This document provides an overview of mean variance optimization and efficient frontier analysis in financial portfolio selection. It introduces key concepts such as quantifying random asset returns using mean and variance, constructing optimal portfolios that maximize return for a given level of risk, and graphing the efficient frontier. The document also covers the two-fund theorem and how introducing a risk-free asset shifts the analysis to focus on excess returns above the risk-free rate.

1. Financial Engineering and Risk Management
Mean Variance Optimization
Martin Haugh Garud Iyengar
Columbia University
Industrial Engineering and Operations Research

2. Overview
Assets and portfolios
Quantifying random asset and portfolio returns: mean and variance
Mean-variance optimal portfolios
Efficient frontier
Sharpe ratio and Sharpe optimal portfolios
Market portfolio
Capital Asset Pricing Model
2

3. Assets and portfolios
Asset ≡ anything we can purchase
Random price P(t)
Random gross return R(t) = P(t+1)
P(t)
Random net return: r(t) = R(t) − 1 = P(t+1)−P(t)
P(t)
Total wealth W > 0 distributed over d assets
wi = dollar amount in asset i: wi > 0 ≡ long, wi < 0 ≡ short
Net return on a position w
rw(t) =
d
i=1 Ri(t)wi −
n
i=1 wi
W
=
d
i=1 ri(t)wi
d
i=1 wi
=
d
i=1
ri(t) ·
wi
W
xi
portfolio vector x = (x1, . . . , xd): each component can be +ve/-ve
xi = fraction invested in asset i ⇒
d
i=1
xi = 1
3

4. How does one deal with randomness?
Random net return on the portfolio rx =
d
i=1 rixi
How does one “quantify” random returns ?
Maximize expected return E[rx]?
Should one worry about spread around the mean?
How does one quantify the spread?
4

5. Random returns on assets and portfolios
Parameters defining asset returns
Mean of asset returns: µi = E[ri(t)]
Variance of asset returns: σ2
i = var(ri(t))
Covariance of asset returns: σij = cov ri(t), rj(t) = ρijσiσj
Correlation of asset returns ρij = cor ri(t), rj(t)
All parameters assumed to be constant over time.
Parameters defining portfolio returns
Expected return on a portfolio x = (x1, . . . , xd)
µx = E[rx(t)] =
d
i=1
E[ri(t)]xi =
d
i=1
µixi
Variance of the return on portfolio x:
σ2
x = var(rx(t)) = var
d
i=1
rixi =
d
i=1
d
j=1
cov ri(t), rj(t) xixj
5

6. Example
d = 2 assets with Normally distributed returns N(µ, σ2
)
r1 ∼ N(1, 0.1) r2 ∼ N(2, 0.5) cor(r1, r2) = −0.25
Parameters
µ1 = 1 µ2 = 2
σ2
1 = var(r1) = 0.1 σ2
2 = var(r2) = 0.5
σ12 = cov(r1, r2) = cor(r1, r2)σ1σ2 = −0.25
√
0.05 = 0.0559
Portfolio: (x, 1 − x)
µx =
d
i=1
µixi = x + 2(1 − x)
σ2
x =
d
i,j=1
σijxixj =
d
i=1
σ2
i x2
i + 2
j>i
σijxixj
= 0.1x2
+ 0.5(1 − x)2
+ 2(0.0559)x(1 − x)
6

7. Diversification reduces uncertainty
d assets each with µi ≡ µ, σi ≡ σ, ρij = 0 for all i = j
Two different portfolios
x = (1, 0, . . . , 0) : everything invested in asset 1
y = 1
d (1, 1, . . . , 1) : equal investment in all assets.
Expected returns of the two portfolios
µx = E[
d
i=1 µixi] = µ1 = µ
µy = E[
d
i=1 µiyi] = 1
d
d
i=1 µi = µ
Both have the same expected return!
Variance of returns of the two portfolios
σ2
x = var(
d
i=1 rixi) = σ2
σ2
y = var(
d
i=1 riyi) =
d
i=1 σ2
(1
d )2
= σ2
d
Diversified portfolio has a much lower variance!
7

