Portfolio Theory
Dr. Andrea Rigamonti
andrea.rigamonti@econ.muni.cz
Lecture 4
Content:
• Returns and risk
• Estimation windows
• Mean and variance of a portfolio
Returns and risk
Simple returns: given the price 𝑃𝑡 at time t, they are
computed as:
𝑅𝑡 =
𝑃𝑡 − 𝑃𝑡−1
𝑃𝑡−1
• Simple returns are asset-additive. At time t, given n asset
returns 𝑅𝑖,𝑡 with corresponding weights 𝑤𝑖,𝑡, the portfolio
return 𝑅 𝑝,𝑡 is equal to the weighted average of the single
assets’ return:
𝑅 𝑝,𝑡 = ෍
𝑖=1
𝑁
𝑤𝑖,𝑡 𝑅𝑖,𝑡
Returns and risk
• Simple returns are NOT time-additive. To compute the
evolution of the value of an asset (or of a portfolio) over 𝑇
periods, given the initial value 𝑉0, we have to use the
formula:
𝑉𝑇 = 𝑉0 + ෍
𝑡=1
𝑇
( 𝑉𝑡−1 𝑅𝑡)
• Equivalently, we can use this formula that directly
computes cumulative returns:
𝑉𝑇 = 𝑉0 + 𝑉0 ෑ
𝑡=1
𝑇
1 + 𝑅𝑡 − 1
Returns and risk
Log returns (or continuous returns):
𝑟𝑡 = ln
𝑃𝑡
𝑃𝑡−1
= ln 𝑃𝑡 − ln(𝑃𝑡−1)
• Log returns are NOT asset-additive: they cannot be used to
compute portfolio return as done with simple returns. But if
returns are close to 0, they provide a good approximation:
𝑅 𝑝,𝑡 ≈ ෍
𝑖=1
𝑁
𝑤𝑖,𝑡 𝑟𝑖,𝑡
• Log returns are time-additive: the cumulative return from
time 1 to time T can be obtained with a simple sum:
𝑐𝑢𝑚𝑟𝑒𝑡 = ෍
𝑡=1
𝑇
𝑟𝑡
Returns and risk
Log returns do not have the same intuitive interpretation
of simple returns (in addition to not being asset-additive).
If you are working with log returns, you might want to
convert them back to simple returns when evaluating the
results. This can be done with a simple formula:
𝑅 = exp 𝑟 − 1
Simple returns can be converted to log returns in this way:
𝑟 = ln(𝑅 + 1)
Returns and risk
When dividends are paid, all else equal, the price of the
shares decrease by an amount equal to the dividend.
Unadjusted returns register a loss that is not real, as we are
fully compensated for the decrease in the price by receiving
an equal amount of wealth in form of dividend.
We adjust by assuming that the dividend is immediately
reinvested in the same stocks. The return adjusted for the
dividend 𝐷𝑖𝑣 𝑡 paid at time t is:
𝑅𝑡 =
𝑃𝑡 + 𝐷𝑖𝑣 𝑡 − 𝑃𝑡−1
𝑃𝑡−1
Returns and risk
We can think at the probability distribution as a function
that assigns a probability 𝑝 𝑅 that each possible return R of
an asset will occur.
Expected (or mean) return: the expected value of the
returns, computed as a weighted average of the possible
returns, where the weights correspond to the probabilities:
𝐸𝑥𝑝𝑒𝑐𝑡𝑒𝑑 𝑅𝑒𝑡𝑢𝑟𝑛 = 𝐸[𝑅] = ෍
𝑅
𝑝 𝑅 𝑅
The simplest approach is to compute the expected return as
a sample estimate using the T past returns:
𝐸[𝑅] = 𝑅 =
1
𝑇
෍
𝑖=1
𝑇
𝑅𝑡
Returns and risk
In this context, the variance is the expected squared deviation
from the mean:
𝑉𝑎𝑟(𝑅) = 𝐸[(𝑅 − 𝐸[𝑅])2
] = ෍
𝑅
𝑝 𝑅(𝑅 − 𝐸[𝑅])2
To measure volatility the standard deviation (SD) is preferable
to the variance because it is in the same unit of the returns.
