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Preliminaries Analysis with no risk-free asset Analysis with risk-free asset Conclusion
Lecture 4: Optimal Risky PortfoliosSAPM [Econ F412/FIN F313 ]
Ramana Sonti
BITS Pilani, Hyderabad Campus
Term II, 2014-15
1/27 Lecture 4: Optimal Risky Portfolios Ramana Sonti
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Preliminaries Analysis with no risk-free asset Analysis with risk-free asset Conclusion
Agenda
1 PreliminariesIntroduction and review
2 Analysis with no risk-free assetTwo risky assetsThree (and more) risky assetsMore on efficient frontiers
3 Analysis with risk-free assetTwo risky assets and a risk-free assetThree (and more) risky assets and a risk-free asset
4 ConclusionOptimal complete portfoliosFinal thoughts
2/27 Lecture 4: Optimal Risky Portfolios Ramana Sonti
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Preliminaries Analysis with no risk-free asset Analysis with risk-free asset Conclusion
Introduction and review
The basic problem
Investor has available investment alternatives n risky assets: random returns 1 risk-free asset: constant return
Question: How should the investor allocate money amongst theavailable choices?
Basic assumptions: Investors care about
Average return: Expected return, E(r) Risk: Variance of return, 2(r), or Standard deviation (r)
Investors are greedy: like more return with less risk Investors are risk averse: as the risk level increases, the extra return
needed to compensate them for taking an unit of extra risk increases
3/27 Lecture 4: Optimal Risky Portfolios Ramana Sonti
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Preliminaries Analysis with no risk-free asset Analysis with risk-free asset Conclusion
Introduction and review
Portfolio math review
For a portfolio of two assets (with weights w1 and w2), Portfolio expected return: E(rp) = w1E(r1) + w2E(r2) Portfolio variance: 2p = w2121 + w2222 + 2w1 w21,2
In general, for a portfolio of n assets, Portfolio expected return: E(rp) =
ni=1
wiE(ri)
Portfolio variance: 2p =ni=1
w2i 2i +
ni=1
nj=1i,j
wiwji,j
In more elegant matrix notation, Portfolio expected return: E(rp) = w Portfolio variance: 2p = ww Note: is the vector of expected returns, and is the
variance-covariance matrix of asset returns of the set of assets heldaccording to weights in w
4/27 Lecture 4: Optimal Risky Portfolios Ramana Sonti
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Preliminaries Analysis with no risk-free asset Analysis with risk-free asset Conclusion
Two risky assets
Two risky assets: Example
Two assets X and Y with expected returns, variances, andcovariance (assumed correlation is ) as follows:
=[0.100.20
] =
[0.0049 (0.07)(0.10)
(0.07)(0.10) 0.01
] What is the expected return and standard deviation of a w : 1 w
portfolio of X and Y? Portfolio expected return: E(rp) = w(0.10) + (1 w)(0.20) Portfolio variance:2p = w
2(0.0049) + (1 w)2(0.01) + 2w(1 w)(0.07)(0.10) For a given value of , we can trace out the expected return and
standard deviation of different portfolio combinations of X and Y,e.g., 100:0, 60:40,
5/27 Lecture 4: Optimal Risky Portfolios Ramana Sonti
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Preliminaries Analysis with no risk-free asset Analysis with risk-free asset Conclusion
Two risky assets
Diversification in action: Two risky assets
0 0.02 0.04 0.06 0.08 0.1 0.12
0.1
0.12
0.14
0.16
0.18
0.2
Portfolio standard deviation
Port
folio
expe
cted
retu
rn
X
Y
=1.0=0.5=0=-0.5=-1.0
6/27 Lecture 4: Optimal Risky Portfolios Ramana Sonti
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Preliminaries Analysis with no risk-free asset Analysis with risk-free asset Conclusion
Two risky assets
Mean-Standard deviation diagram
The mean-standard deviation diagram shows the risk-returncharacteristics of all portfolios that can be formed using the twoassets. In other words, it traces out the investment opportunity set
When correlation is +1.0 or -1.0, the mean-standard deviationdiagram is linear
When correlation is between the two extremes, the mean-standarddeviation diagram is a hyperbola
As correlation decreases, the diagram curves in to the left, i.e., wehave portfolios which have lesser risk for the same expected return
