Portfolio Theory
MGFB Principles of Finance
Jincheng Tong
University of Toronto
March 1, 2025
1 / 26
Why Is This Important?
- We as investors want to achieve the highest possible expected return for a given level of
risk
- Modern Portfolio Theory (MPT) allows us to find the optimal way to allocate our money
to achieve this goal
- Whether you are managing your own or someone else’s money, MPT tells you how to
build the “best” portfolio
- MPT and its o!springs are still widely studied and used
- MPT is a stepping stone to the very popular asset pricing models such as the CAPM
2 / 26
Intuition Behind MPT
- Investing is a trade-o! between risk and return
- Rather than selecting assets on their own merits, we should consider how returns of every
asset correlate with returns of every other asset in the portfolio
- We can combine assets to achieve the highest possible expected return for a given level of
risk
- Or alternatively, we can combine the assets to achieve the lowest possible level of risk for
a given level of expected return
- MPT can be thought of as a form of diversification: minimizing risk without hurting
expected return
3 / 26
Two Risky Assets
- Let’s start with two risky assets, A and B
- The expected return and variance on a portfolio of two securities A and B are given by
E ( RP )
= wA E ( R A ) + wB E ( R B )
σ2 (RP ) = wA2 σ2 (RA ) + wB2 σ2 (RB ) + 2wA wB σ (RA )σ (RB )ρ(RA , RB )
wA = 1 → wB
- We can vary weights wA and wB and plot the expected return and standard deviation all possible
portfolios of the two risky assets, A and B
- Risk of the portfolios we obtain this way will importantly depend on correlation, ρ(RA , RB )
4 / 26
Two Risky Assets
zero
A Bhave
Assume
correlation
- Suppose ρ(RA , RB ) = 0
- E (RA ) = 10%, σ (RA ) = 20%, and E (RB ) = 5%, σ (RA ) = 10%
- Compute E (RP ) and σ (RP ) for portfolios of assets A and B
- Plot these portfolios on the graph: (x , y ) = (σ (RP ), E (RP ))
wA
0
0.2
0.4
0.6
0.8
1
wB
1
0.8
0.6
0.4
0.2
0
E ( RP )
5%
6%
7%
8%
9%
10%
σ ( RP )
10.00%
8.94%
10.00%
12.65%
16.12%
20.00%
5 / 26
Two Risky Assets
- Suppose ρ(RA , RB ) = 0
- E (RA ) = 10%, σ (RA ) = 20%, and E (RB ) = 5%, σ (RA ) = 10%
- Compute E (RP ) and σ (RP ) for portfolios of assets A and B
- Plot these portfolios on the graph: (x , y ) = (σ (RP ), E (RP ))
Rr a xtoositxs it
itTitl
adf.fi
T
np
wA
0
0.2
0.4
0.6
0.8
1
wB
1
0.8
0.6
0.4
0.2
0
E ( RP )
5%
6%
7%
8%
9%
10%
σ ( RP )
10.00%
8.94%
10.00%
12.65%
16.12%
20.00%
6 / 26
Two Risky Assets
Portfolio Oppurtunity Set for Assets A and B with Zero Correlation
12
wA = 1, wB = 0
10
with
better
this
ago
wA = 0.8, wB = 0.2
ther mis
wA = 0.6, wB = 0.4
E (RP ) (%)
8
wA = 0.4, wB = 0.6
20,80
ppm
6
I
Yskispositivelyrelatedtohigherreturn
wA = 0.2, wB = 0.8
wA = 0, wB = 1
4
2
inefficient
0
0
2
4
6
8
new
Linnestinthisleg
10
12
14
16
18
20
22
σ (RP ) (%)
of
at 2optionsat10
level risk
there
7 / 26
Two Risky Assets: Importance of Correlation
afterreatin movetogether
- Suppose ρ(RA , RB ) = 1
- Expected returns and volatility of A and B remain unchanged
- Compute E (RP ) and σ (RP ) for portfolios of assets A and B
- How do your portfolios change from the case when ρ(RA , RB ) = 0?
wA
0
0.2
0.4
0.6
0.8
1
wB
1
0.8
0.6
0.4
0.2
0
E ( RP )
5%
6%
7%
8%
9%
10%
σ ( RP )
10%
12%
14%
16%
18%
20%
8 / 26
Two Risky Assets: Importance of Correlation
- Suppose ρ(RA , RB ) = 1
- Expected returns and volatility of A and B remain unchanged
- Compute E (RP ) and σ (RP ) for portfolios of assets A and B
- How do your portfolios change from the case when ρ(RA , RB ) = 0?
