENDS ON HOW
CONSERVATIVE OR AGGRESSIVE IS THE INVESTOR;
2) FOR MOST OF THE RANGE OF RETURN TARGETS, REAL
ESTATE IS A SIGNIFICANT SHARE. (COMPARE THESE SHARES TO
THE AVERAGE U.S. PENSION FUND REAL ESTATE ALLOCATION
WHICH IS LESS THAN 5%. THIS IS WHY PORTFOLIO THEORY HAS
BEEN USED TO TRY TO GET INCREASED PF ALLOCATION TO
REAL ESTATE.)
3) THE ROBUSTNESS OF REAL ESTATE'S INVESTMENT APPEAL IS
DUE TO ITS LOW CORRELATION WITH BOTH STOCKS & BONDS,
THAT IS, WITH ALL OF THE REST OF THE PORTFOLIO. (NOTE IN
PARTICULAR THAT OUR INPUT ASSUMPTIONS IN THE ABOVE
EXAMPLE NUMBERS DID NOT INCLUDE A PARTICULARLY HIGH
RETURN OR PARTICULARLY LOW VOLATILITY FOR THE REAL
ESTATE ASSET CLASS. THUS, THE LARGE REAL ESTATE SHARE
IN THE OPTIMAL PORTFOLIO MUST NOT BE DUE TO SUCH
ASSUMPTIONS.)
31 21.2.7 SUPPOSE WE EXPAND THE PORTFOLIO CHOICE
SET BY ADDING ADDITIONAL SUB-CLASSES OF ASSETS…
For example, suppose we add the following expectations for an additional
sub-class of stocks (small stocks) and an additional sub-class of real
estate (REITs)…
Exhibit 21-9a: Possible Risk & Return Expectations for 5 Asset Classes
Large Stocks
Expected Return (E[r]) 10.00% Small
Stocks
12.00% Bonds REITs 6.00% 10.00% Private Real
Estate
7.00% Volatility 15.00% 20.00% 8.00% 15.00% 10.00% 100.00% 60.00% 30.00% 45.00% 25.00% 100.00% 0.00% 70.00% 25.00% 100.00% 20.00% 15.00% 100.00% 40.00% Correlation with:
Large Stocks
Small Stocks
Bonds
REITs
Private Real Estate 100.00% 32 100% ASSET COMPOSITION OF THE EFFICIENT FRONTIER
(based on Exhibit 21-9a expectations)
Private Real
Estate 75% REITs
Bonds 50% Small
Stocks 25% Large
Stocks
0%
6.0% 7.5% 9.0% 10.5% 12.0% INVESTOR RETURN TARGET 33 Section 21.3
VII. INTRODUCING A "RISKLESS ASSET"...
IN A COMBINATION OF A RISKLESS AND A RISKY ASSET, BOTH
RISK AND RETURN ARE WEIGHTED AVERAGES OF RISK AND
RETURN OF THE TWO ASSETS:
Recall: sP = √[ ω²(si)² + (1-ω)²(sj)² + 2ω(1-ω)sisjCij ]
If sj=0, this reduces to:
sP = √[ ω²(si)² = ωsi SO THE RISK/RETURN COMBINATIONS OF A MIXTURE OF
INVESTMENT IN A RISKLESS ASSET AND A RISKY ASSET LIE ON
A STRAIGHT LINE, PASSING THROUGH THE TWO POINTS
REPRESENTING THE RISK/RETURN COMBINATIONS OF THE
RISKLESS ASSET AND THE RISKY ASSET. 34 If either i or j is riskless . . .
E[rj] E[ri] si sj Volatility
35 Î IN PORTFOLIO ANALYSIS, THE "RISKLESS ASSET"
REPRESENTS BORROWING OR LENDING BY THE INVESTOR…
BORROWING IS LIKE "SELLING SHORT" OR HOLDING A NEGATIVE
WEIGHT IN THE RISKLESS ASSET. BORROWING IS "RISKLESS"
BECAUSE YOU MUST PAY THE MONEY BACK “NO MATTER
WHAT”.
LENDING IS LIKE BUYING A BOND OR HOLDING A POSITIVE
WEIGHT IN THE RISKLESS ASSET. LENDING IS "RISKLESS"
BECAUSE YOU CAN INVEST IN GOVT BONDS AND HOLD TO
MATURITY. 36 SUPPOSE YOU COMBINE RISKLESS BORROWING OR LENDING
WITH YOUR INVESTMENT IN THE RISKY PORTFOLIO OF STOCKS
& REAL ESTATE.
YOUR OVERALL EXPECTED RETURN WILL BE:
rW = vrP + (1-v)rf
AND YOUR OVERALL RISK WILL BE:
sW = vsP + (1-v)0 = vsP
Where: v = Weight in risky portfolio
rW, sW = Return, Std.Dev., in overall wealth
rP, sP = Return, Std.Dev., in risky portfolio
rf = Riskfree Interest Rate v NEED NOT BE CONSTRAINED TO BE LESS THAN UNITY.
v CAN BE GREATER THAN 1 ("leverage" , "borrowing"), OR
v CAN BE LESS THAN 1 BUT POSITIVE ("lending", investing in bonds,
in addition to investing in the risky portfolio).
THUS, USING BORROWING OR LENDING, IT IS POSSIBLE TO
OBTAIN ANY RETURN TARGET OR ANY RISK TARGET. THE
RISK/RETURN COMBINATIONS WILL LIE ON THE STRAIGHT LINE
PASSING THROUGH POINTS rf AND rP. 37 NUMERICAL EXAMPLE
SUPPOSE:
RISKFREE INTEREST RATE = 5%
STOCK EXPECTED RETURN = 15%
STOCK STD.DEV. = 15%
________________________________________________________
IF RETURN TARGET = 20%,
BORROW $0.5
INVEST $1.5 IN STOCKS (v = 1.5).
EXPECTED RETURN WOULD BE:
(1.5)15% + (-0.5)5% = 20%
RISK WOULD BE
(1.5)15% + (-0.5)0% = 22.5%
________________________________________________________
IF RETURN TARGET = 10%,
LEND (INVEST IN BONDS) $0.5
INVEST $0.5 IN STOCKS (v = 0.5).
EXPECTED RETURN WOULD BE:
(0.5)15% + (0.5)5% = 10%
RISK WOULD BE
(0.5)15% + (0.5)0% = 7.5% ___________________________________________________________ 38 NOTICE THESE POSSIBILITIES LIE ON A STRAIGHT LINE IN
RISK/RETURN SPACE . . .
RISK & RETURN COMBINATIONS USING STOCKS & RISKLESS BORROWING OR LENDI
35%
30%
EX 25%
PC
TE
D 20%
RE
TU 15%
R
N
10%
5% BORROW V=150% LEND
V=100%
V=50%
V = WEIGHT IN STOCKS
V=0 0%
0% 7.5% 15%
RISK (STD.DEV.) 22.5% 39 BUT NO MATTER WHAT YOUR RETURN TARGET, YOU CAN DO
BETTER BY PUTTING YOUR RISKY MONEY IN A DIVERSIFIED
PORTFOLIO OF REAL ESTATE & STOCKS . . .
SUPPOSE:
REAL ESTATE EXPECTED RETURN = 10%
REAL ESTATE STD.DEV. = 10%
CORRELATION BETWEEN STOCKS & REAL ESTATE = 25%
THEN 50% R.E. / STOCKS MIXTURE WOULD PROVIDE:
EXPECTED RETURN ...

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- Fall '16
- Alex
- Capital Asset Pricing Model, Modern portfolio theory