As we change the correlation, only portfolio variance (standarddeviation) changes, not the expected return
Note that we cannot choose the correlation between any pair ofrisky assets, it is what it is
7/27 Lecture 4: Optimal Risky Portfolios Ramana Sonti
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Preliminaries Analysis with no risk-free asset Analysis with risk-free asset Conclusion
Three (and more) risky assets
Three risky assets
Lets fix the correlation between X and Y at = 0.10, and add a thirdrisky asset, Z, with the following properties
= 0.100.200.15
=
0.0049 0.0007 00.0007 0.01 0.01080 0.0108 0.0144
What is the investment opportunity set now? With two risky assets, the opportunity set was a curve With three risky assets, the opportunity set is a whole area
Question: Is there an optimal combination of the three assets? Yes, the mean variance frontier plots the optimal combinations of the
three assets
8/27 Lecture 4: Optimal Risky Portfolios Ramana Sonti
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Preliminaries Analysis with no risk-free asset Analysis with risk-free asset Conclusion
Three (and more) risky assets
Mean variance frontier: Three risky assets
0.05 0.06 0.07 0.08 0.09 0.1 0.11 0.12 0.130
0.05
0.1
0.15
0.2
0.25
Portfolio standard deviation
Port
folio
expe
cted
retu
rn
X
Y
Z
Efficient Frontier
MVP
9/27 Lecture 4: Optimal Risky Portfolios Ramana Sonti
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Preliminaries Analysis with no risk-free asset Analysis with risk-free asset Conclusion
More on efficient frontiers
Efficient frontier: Analytical solution Question: Of the infinite number of portfolios that can be formed ofX, Y and Z, which ones are optimal? For a given expected return m, what combination of the assets results
in the lowest variance (standard deviation)?
minw 2p = w
w
subject to E(rp) = w = m andni=1
wi = 1
Alternatively, for a given level of variance s, what combination of theassets results in the highest expected return?
maxw E(rp) = w
subject to 2p = ww = s and
ni=1
wi = 1
The bad news: General analytical solution is quite complicated The good news: we can use Excels Solver quite effectively to
numerically generate the efficient frontier
10/27 Lecture 4: Optimal Risky Portfolios Ramana Sonti
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Preliminaries Analysis with no risk-free asset Analysis with risk-free asset Conclusion
More on efficient frontiers
Two fund separation
Two fund separation : Any efficient portfolio can be obtained as thecombination of two efficient portfolios
The efficient portfolio with E(rp) = 16% consists of wX = 0.54,wY = 0.74, wZ = 0.28, and that with E(rp) = 22% consists ofwX = 0.2625, wY = 1.6625, wZ = 0.925. Using this information, howdo we derive the efficient portfolio with E(rp) = 20%? Step 1: Given that 20 = 13 (16) + 23 (22), we deduce weights as 13 and 23 Step 2: Now we get the weights of the 20% expected return efficient
portfolio as wX = 13 (0.54) + 23 (0.2625) = 0.355 wY = 13 (0.74) + 23 (1.6625) = 1.355 wZ = 13 (0.28) + 23 (0.925) = 0.71
Step 3: The mean and std. devn. of this portfolio are 20% and 0.0783try this calculation!), which puts it right on the efficient frontier, asadvertised!
11/27 Lecture 4: Optimal Risky Portfolios Ramana Sonti
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Preliminaries Analysis with no risk-free asset Analysis with risk-free asset Conclusion
More on efficient frontiers
Two fund separation: Example
0.06 0.07 0.08 0.09 0.1
0.14
0.16
0.18
0.2
0.22
0.24
Portfolio standard deviation
Port
folio
expe
cted
retu
rn
EP 1: 0.54,0.74,-0.28
EP 2: 0.2625,1.6625,-0.925
EP 3: 0.355,1.355,-0.71
Efficient Frontier
MVP
This suggests the following recipe: Step 1: Use Solver to solve for any two efficient portfolios Step 2: Form combinations of the two efficient portfolios to trace the
entire efficient frontier
12/27 Lecture 4: Optimal Risky Portfolios Ramana Sonti
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Preliminaries Analysis with no risk-free asset Analysis with risk-free asset Conclusion
More on efficient frontiers
Constraints on investment Say we want to solve a constrained version of the problem, e.g., no
short sales allowed for mutual funds The constrained efficient frontier lies completely within the
unconstrained efficient frontier (why?)