8
6 Rp
T.iiifi.T.name i
I
wA
0
0.2
0.4
0.6
0.8
1
wB
1
0.8
0.6
0.4
0.2
0
E ( RP )
5%
6%
7%
8%
9%
10%
σ ( RP )
10%
12%
14%
16%
18%
20%
9 / 26
Two Risky Assets: Importance of Correlation
Portfolio Opportunity Set for Assets A and B with Perfect Positive Correlation
12
wA = 1, wB = 0
10
wA = 0.8, wB = 0.2
before
E (RP ) (%)
8
than
wA = 0.6, wB = 0.4
wA = 0.4, wB = 0.6
6
pithturn
wA = 0.2, wB = 0.8
wA = 0, wB = 1
4
I
1
2
0
0
2
4
6
8
10
12
14
16
18
20
22
σ (RP ) (%)
10 / 26
Two Risky Assets: Importance of Correlation
- Correlation has no e!ect on the expected return of a portfolio.
- However, the volatility of the portfolio will di!er depending on the correlation.
- The lower the correlation, the lower the volatility we can obtain. As the correlation
decreases, the volatility of the portfolio falls.
- The curve showing the portfolios will bend to the left to a greater degree as shown on the
next slide.
- The degree of curvature reflects the diversification e!ect: the lower the correlation
between the two securities, the greater the diversification.
11 / 26
Two Risky Assets: Importance of Correlation
Portfolio Opportunity Sets with Di!erent Correlation Coe”cients
12
ρ = 0 (Zero Correlation)
ρ = 1 (Perfect Positive)
ρ = →1 (Perfect Negative)
10
E (RP ) (%)
to diversification
1
8
6
6
1
4
free C
risk
due
higherreturn
2
0
0
2
stratification
4
1
6L
8
18remin portfoliorisk
tp.tl
10
12
14
16
18
20
22
σ (RP ) (%)
12 / 26
The E!ect of Diversification
- Diversification can substantially reduce the variability of returns without an equivalent
reduction in expected returns.
- This reduction in risk arises because worse than expected returns from one asset are o!set
by better than expected returns from another.
- Recall that the variance (risk) of a single security’s return can be broken down into:
- Systematic (market) risk – Economy-wide random events that a!ect almost all assets to a
certain degree
- Unsystematic (diversifiable) risk – Random events that a!ect single security or small groups
of securities
- The E!ect of Diversification:
- unsystematic risk will significantly diminish in large portfolios
- systematic risk is not a!ected by diversification since it a!ects all securities in any large
portfolio
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Two Risky Assets: Minimum Variance Portfolio needtoknowforexam
- Minimum variance portfolio (MVP) is the portfolio with the lowest level of risk among all
possible portfolios of the two assets, A and B
- To find what fraction of wealth to invest in asset A to obtain this MVP, we can solve
weight
2
2risk
min σP = min wA σA2 + (1 → wA )2 σB2 + 2wA (1 → wA )σA σB ρAB
w
w
A
A
movement
co
1waWB
- To find wA that minimizes portfolio variance, we set to zero the derivative with respect to
wA
εσP2
= 2wA σA2 → 2(1 → wA )σB2 + 2σA σB ρAB → 4wA σA σB ρAB = 0, or
εwA
Riskminimizationpronum
wAMVP =
σB2 → σA σB ρAB
σA2 + σB2 → 2σA σB ρAB
dependo
It
14 / 26
Two Risky Assets: Minimum Variance Portfolio
- In our example, ρAB = 0, σA = 0.2, and σB = 0.1.
- Find the minimum variance portfolio of assets A and B .
- What is the expected return and the standard deviation of this MVP?
The weights of the MVP portfolio can be computed as
wAMVP =
wBMVP = 1 → 0.2 = 0.8.
RB
seer wine
run wa
0.20
20 axis 0are
The expected return and standard deviation of thiscan
portfolio
10.27 10.17 2 0.20.1 0
our
10.1
w
σB2 → σA σB ρAB
0.12
=
= 0.2,
2
2
2
2
0.2
+
0.1
σA + σB → 2σA σBWpl
ρAB WY 0.8
attention 8.941
E (RMVP ) = wA E (RA ) + wB E (RB )
= 0.2 ↑ 0.1 + 0.8 ↑ 0.05 = 0.06 = 6%
!