𝑆𝐷(𝑅) = 𝑉𝑎𝑟(𝑅)
Given the sample mean, we can compute the sample variance:
𝑉𝑎𝑟(𝑅) =
1
𝑇 − 1
෍
𝑖=1
𝑇
(𝑅𝑡−𝑅)2
Returns and risk
A higher variance implies higher risk, as returns are more
likely to be very different from the mean.
Returns and risk
We can evaluate the precision of our estimate by
computing the standard error, which is the standard
deviation of the estimated value of the mean of the
actual distribution around its true value.
If returns are identically and independently distributed,
and the distribution remains the same over time, we
calculate the standard error SE of the estimate of the
expected return as:
𝑆𝐸 =
𝑆𝐷
𝑇
In practice, the true value of the standard deviation SD is
unknown, and its sample estimate is used.
Estimation windows
• From the formula it appears that a bigger sample size
would make the estimate more reliable.
• This is true up to a certain point, as the return distribution
(and therefore the true value of the parameters) change
over time.
• Trade-off: using longer estimation windows allows for a
larger sample size, but also includes observations that
might be too old and no longer useful.
• There is no hard rule to determine the ideal estimation
window length. Usually, up to 5 years are suggested with
daily data, 10 or 15 years for weekly data, and a few
decades with monthly data.
Estimation windows
When computing sample estimates, a rolling window or an
expanding window can be used.
• A rolling window of length T uses observations from
period 1 to T for the first estimation, then from 2 to T+1
for the second, and so on. It has the advantage of
gradually getting rid of older observations.
• An expanding window with initial length T uses
observations from period 1 to T for the first estimation,
then from 1 to T+1 for the second, and so on. It allows for
more out-of-sample periods, as one can start with a low T.
In general, a rolling window is recommendable if the
available time series allows for a large T.
Mean and variance of a portfolio
The portfolio weights represent the fraction of the total
investment in the portfolio held in each individual stock:
𝑤𝑖 =
𝑉𝑎𝑙𝑢𝑒 𝑜𝑓 𝑖𝑛𝑣𝑒𝑠𝑡𝑚𝑒𝑛𝑡 𝑖
𝑇𝑜𝑡𝑎𝑙 𝑣𝑎𝑙𝑢𝑒 𝑜𝑓 𝑡ℎ𝑒 𝑝𝑜𝑟𝑡𝑓𝑜𝑙𝑖𝑜
Weights can also be negative, if short selling is allowed.
Given a portfolio of N assets with returns 𝑅𝑖 and weights 𝑤𝑖,
the return on the portfolio, 𝑅 𝑝, is the weighted average of
the returns on the investments in the portfolio:
𝑅 𝑝 = ෍
𝑖=1
𝑁
𝑤𝑖 𝑅𝑖
Mean and variance of a portfolio
The expected return of a portfolio is the weighted average of
the expected returns of the investments in it, using the
portfolio weights:
𝐸[𝑅 𝑝] = ෍
𝑖=1
𝑁
𝑤𝑖 𝐸[𝑅𝑖]
The Variance of a two-stock Portfolio is given by:
𝑉𝑎𝑟(𝑅 𝑝) = 𝑤1
2
𝑉𝑎𝑟(𝑅1) + 𝑤2
2
𝑉𝑎𝑟(𝑅2) + 2𝑤1 𝑤2 𝐶𝑜𝑣(𝑅1, 𝑅2)
Computing the variance of portfolios of more than two assets
requires some matrix algebra to be feasible.