0.06 0.07 0.08 0.09 0.1
0.1
0.12
0.14
0.16
0.18
0.2
Portfolio standard deviation
Port
folio
expe
cted
retu
rn
Unconstrained frontier
Constrained frontier
13/27 Lecture 4: Optimal Risky Portfolios Ramana Sonti
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Preliminaries Analysis with no risk-free asset Analysis with risk-free asset Conclusion
More on efficient frontiers
Adding a fourth...and a fifth...asset
0.04 0.06 0.08 0.1 0.120
0.05
0.1
0.15
0.2
0.25
Portfolio standard deviation
Port
folio
expe
cted
retu
rn
More assets
14/27 Lecture 4: Optimal Risky Portfolios Ramana Sonti
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Preliminaries Analysis with no risk-free asset Analysis with risk-free asset Conclusion
More on efficient frontiers
A second look at diversification In general, as we keep adding assets to a portfolio, the efficient
frontier moves to the left Question: Will the efficient frontier ever touch the vertical axis, i.e., is
there a combination of the risky assets that is completely risk-free? Answer: Generally, no
Start with portfolio variance:2p =ni=1
w2i 2i +
ni=1
nj=1i,j
wiwji,j
Assume all assets have a variance v, and each pair of assets acovariance c
This means 2p = n 1n2 v + n(n 1) 1n2 c = 1n v + n1n c As n , 1n 0, and n1n 1, which implies 2p c
Conclusion 1: For a well-diversified portfolio, covariances matter, notso much the variances
Conclusion 2: If c is not zero, we cannot diversify away all risk: leadsto the idea of diversifiable (idiosyncratic) vs. undiversifiable(systematic) risk
15/27 Lecture 4: Optimal Risky Portfolios Ramana Sonti
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Preliminaries Analysis with no risk-free asset Analysis with risk-free asset Conclusion
More on efficient frontiers
Diversification in well-diversified portfolios
0 20 40 60 80 100No. of stocks
Port
folio
var
ianc
e
Diversifiable risk
Systematic risk
16/27 Lecture 4: Optimal Risky Portfolios Ramana Sonti
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Preliminaries Analysis with no risk-free asset Analysis with risk-free asset Conclusion
Two risky assets and a risk-free asset
Two risky assets and a risk-free asset
Let us add a risk-free asset to our two assets X and Y. Further,assume rf = 0.05. What happens? The opportunity set now expands dramatically The investor can now combine the risk-free asset and X, and choose
any portfolio along the line CALX The investor can also combine the risk-free asset and Y, and choose
any portfolio along the line CALY Better yet, the investor can combine the risk-free asset and any
portfolio of X and Y pivoting about the risk-free asset until the CALwhich passes through the risk-free asset and the portfolio labelledMVE
The portfolio of X and Y at the point of tangency is called the MeanVariance Efficient (MVE) portfolio. It provides the maximumreward-to-risk ratio, a.k.a. Sharpe ratio
17/27 Lecture 4: Optimal Risky Portfolios Ramana Sonti
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Preliminaries Analysis with no risk-free asset Analysis with risk-free asset Conclusion
Two risky assets and a risk-free asset
Two risky assets: Tangency and MVE
0 0.02 0.04 0.06 0.08 0.1 0.120
0.05
0.1
0.15
0.2
0.25
Portfolio standard deviation
Port
folio
expe
cted
retu
rn
X
Y
CAL X
CAL Y
MVE
CAL
18/27 Lecture 4: Optimal Risky Portfolios Ramana Sonti
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Preliminaries Analysis with no risk-free asset Analysis with risk-free asset Conclusion
Two risky assets and a risk-free asset
Analytics of the MVE (2 asset case) The problem to be solved is
maxw
E(rp) rfp
,
where E(rp) = wE(rX) + (1 w)E(rY )and p =
[w22X + (1 w)22Y + 2w(1 w)X,Y
] 12
The solution to this problem is
wX =
[E(rX) rf
]2Y
[E(rY ) rf
]X,Y[
E(rX) rf]2Y +
