Week 1: Basics of risk and return, portfolio theory
Webcast portfolio theory 101:
When looking at the figure, we can characterize the assets by looking at their expected return
(y-axis) and their risk, measured by the standard deviation (x-axis).
In this case we have two assets. The combination between these assets can take three forms:
- Going long in both assets → positive portfolio weights (the portfolio is somewhere in
between assets A and B in the figure
- Going long in asset A and short in asset B → portfolio weight of asset A is > 1 and B < 0
(the portfolio is somewhere along the extension line beyond the point of asset A)
- Going long in B and short in A → portfolio weight of B is > 1 and A < 0 (the portfolio is somewhere
along the extension line beyond the point of asset B)
In this figure, this portfolio would be inefficient.
In this case the correlation between the assets is equal to 0. This means that the movement of one
asset doesn’t influence the other asset. As we can see from the figure, this still means that
diversifying the portfolio decreases the risk. This can be seen by the lower associated risk of the
minimum risk portfolio compared to the risk of the assets individually.
The second figure indicates that a correlation of -1 (perfect negative correlation) leads to a minimum
risk portfolio with risk equal to 0. This means that a negative correlation between assets leads to
lower risk of the portfolio.
The third figure indicates that a correlation of +1 (perfect correlation) is less beneficial for an investor.
The straight line between the assets shows that there are no diversification benefits left.
When we introduce a risk-free rate, a line can be drawn tangent to the bullet. This line is called the
efficient frontier. Every portfolio combination that is on this straight line is an efficient portfolio. The
point where the straight line meets the bullet is the tangency portfolio.
If we now look at a situation where the correlation is equal to -0.3 (figure), we can see that the new
tangency portfolio has lower risk compared to the tangency portfolio when the correlation was equal
to 0.
Pennacchi §2.3: The efficient frontier
The bullet in the figures above can be described as the “risk versus expected return investment
opportunity set.” This set represents all possible ways of combining various individual assets to
generate alternative combinations of portfolio mean and variance (or standard deviation). This set
includes inefficient portfolios (those in the interior of the opportunity set) as well as efficient
portfolios (those on the “frontier” of the set). Efficient portfolios are those that make best use of the benefits
of diversification. As we shall later prove, efficient portfolios have the attractive characteristic that any two
efficient portfolios can be used to create any other efficient portfolio.
§2.3.1 A simple example
The expected return of a portfolio is calculated with:
𝑅𝑝 = ω𝑅¯𝐴 + (1 − ω)𝑅¯𝐵
- 𝑅𝑝 = Expected return of the portfolio
- ω = weight invested in asset A
- 𝑅¯𝐴 = return of asset A
- 𝑅¯𝐵 = return of asset B
,The standard deviation of the return on the portfolio is:
1/2
2 2 2 2
σ𝑝 = ⎡⎢ω σ 𝐴 + 2ω (1 − ω) σ𝐴σ𝐵ρ + (1 − ω) σ 𝐵⎤⎥
⎣ ⎦
Let us consider some special cases regarding the correlation between the two assets. Suppose ρ = 1, so that
the two assets are perfectly positively correlated. Then the portfolio standard deviation equals:
⎡ ⎤
σ𝑝 = ⎢ω σ 𝐴
+ (1 − ω)σ𝐵 ⎥
⎣ ⎦
Next, suppose ρ = −1, so that the assets are perfectly negatively correlated. Then
⎡ ⎤
σ𝑝 = ⎢ω σ 𝐴
− (1 − ω)σ𝐵 ⎥
⎣ ⎦
This formula can be used to calculate the proportion of portfolio A:
(σ𝐵 ± σ𝑝) (σ𝐴 ± σ𝑝)
ω = (σ𝐵 − σ𝐴)
or for portfolio B: ω = (σ𝐴 − σ𝐵)
For either the ρ = 1 or ρ = −1 case, an investor would always choose a portfolio represented by the positively
sloped lines (upper half of the bullet) because they give the highest average portfolio return for any given level
of portfolio risk. These lines represent the so-called efficient portfolio frontier. The exact portfolio chosen by
the individual would be where her indifference curve is tangent to the frontier.