σ (Tish
RMVP ) =a wA2 σ2 (RA ) + wB2 σ2 (RB )0+03
2wA wB σ (RA )σ (RB )ρ(RA , RB )
↓
↓ seaioaiit.ismjYiginti'soal risk
2
2
2
2
= 0.2 ↑ 0.2 + 0.8 ↑ 0.1 wa
= 0.008 = 0.0894 = 8.94%
minimize
g
p ortfolio
would
getariskfree
15 / 26
Two Risky Assets: Minimum Variance Portfolio
Portfolio Opportunity Set for Assets A and B with Zero Correlation
12
wA = 1, wB = 0
10
return
E (RP ) (%)
8
fffett
6
goal
nota
doorstar
4
illiterate
curve
in
2
0
risk
between
relationship
positive
0
2
4
6
8
10
wA = 0, wB = 1
Minimum Variance Portfolio
wA = 0.2, wB = 0.8
E (RP ) = 6%, σ (RP ) = 8.94%
12
14
16
18
20
22
σ (RP ) (%)
16 / 26
Two Risky Assets: E”cient Portfolios
- In the two asset case, MVP is the portfolio with the lowest possible risk and therefore
defines the lowest expected return we should be willing to accept
- We would never invest in portfolios with a lower expected return than the expected return
o!ered by the MVP. For example, would you hold asset B by itself?
- Rather than investing in such ine”cient portfolios, we would invest in the e!cient
portfolios: they o!er the highest achievable expected return for a given level of risk
17 / 26
Two Risky Assets: E”cient Portfolios
Portfolio Opportunity Set for Assets A and B with Zero Correlation
12
wA = 1, wB = 0
10
E”cient Portfolios
E (RP ) (%)
8
l
l
l
6
wA = 0, wB = 1
l
Minimum Variance Portfolio
wA = 0.2, wB = 0.8
E (RP ) = 6%, σ (RP ) = 8.94%
l
Ine”cient Portfolios
4
2
0
0
2
4
6
8
10
12
14
16
l
18
20
22
σ (RP ) (%)
18 / 26
Two Risky Assets: Allowing for Short Sales
- So far, we did not allow for the portfolio weights to be negative
- Can portfolio weights be negative? Yes!
- Long position in asset i: wi > 0
- Short position in asset i: wi < 0
- In a short sale, investor is borrowing a security from a broker and selling it, with the
understanding that it must later be bought back (hopefully at a lower price) and returned
to the broker
- When an investor goes long on an investment, it means that he or she has bought a stock
believing its price will rise in the future
- Conversely, when an investor goes short, he or she is anticipating a decrease in share price
19 / 26
Two Risky Assets: Allowing for Short Sales
EIRA 10 ERB
Portfolio Opportunity Set with and without Short Sales
14
feirkisrt.no this
Expected return, %
12
wA = 1.2, wB = →0.2
10
morerisk
morereturn
8
MVP
6
wA = →0.2, wB = 1.2
4
RP
2
0
o2 FRA
ez ERB
0.210 1.2 5 4
0
2
4
6
8
10
12
14
16
Standard deviation, %
18
20
22
24
20 / 26
Incorporating A Risk Free Asset
- So far, we have considered the risk and return possibilities that result from combining
risky assets into portfolios
- Let us introduce a risk-free asset with expected return Rf and σRf = 0
- We can create previously unattainable portfolios P by investing wM in a portfolio M on
the e”cient frontier and the rest in the risk-free asset
E ( R P ) = wM E ( R M ) + ( 1 → w M ) R f = R f + wM ( E ( R M ) → R f )
2
Var (RP ) = wM
Var (RM ), which gives wM = σ (RP )/σ(RM )
- Combining the equations gives the formula for a line from Rf to M
E ( RM ) → Rf
repian E (RP ) = Rf + σ(R ) σ(RP )
M
- The slope of this line is the ratio of risk premium to standard deviation, or the Sharpe
ratio
opener on this
patamonSecurity
spsert 0.36
freeasset
E ( RM ) → Rf
go.fnnanmon.mt.vethanrisk
Important
Sharpe ratio =
σ ( RM )
O
hterthebetter
riskfreeassets
21 / 26
Risky Portfolio and A Risk Free Asset
Haility
riskofthissecurity
returnamarton
amount risk
of
- To earn the highest expected return for any level of volatility we will select M that
generates the steepest line (the highest Sharpe ratio) when combined with Rf