[E(rY ) rf
]2X
[E(rX) rf + E(rY ) rf
]X,Y
wY = 1 wX
Try using this formula in our example to get wX = 0.3607 andwY = 0.6393
19/27 Lecture 4: Optimal Risky Portfolios Ramana Sonti
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Preliminaries Analysis with no risk-free asset Analysis with risk-free asset Conclusion
Three (and more) risky assets and a risk-free asset
Three risky assets and a risk-free asset Let us add the risk-free asset with rf = 0.05 to our three risky assetsX, Y and Z
Now, Sharpe ratio is maximized at the point where the CAL istangent to the efficient frontier
The MVE, or tangency portfolio iswX = 0.2274,wY = 1.7793,wZ = 1.0067
0 0.02 0.04 0.06 0.08 0.1 0.120
0.05
0.1
0.15
0.2
0.25
Portfolio standard deviation
Port
folio
expe
cted
retu
rn
MVE
CAL
H0.2274, 1.7793,-1.0067L
20/27 Lecture 4: Optimal Risky Portfolios Ramana Sonti
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Preliminaries Analysis with no risk-free asset Analysis with risk-free asset Conclusion
Three (and more) risky assets and a risk-free asset
Analytical solution (3 asset case) Set up the following equation system
0.0049x + 0.0007y + 0.0000z = 0.10 0.05 = E(rX) rf0.0007x + 0.0100y + 0.0108z = 0.20 0.05 = E(rY ) rf0.0000x + 0.0108y + 0.0144z = 0.15 0.05 = E(rZ) rf
Solve: x = 4.8186, y = 37.6984, z = 21.3294 Rescale weights to add up to 1, i.e., divide by the sum of the three
numbers, 21.1876
wX =4.818621.1876
= 0.2274,
wY =37.698421.1876
= 1.7793,
wZ =21.329421.1876
= 1.0067,
which are the same values as before!21/27 Lecture 4: Optimal Risky Portfolios Ramana Sonti
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Preliminaries Analysis with no risk-free asset Analysis with risk-free asset Conclusion
Three (and more) risky assets and a risk-free asset
The MVE portfolio
It turns out that the MVE portfolio has an interesting propertyE(rX) rf
Cov(rX, rMVE)=
E(rY ) rfCov(rY , rMVE)
=E(rZ) rf
Cov(rZ , rMVE)
Intuition: In the MVE portfolio, the marginal reward-to-risk ratio isequalized across all risky assets, and there is nothing to be gainedby changing the portfolio composition further
It is this important property underlies our earlier analytical method
22/27 Lecture 4: Optimal Risky Portfolios Ramana Sonti
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Preliminaries Analysis with no risk-free asset Analysis with risk-free asset Conclusion
Three (and more) risky assets and a risk-free asset
The MVE portfolio...Contd. Sketch of proof:
Say we start with the MVE portfolio T of X, Y, and Z (expected returnof T and variance of 2T )
Suppose we try to improve upon the Sharpe ratio of this portfolio asfollows: For every $1 invested in T, buy $X amount of X (financed byborrowing at rf ), and, sell $Y amount of Y from this portfolio (proceedsinvested at rf )
New (dollar) expected return N = T + X(X rf ) Y (Y rf ) New (dollar) variance 2N 2T + 2XCov(rX , rT ) 2YCov(rY , rT ) Set 2N 2T = 0, by choosing X = Y Cov(rY ,rT )Cov(rX ,rT ) , i.e., we have rigged this
to make sure the portfolio variance does not change due to themodifications we made with X and Y
But since we started with the MVE portfolio, we must haveX(X rf ) Y (Y rf ) = 0, otherwise Portfolio T must not have beenMVE to begin with
Combining the last two equations, we get the desired propertyE(rX )rfCov(rX ,rT )
=E(rY )rfCov(rY ,rT )
23/27 Lecture 4: Optimal Risky Portfolios Ramana Sonti
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Preliminaries Analysis with no risk-free asset Analysis with risk-free asset Conclusion
Optimal complete portfolios
Optimal complete portfolios
Suppose we have two investors: one with a coefficient of riskaversion A = 15, and another with A = 40. How will their money beallocated among the risk-free asset and the three risky assets?