8. Markowitz mean-variance portfolio selection
Markowitz (1954) proposed that
“Return” of a portfolio ≡ Expected return µx
“Risk” of a portfolio ≡ volatility σx
Efficient frontier
0 20 40 60 80 100 120 140 160 180
−60
−40
−20
0
20
40
60
80
100
120
volatility (%)
return(%)
Efficient frontier ≡ max return
for a given risk
How does one characterize the
efficient frontier?
How does one compute effi-
cient/optimal portfolios?
8

9. Mean variance formulations
Minimize risk ensuring return ≥ target return
minx σ2
x
s.t. µx ≥ r
≡ minx
d
i=1
d
j=1 σijxixj
s.t.
d
i=1 µixi ≥ r
d
i=1 xi = 1.
Maximize return ensuring risk ≤ risk budget
maxx µx
s.t. σ2
x ≤ ¯σ2
≡ maxx
d
i=1 µixi
s.t.
d
i=1
d
j=1 σijxixj ≤ ¯σ2
,
d
i=1 xi = 1.
Maximize a risk-adjusted return
maxx µx − τσ2
x ≡ maxx
d
i=1 µixi − τ
d
i=1
d
j=1 σijxixj
s.t.
d
i=1 xi = 1.
τ ≡ risk-aversion parameter
9

10. Financial Engineering and Risk Management
Efficient frontier
Martin Haugh Garud Iyengar
Columbia University
Industrial Engineering and Operations Research

11. Mean-variance for 2-asset market
d = 2 assets
Asset 1: mean return µ1 and variance σ2
1
Asset 2: mean return µ2 and variance σ2
2
Correlation between asset returns ρ
Portfolio: (x, 1 − x)
µx =
d
i=1
µixi = µ1x + µ2(1 − x)
σ2
x =
d
i,j=1
σijxixj =
d
i=1
σ2
i x2
i + 2
j>i
σijxixj
= σ2
1x2
+ σ2
2(1 − x)2
+ 2ρσ1σ2x(1 − x)
2

12. Mean-variance for 2-asset market
Minimize risk formulation for the mean-variance portfolio selection problem
minx σ2
1x2
+ σ2
2(1 − x)2
+ 2ρσ1σ2x(1 − x)
s.t. µ1x + µ2(1 − x) = r
Expected return constraint: x = r−µ2
µ1−µ2
Variance:
σ2
r = σ2
1
r − µ2
µ1 − µ2
2
+ σ2
2
µ1 − r
µ1 − µ2
2
+ 2ρσ1σ2
r − µ2
µ1 − µ2
µ1 − r
µ1 − µ2
= ar2
+ br + c
Explicit expression for the variance as a function of target return r.
3

13. Efficient frontier
0 2 4 6 8 10 12 14
−6
−4
−2
0
2
4
6
8
10
12
volatility (%)
return(%)
σmin
rmin
Efficient
Inefficient
Only the top half is efficient! why did we get the bottom?
How does one solve the d asset problem?
4

14. Computing the optimal portfolio
Mean-variance portfolio selection problem
σ2
(r) = minx
d
i=1
d
j=1 σijxixj
s.t.
d
i=1 µixi = r
d
i=1 xi = 1.
Form the Lagrangian with Lagrange multipliers u and v
L =
d
i=1
d
j=1
σijxixj − v
d
i=1
µixi − r − u
d
i=1
xi − 1
Setting ∂L
∂xi
= 0 for i = 1, . . . , d gives d equations
2
d
j=1
σijxj − vµi − u = 0, for all i = 1, . . . d (∗)
Can solve the d + 2 equations in d + 2 variables: x1, . . . , xd, u and v.
Theorem. A portfolio x is mean-variance optimal if, and only if, it is feasible and
there exists u and v satisfying (∗).
5

15. Computing the optimal portfolio
Matrix formulation









2σ11 2σ12 . . . 2σ1d −µ1 −1
2σ21 2σ22 . . . 2σ2d −µ2 −1
...
...
...
...
...
...
2σd1 2σd2 . . . 2σdd −µd −1
µ1 µ2 . . . µd 0 0
1 1 . . . 1 0 0