When correlation between the assets is imperfect (−1 <ρ< 1), the relationship between portfolio expected
return and standard deviation is not linear, but is hyperbolic. In this case, it is no longer possible to create a
riskless portfolio, so that the portfolio having minimum standard deviation is one where σ𝑝 > 0.
Every portfolio on the mean-variance frontier can be replicated by a combination of any two frontier
portfolios; and an individual will be indifferent between choosing among the n financial assets, or choosing a
combination of just two frontier portfolios.
This remarkable finding has an immediate practical implication. If all investors have the same beliefs regarding
the distribution of asset returns, then they can form their individually preferred frontier portfolios by trading in
as little as two frontier portfolios. For example, if a security market offered two mutual funds, each invested in
a different frontier portfolio, any mean-variance investor could replicate his optimal portfolio by appropriately
dividing his wealth between only these two mutual funds.
Week 1, #1: Portfolio theory (A)
Portfolio theory is normative: which portfolio of assets is best for a given set of preferences. It focuses on a
single agent, and therefore needs fewer assumptions than a model of the entire market.
- A1. Agents prefer more over less (nonsatiation)
- A2. Agents dislike risk (risk-aversion).
These are necessary to determine the 2 parts of the trade-off; what’s good and what’s bad. Additionally, we
assume about the investors:
- A3. They maximize utility, and do so for 1 period.
- A4. Utility is a function of expected return and variance, and nothing else.
A3 and A4 combine to form a central tenet of portfolio theory: agents minimize risk at a given level of return,
or maximize return at a given level of risk. If A5 to A7 hold, the two approaches give the exact same set of
portfolios (the efficient set). The exact amount of risk aversion will determine which portfolio of that set is
ultimately chosen.
For the market conditions, we assume:
- A5. No distortion from costs, transaction fees, inflation or taxes.
, - A6. All information is available at no cost.
- A7. All investments are infinitely divisible.
The return of a portfolio is the return of the assets in the portfolio times the portfolio weights (ω), which gives:
𝑟𝑝 = ω' 𝑟
- 𝑟𝑝 = return of the portfolio
- ω ' = asset weight in portfolio. 'in this formula is used as the transpose sign, so it’s a transposed
vector.
- 𝑟 = return of the asset.
In matrix notation this would be: (In this example there are two assets
in the portfolio).
The variance of the portfolio is determined by the weights and the covariance matrix of the assets.
2
σ 𝑝
= ω' 𝑉ω
2
- σ 𝑝
= variance of the portfolio
- ω' = asset weight in portfolio
- 𝑉ω =
In matrix notation this would be:
If we assume there is no risk-free asset (actually a more realistic assumption than it seems) we should add the
constraint that portfolio weights sum to 1:
ω' 𝑒 = 1
- 𝑒 = is a vector consisting solely of 1’s
Optimization can be achieved by constructing the Lagrangian:
1
𝑚𝑖𝑛 ω' 𝑉ω + λ⎡⎢𝑅𝑝 − ω'𝑟𝑝⎤⎥ + γ[1 − ω'𝑒]
2 ⎣ ⎦
𝑅𝑝 indicates the required return. 𝑟𝑝 is the actual return
This formula contains two restrictions:
-⎡⎢𝑅 − ω'𝑟⎤⎥ indicates the difference between the required return and the actual return. This
⎣ 𝑝 ⎦
difference is multiplied by λ, which is a Lagrange multiplier
- [1 − ω'𝑒] indicates 1 minus the sum of the portfolio weights. This difference is multiplied by γ, which
is also a Lagrange multiplier
What is between both pairs of brackets should be equal to 0.
We minimize the variance by adjusting the weights, and subject to the restrictions that the portfolio return
equals 𝑅𝑝 and the weights sum to 1. We then construct the first order conditions (set derivatives to zero) w.r.t.
ω and the Lagrange multipliers λ and γ.
To tackle a Lagrangian, we need to find the so-called first order conditions. This means that we take the
derivative of the Lagrangian with respect to every input parameter and Lagrangian multipliers λ & γ and set it
equal to 0.