Key insight 1: Risk aversion does not play any role in determiningthe optimal risky portfolio, a.k.a. the MVE portfolio.
Key insight 2: All investors, irrespective of risk aversion, choose tobe on the CAL with the highest slope. In other words, all investorshold portfolios of the risk-free asset and the MVE portfolio. This istwo-fund separation in the presence of a risk-free asset
In summary, all investors follow a two step process Step 1: Come up with the MVE portfolio (common to all) the
investment decision Step 2: Decide on the split of their money between T-bills and the MVE
portfoliothe financing decision
24/27 Lecture 4: Optimal Risky Portfolios Ramana Sonti
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Preliminaries Analysis with no risk-free asset Analysis with risk-free asset Conclusion
Optimal complete portfolios
Investor with A = 15
The MVE portfolio consists ofwX = 0.2274,wY = 1.7793,wZ = 1.0067. The expected return andstandard deviation of this portfolio are 0.2276 and 0.0916respectively
Recall that the relative split between the risk-free asset and the MVEportfolio is given by our old formulaw = E(rMVE)rfA2MVE
= 0.22760.0515(0.0916)2 = 141.25%
If the investor had a capital of $1 M to start with, she would borrow$412,500 (at the risk-free rate), and invest the total, i.e. $1.4125 M inthe MVE portfolio, i.e. split this amount among X, Y, and Z accordingto the proportions wX = 0.2274,wY = 1.7793,wZ = 1.0067
Exercise: Work out the allocation for an investor with A = 40, andcompare it to this allocation. It is useful to understand thedifferences between the two allocations
25/27 Lecture 4: Optimal Risky Portfolios Ramana Sonti
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Preliminaries Analysis with no risk-free asset Analysis with risk-free asset Conclusion
Optimal complete portfolios
The big picture
0 0.04 0.08 0.12 0.16 0.2 0.24 0.280
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
Portfolio standard deviation
Port
folio
expe
cted
retu
rn
MVE
Risky asset efficient frontier
CAL
A=40
A=15
T-bills
Both investors choose portfolios of the same two assets: T-bills and the MVE Locate on the CAL according to their risk aversion
26/27 Lecture 4: Optimal Risky Portfolios Ramana Sonti
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Preliminaries Analysis with no risk-free asset Analysis with risk-free asset Conclusion
Final thoughts
Final thoughts Markowitz optimization is elegant and easy to implement if we are
willing to assume that all investors care about are the mean andvariance of risky assets This is strictly true if (a) all investors really have quadratic utility
functions, or if (b) asset returns are always normally distributed Both assumptions are a bit problematic. Nevertheless, as a first
approximation, they will do
This procedure suggests that an investment advisor shouldrecommend the same optimal risky portfolio to different investors,irrespective of their risk aversion Ignores real-world issues such as taxation This is a one-period problem,and not a multi-period formulation. Here,
the long run is essentially a series of short runs
We have still not talked about the inputs for this optimization, i.e.,expected returns and variances/covariances This turns out to be the most problematic thing to do, and is the stuff of
asset pricing, which we shall study next
27/27 Lecture 4: Optimal Risky Portfolios Ramana Sonti
PreliminariesIntroduction and review
Analysis with no risk-free assetTwo risky assetsThree (and more) risky assetsMore on efficient frontiers
Analysis with risk-free assetTwo risky assets and a risk-free assetThree (and more) risky assets and a risk-free asset
ConclusionOptimal complete portfoliosFinal thoughts