A









x1
x2
...
xd
v
u









=









0
0
...
0
r
1









b
Therefore 








x1
x2
...
xd
v
u









= A−1
b
6

16. Two fund theorem
Fix two different target returns: r1 = r2
Suppose
x(1)
= (x
(1)
1 , . . . , x
(1)
d ) optimal for r1: Lagrange multipliers (v1, u1)
x(2)
= (x
(2)
1 , . . . , x
(2)
d ) optimal for r2: Lagrange multipliers (v2, u2)
Consider any other return r
Choose β = r−r1
r2−r1
Consider the position: y = (1 − β)x(1)
+ βx(2)
y is a portfolio
d
i=1
yi = (1 − β)
d
i=1
x
(1)
i + β
d
i=1
x
(2)
i = (1 − β) + β = 1
y is feasible for target return r
d
i=1
µiyi = (1 − β)
d
i=1
µix
(1)
i + β
d
i=1
µix
(2)
i = (1 − β)r1 + βr2 = r
7

17. Two fund theorem (contd)
Set v = (1 − β)v1 + βv2 and u = (1 − β)u1 + βu2.
2
d
j=1
σijyj − vµi − u =
d
j=1
2σij((1 − β)x
(1)
j + βx
(2)
j )
− µi((1 − β)v1 + βv2) − ((1 − β)u1 + βu2)
= (1 − β) 2
d
j=1
σijx
(1)
j − v1µi − u1
+ β 2
d
j=1
σijx
(2)
j − v2µi − u2 = 0
y is optimal for target return r!
Theorem All efficient portfolios can be constructed by diversifying between any
two efficient portfolios with different expected returns.
Why are there so many funds in the market?
8

18. Efficient frontier
The optimal portfolio for target return r
y∗
=
r2 − r
r2 − r1
x(1)
+
r − r1
r2 − r1
x(2)
= r
x(2)
− x(1)
r2 − r1
g
+
r2x(1)
− r1x(2)
r2 − r1
h
y∗
i = rgi + hi, i = 1, . . . , d.
Therefore
σ2
(r) =
d
i=1
d
j=1
σij(rgi + hi)(rgj + hj)
= r2
d
i=1
d
j=1
σijgigj + 2r
d
i=1
d
j=1
σijgihj +
d
i=1
d
j=1
σijhihj
The d-asset frontier has the same structure as the 2-asset frontier.
9

19. Efficient frontier
0 2 4 6 8 10 12 14
−6
−4
−2
0
2
4
6
8
10
12
volatility (%)
return(%)
σmin
rmin
Efficient
Inefficient
10

20. Financial Engineering and Risk Management
Mean-variance with a risk-free asset
Martin Haugh Garud Iyengar
Columbia University
Industrial Engineering and Operations Research

21. Mean Variance with a risk-free asset
New asset: pays net return rf with no risk (deterministic return)
x0 = fraction invested in the risk-free asset
Mean-variance problem: x0 does not contribute to risk.
max rf x0 +
d
i=1 µixi − τ
d
i=1
d
j=1 σijxixj
s. t. x0 +
d
i=1 xi = 1.
Only meaningful for r ≥ rf
Substituting x0 = 1 −
d
i=1 xi we get
max rf +
d
i=1(µi − rf )xi − τ
d
i=1
d
j=1 σijxixj
ˆµi = µi − rf = excess return of asset i
2

22. Mean-variance optimal portfolio
Taking derivatives we get
ˆµi − 2τ
d
j=1
σijxj = 0, i = 1, . . . , d.
Matrix formulation
2τ





σ11 σ12 . . . σ1d
σ21 σ22 . . . σ2d
...
...
...
...
σd1 σd2 . . . σdd





V





x1
x2
...
xd





=



ˆµ1
...
ˆµd



ˆµ
⇒ x(τ) =
1
2τ
V−1
ˆµ
The family of frontier portfolios as a function of τ:
1 −
d
i=1
xi(τ), x(τ) : τ ≥ 0
3