- The steepest possible line is at the point where it just touches the e”cient frontier, so the
portfolio M we choose is the tangent portfolio
- The tangent line from Rf to this M is known as the capital market line (CML) (also
called the capital allocation line) and is described by
E ( RM ) → Rf
E ( RP ) = Rf +
σ ( RP )
σ ( RM )
- In the case when we have a risk-free asset and risky assets, it is not optimal to invest in
any portfolio that is NOT on the CML
- Which portfolio on the CML you will invest in depends on how much risk you are willing
to take
22 / 26
A Risky Portfolio and A Risk Free Asset
Portfolio Opportunity Set with Risk-Free Asset
14
Expected return, %
12
A
10
it
8
in
MVP
6
B
4
2
Rf
0
0
1
2
4
6
8
10
12
14
16
Standard deviation, %
18
20
22
24
22 / 26
A Risky Portfolio and A Risk Free Asset
Portfolio Opportunity Set with Risk-Free Asset
14
Expected return, %
12
Wa100
A
10
az 2
8
6
x
4
hrnantisan
2
Rf
0
0
Janie
x
2
4
6
8
10
12
14
16
Standard deviation, %
18
20
22
24
22 / 26
A Risky Portfolio and A Risk Free Asset
Portfolio Opportunity Set with Risk-Free Asset
14
Expected return, %
12
I
10
shier
A
C
8
6
4
I
2
Rf
0
0
2
4
6
8
10
12
14
16
Standard deviation, %
18
20
22
24
22 / 26
A Risky Portfolio and A Risk Free Asset
Portfolio Opportunity Set with Risk-Free Asset
14
Expected return, %
12
F
8
eminent
6
I
A
highersharperation
10
C
Iffit
Tangent Portfolio
4
flee
2
Rf
0
0
2
4
6
8
highestslopeefficient higher
10
12
14
16
Standard deviation, %
18
sharperatio
20
22
24
23 / 26
not onexam
How Can We Obtain Portfolio M?
- To arrive at portfolio M we look for a line from Rf to a point on the e”cient frontier with the
highest slope
- The slope is equal to the Sharpe ratio: so we maximize the Sharpe ratio
- In case of two risky assets, Sharpe ratio is:
SR =
E ( RM ) → Rf
wA E ( R A ) + ( 1 → wA ) E ( R B ) → R f
= !
σM
w 2 σ2 + (1 → w )2 σ2 + 2w (1 → w )σ σ ρ
A A
A
B
A
A
A B AB
- To figure out the weight of asset A, wA , in the market portfolio M, we maximize the Sharpe ratio
by setting
εSR
=0
εwA
- Once you know optimal wA , you know that wB = 1 → wA
- Note: the mathematics of the problem quickly becomes more complicated as we add more risky
assets
24 / 26
Two Fund Separation
- The portfolios of all risk averse investors thus consist of combinations of two funds : the
risk-free asset and the risky portfolio M
- The degree of risk aversion influences only the relative weight of the two funds, not their
composition
- If you want to achieve a greater return than the tangency portfolio, you can borrow at rate
Rf and invest everything in M (Jane’s portfolio)
- Investing a positive amount in both Rf and M will give you a lower return than the tangent
portfolio (Jack’s portfolio)
- Lower risk portfolios are obtained by increasing the weight of the risk free asset rather than
by re-optimizing the fund of risky assets
- Portfolio problem is then separated into two separate tasks :
1. Determine M by maximizing the Sharpe ratio
2. Choose combination of M and Rf that agrees with your preferences
25 / 26
Summary
- In the case with just risky assets, the optimal decision is to invest in one of the portfolios
on the e”cient frontier
- In the case with both risky assets and a risk-free asset, the optimal decision is to invest in
a portfolio on the capital market line
- Which portfolio on the e”cient frontier (in the case with only risky assets) or on the
capital market line (in the case with both risky assets and a risk-free asset) you end up
investing in depends on how much risk you are willing to take
26 / 26