Mean and variance of a portfolio
Consider N assets. We define the vector of returns R, the vector
of mean µ, the vector of weights w and the covariance matrix ∑
as:
R =
𝑅1
𝑅2
⋮
𝑅 𝑁
µ =
µ1
µ2
⋮
µ 𝑁
=
𝐸[𝑅1]
𝐸[𝑅2]
⋮
𝐸[𝑅 𝑁]
w=
𝑤1
𝑤2
⋮
𝑤 𝑁
∑=
𝜎1
2 𝜎1,2 … 𝜎1,𝑁
𝜎1,2
⋮
𝜎2
2
⋮
…
⋱
𝜎2,𝑁
⋮
𝜎1,𝑁
𝜎2,𝑁 … 𝜎 𝑁
2
The variance of a portfolio is: 𝑉𝑎𝑟 𝑅 𝑝 = 𝒘′∑𝒘
Mean and variance of a portfolio
Obviously, we can also express with matrix notation the
formulas for the portfolio return and expected return:
𝑅 𝑝 = 𝒘′𝑹
µ 𝑝 = 𝒘′µ
Example with a portfolio of three assets: A, B and C.
𝑅 𝑝 = 𝒘′ 𝑹 = 𝑤 𝐴 𝑤 𝐵 𝑤 𝐶
𝑅 𝐴
𝑅 𝐵
𝑅 𝐶
= 𝑤 𝐴 𝑅 𝐴 + 𝑤 𝐵 𝑅 𝐵 + 𝑤 𝐶 𝑅 𝐶
µ 𝑝 = 𝒘′µ= 𝑤 𝐴 𝑤 𝐵 𝑤 𝐶
µ 𝐴
µ 𝐵
µ 𝐶
= 𝑤 𝐴µ 𝐴 + 𝑤 𝐵µ 𝐵 + 𝑤 𝐶µ 𝐶
Mean and variance of a portfolio
𝑉𝑎𝑟 𝑅 𝑝 = 𝒘′∑𝒘 = 𝑤 𝐴 𝑤 𝐵 𝑤 𝐶
𝜎𝐴
2
𝜎𝐴𝐵
𝜎𝐴𝐵
𝜎 𝐵
2
𝜎𝐴𝐶
𝜎 𝐵𝐶
𝜎𝐴𝐶 𝜎 𝐵𝐶 𝜎 𝐶
2
𝑤 𝐴
𝑤 𝐵
𝑤 𝐶
= 𝑤 𝐴 𝜎𝐴
2
+ 𝑤 𝐵 𝜎𝐴𝐵 + 𝑤 𝐶 𝜎𝐴𝐶 𝑤 𝐴 𝜎𝐴𝐵 + 𝑤 𝐵 𝜎 𝐵
2
+ 𝑤 𝐶 𝜎 𝐵𝐶 𝑤 𝐴 𝜎𝐴𝐶 +𝑤 𝐵 𝜎 𝐵𝐶 𝑤 𝐶 𝜎 𝐶
2
𝑤 𝐴
𝑤 𝐵
𝑤 𝐶
= 𝑤 𝐴
2
𝜎𝐴
2
+ 𝑤 𝐴 𝑤 𝐵 𝜎𝐴𝐵 + 𝑤 𝐴 𝑤 𝐶 𝜎𝐴𝐶 + 𝑤 𝐴 𝑤 𝐵 𝜎𝐴𝐵 + 𝑤 𝐵
2
𝜎 𝐵
2
+ 𝑤 𝐵 𝑤 𝐶 𝜎 𝐵𝐶 + 𝑤 𝐴 𝑤 𝐶 𝜎𝐴𝐶 + 𝑤 𝐵 𝑤 𝐶 𝜎 𝐵𝐶 + 𝑤 𝐶
2
𝜎 𝐶
2
= 𝑤 𝐴
2
𝜎𝐴
2
+ 𝑤 𝐵
2
𝜎 𝐵
2
+ 𝑤 𝐶
2
𝜎 𝐶
2
+ 2𝑤 𝐴 𝑤 𝐵 𝜎𝐴𝐵 + 2𝑤 𝐴 𝑤 𝐶 𝜎𝐴𝐶 + 2𝑤 𝐵 𝑤 𝐶 𝜎 𝐵𝐶
With N=3 the formula without matrix notation is already long,
and it quickly becomes unfeasible to express it as N grows.
The formula with matrix notation is always 𝑉𝑎𝑟 𝑅 𝑝 = 𝒘′∑𝒘
with any N, and it is very easy to code such formula in, e.g., R.