23. One-fund theorem
The positions in the risky assets in the frontier portfolio
x =
1
2τ
V−1
ˆµ
do not add up to 1.
Define a portfolio of risky assets by dividing x by the sum of its components.
s∗
=
1
d
i=1 xi
x =
1
1
2τ
d
i=1(V−1
ˆµ)i
1
2τ
V−1
ˆµ
The portfolio s∗
is independent of τ! Since
d
i=1 xi = 1 − x0, x = (1 − x0)s∗
.
Family of frontier portfolios = {(x0, (1 − x0)s∗
) : x0 ∈ R}
Theorem All efficient portfolios in a market with a risk-free asset can be
constructed by diversifying between the risk-less asset and the single portfolio s∗
.
4

24. Efficient frontier with risk-free asset
Return and risk of portfolio s∗
: µ∗
s =
d
i=1 µis∗
i , σ∗
s =
d
i=1 j=1 σ2
ijs∗
i s∗
j
Return on a generic frontier portfolio: x0 in risk-free and (1 − x0) in s∗
µx = x0rf + (1 − x0)µ∗
s σx = (1 − x0)σ∗
s
Efficient Frontier
0 1 2 3 4 5 6 7 8 9 10
0
1
2
3
4
5
6
7
8
9
10
(0, rf ) ≡ risk-free asset
volatility (%)
return(%)
(σ∗
s , µ∗
s) ≡ s∗
Straight line with an intercept rf
at σ = 0 and slope
m =
µs − rf
σs
How does this relate to the fron-
tier with only risky assets?
Does the portfolio s∗
have an
economic interpretation?
5

25. Efficient frontier with risk-free asset
0 1 2 3 4 5 6 7 8 9
1
2
3
4
5
6
7
8
9
volatility (%)
return(%)
θ
(σ∗
s , µ∗
s) ≡ s∗
rf
s∗
must be an efficient risky portfolio
The efficient frontier with a risk-free
asset must be tangent to the efficient
frontier with only risky assets.
The portfolio s∗
maximizes the angle θ or equivalently
tan(θ) =
d
i=1 µixi − rf
d
i=1
d
j=1 σijxixj
=
expected excess return
volatility
6

26. Sharpe Ratio
Definition. The Sharpe ratio of a portfolio or an asset is the ratio of the
expected excess return to the volatility. The Sharpe optimal portfolio is a
portfolio that maximizes the Sharpe ratio.
The portfolio s∗
is a Sharpe optimal portfolio
s∗
= argmax
x:
d
i=1
xi =1
µx − rf
σx
Investors diversify between the risk-free asset and the Sharpe optimal portfolio.
The investment in the various risky assets are in fixed proportions ...
prices/returns should be correlated! This insight leads to the Capital Asset
Pricing Model.
7

27. Financial Engineering and Risk Management
Capital Asset Pricing Model
Martin Haugh Garud Iyengar
Columbia University
Industrial Engineering and Operations Research

28. Market Portfolio
Definition. Let Ci, i = 1, . . . , d, denote the market capitalization of the d
assets. Then the market portfolio x(m)
is defined as follows.
x
(m)
i =
Ci
d
j=1 Cj
, i = 1, . . . , d.
Let µm denote the expected net rate of return on the market portfolio, and let
σm denote the volatility of the market portfolio.
Suppose all investors in the market are mean-variance optimizers. Then all of
them invest in the Sharpe optimal portfolio s∗
. Let
w(k)
= wealth of the k-th investor
x
(k)
0 = fraction of wealth of the k-investor in the risk-free asset
Then
Ci =
k
w(k)
(1 − x
(k)
0 )s∗
i
The market portfolio x(m)
= Sharpe optimal portfolio s∗
!
2

29. Capital Market Line
Capital market line is another name for the efficient frontier with risk-free asset
Recall: Efficient frontier = line though the points (0, rf ) and (σm, µm)
Slope of the capital market line
mCML =
µm − rf
σm
= maximum achievable Sharpe ratio
mCML is frequently called the price of risk. It is used to compare projects.
Example. Suppose the price of a share of an oil pipeline venture is $875. It is
expected to yield $1000 in one year, but the volatility σ = 40%. The current
interest rate rf = 5%, the expected rate of return on the market portfolio
µm = 17% and the volatility of the market σm = 12%. Is the oil pipeline worth
considering?
roil =
1000
875
− 1 = 14% ¯r = rf +
µm − rf
σm
σ = 45%
Not worth considering!
3