- First order condition w.r.t. ω (the portfolio weights):
𝑉 ω − λ𝑟𝑝 − γ𝑒 = 0
We want to have an expression for omega. To achieve, we need to get rid of the Lagrange multipliers
, - First order condition w.r.t. λ. If we take the first derivative with respect to the lagrange multiplier λ, we
get the original restriction back:
𝑅𝑝 − ω'𝑟 = 0
- First order condition w.r.t. γ. If we take the first derivative with respect to the lagrange multiplier γ, we
get the original restriction back:
1 − ω'𝑒 = 0
Solving the first one of the portfolio weights gives omega star:
* −1 −1
ω = λ𝑉 𝑟𝑝 + γ𝑉 𝑒
We then use the restrictions to create the equations that will allow us to eliminate λ & γ.
- If we multiply the above by 𝑟𝑝', we get an expression for the required return (first restriction):
* −1 −1
ω 𝑟𝑝' = λ𝑟𝑝'𝑉 𝑟𝑝 + γ𝑟𝑝'𝑉 𝑒
- If we multiply by 𝑒' we get an equation that equals 1 (see second restriction):
* −1 −1
ω 𝑒' = λ𝑒'𝑉 𝑟𝑝 + γ𝑒'𝑉 𝑒
Two equations with the same two unknowns (λ & γ) is solvable. This leads to:
δ𝑟𝑝−α ς−α𝑟𝑝
λ= 2 γ= 2
ςδ−α ςδ−α
−1 −1
- α = 𝑟𝑝'𝑉 𝑒 = 𝑒'𝑉 𝑟𝑝
−1
- ς = 𝑟𝑝'𝑉 𝑟𝑝
−1
- δ = 𝑒'𝑉 𝑒
* −1 −1
Substituting λ & γ in ω = λ𝑉 𝑟𝑝 + γ𝑉 𝑒 gives:
* δ𝑟𝑝−α −1 ς−α𝑟𝑝 −1
ω = 2 𝑉 𝑟𝑝 + 2 𝑉 𝑒
ςδ−α ςδ−α
Collecting terms in 𝑟𝑝 gives:
*
ω = 𝑎 + 𝑏𝑟𝑝
−1 −1 −1 −1
ς𝑉 𝑒−α𝑉 𝑟𝑝 δ𝑉 𝑟𝑝−α𝑉 𝑒
where 𝑎 = 2 and 𝑏 = 2
ςδ−α ςδ−α
so after rearranging terms we can express the optimal portfolio weights in terms of data we have:
- 𝑅𝑝, the required return
- 𝑟, average return of the assets
−1
- 𝑉 , the inverse of the covariance matrix of the assets.
The solution of this is not pretty as we can see from the formula below:
−1 −1 −1 −1
* 𝑒𝑉 𝑒𝑅𝑝−𝑟'𝑉 𝑒 −1 𝑟'𝑉 𝑟−𝑟'𝑉 𝑒𝑅𝑝 −1
ω = −1 −1 −1 2 𝑉 𝑟+ −1 −1 −1 2 𝑉 𝑒
𝑟'𝑉 𝑟 𝑒𝑉 𝑒 − (𝑟'𝑉 𝑒) 𝑟'𝑉 𝑟 𝑒𝑉 𝑒 − (𝑟'𝑉 𝑒)
−1
- 𝑟'𝑉 𝑟 is called zeta (ζ) → calculated by taking the sum of all products (sumproduct) of
−1
𝑟 & 𝑟' 𝑉 for every portfolio
−1
- 𝑒'𝑉 𝑒 is called delta (δ) → calculating by summing the entire inverse covariance matrix
−1 −1
- 𝑟'𝑉 𝑒 is called alpha (α) → is sum of all the 𝑟' 𝑉
−1
In story the calculation of the portfolio weight is the done by (starting at the top): delta (𝑒'𝑉 𝑒 ) times the
−1 −1 −1
required return (𝑅𝑝) minus alpha (𝑟'𝑉 𝑒). This is divided by zeta (𝑟'𝑉 𝑟) times delta (𝑒'𝑉 𝑒) minus alpha
−1 −1
(𝑟'𝑉 𝑒) squared. The total of this is then multiplied by the inverse of the covariance matrix (𝑉 ) times the