30. Inferring asset returns from market returns
An asset is a portfolio: asset j ≡ x(j)
= (0, . . . , 1, . . . , 0) , 1 in the j-th position.
Diversify between x(j)
and market portfolio x(m)
: γx(j)
+ (1 − γ)x(m)
return µγ = γµj + (1 − γ)µm
volatility σγ = γ2σ2
j + (1 − γ)2σ2
m + 2σjmγ(1 − γ)
0 1 2 3 4 5 6 7 8 9
1
2
3
4
5
6
7
8
9
volatility (%)
return(%)
(σm, µm) ≡ x(m)
risk-free asset ≡ (0, rf )
Efficient frontier with risk−free
Efficient frontier w/o risk−free
Efficient frontier of market + asset 2
4

31. All three curves are tangent at (σm, rm)
Slope of the capital market line
mCML =
µm − rf
σm
Slope of the frontier generated by asset j and market portfolio x(m)
dµγ
dσγ
=
dµγ
dγ
dσγ
dγ
=
µj − µm
γσ2
j −(1−γ)σ2
m+(1−γ)σjm−γσjm
√
γ2σ2
j +(1−γ)2σ2
m+2σjmγ(1−γ)
dµγ
dσγ γ=0
=
µj − µm
σjm−σ2
m
σm
Equating slopes at γ = 0 we get the following result:
µj − rf =
σjm
σ2
m
beta of asset j
(µm − rf )
This pricing formula is called the Capital Asset Pricing Model (CAPM).
5

32. Connecting CAPM to regression
Regress the excess return rj − rf of asset j on the excess market return rm − rf
(rj − rf ) = α + β(rm − rf ) + j
Parameter estimates
coefficient βj =
cov(rj −rf ,rm−rf )
var(rm−rf ) =
σjm
σ2
m
intercept αj = (E[rj] − rf ) − β(E[rm] − rf ) = (µj − rf ) − β(µm − rf ).
residuals j and (rm − rf ) are uncorrelated, i.e. cor( j, rm − rf ) = 0.
CAPM implies that αj = 0 for all assets.
Effective relation: rj − rf = βj(rm − rf ) + j
Decomposition of risk
var(rj − rf ) = β2
j var(rm − rf ) + var( )
σ2
j = β2
j σ2
m
market risk
+ var( )
residual risk
Only compensated for taking on market risk and not residual risk
6

33. Security Market Line
Plot of the historical returns on an asset vs rf + β(µm − rf )
−2 −1.5 −1 −0.5 0 0.5
−8
−6
−4
−2
0
2
4
1
2
3
4
5
6
7
8
Security market line
beta
return(%)
The assets are labeled in the order they appears in the spreadsheet.
All assets should lie on the security line if CAPM holds. So why the discrepancy?
7

34. Assumptions underlying CAPM
All investors have identical information about the uncertain returns.
All investors are mean-variance optimizers (or the returns are Normal)
The markets are in equilibrium.
Leveraging deviations from the security market line
Jensen’s index or alpha
α = (ˆµj − rf ) − βj(ˆµm − rf )
hold long if positive, short otherwise
Sharpe ratio of a stock
sj =
ˆµj − rf
ˆσj
hold long if > mMCM, short otherwise.
8

35. CAPM as a pricing formula
Suppose the payoff from an investment in 1yr is X. What is the fair price for this
investment.
Let rX = X
P − 1 denote the net rate of return on X. The beta of X is given by
βX =
cov(rX , rm)
σ2
m
=
1
P
cov(X, rm)
σ2
m
Suppose CAPM holds. Then µX = E[rX ] must lie on the security market line, i.e.
µX = rf + βX (rm − rf )
E[X]
P
− 1 = rf +
1
P
cov(X, rm)
var(rm)
(µm − rf )
Rearranging terms:
P =
E[X]
1 + rf
−
cov(X, rm)
(1 + rf )var(rm)
(µm − rf )
9