Portfolio Theory
Dr. Andrea Rigamonti
andrea.rigamonti@econ.muni.cz
Seminar 5
Content:
• Initial settings and inputs estimation
• Portfolio optimization
Initial settings and inputs estimation
We want to perform both mean-variance utility maximization
and variance minimization given a target return.
• We use 𝛾 = 2 (lower risk aversion) and 𝛾 = 5 (higher risk
aversion) for the former
• We set Re = 0.5% as target monthly return for the latter
For the inputs estimation we use a rolling window of 120
periods. The portfolio performance will therefore be
computed starting from the 121th period.
Portfolio optimization
First we perform the mean-variance utility maximization.
At each period 𝑡, we compute the optimal weights as
𝒘 𝒎𝒗 =
1
𝛾
∑−𝟏 𝝁
and we compute the resulting portfolio return using
𝑅 𝑝 = 𝒘′ 𝒎𝒗 𝑹 𝒕
where 𝑹 𝒕 is the vector of the single assets returns at period 𝑡.
We will obtain a vector of out-of-sample returns 𝑹 𝒑, of which
we compute the mean, standard deviation, and utility using
𝑈 = 𝑀𝑒𝑎𝑛(𝑹 𝒑) −
𝛾
2
𝑉𝑎𝑟( 𝑹 𝒑)
Portfolio optimization
We do the same for the 1/N portfolio, but the weights are
computed using the formula:
𝒘 𝟏/𝑵 =
1
𝛾
𝟏′𝝁
𝟏′∑𝟏
𝟏
We then consider variance minimization given a target return.
The formula for the optimal weights is:
𝒘 𝒎𝒗 =
𝑅𝑒
𝝁′∑−𝟏 𝝁
∑−𝟏 𝝁
We then compute the mean, variance and Sharpe ratio:
𝑆𝑅 =
𝑀𝑒𝑎𝑛(𝑃𝑜𝑟𝑡𝑓𝑜𝑙𝑖𝑜 𝑒𝑥𝑐𝑒𝑠𝑠 𝑟𝑒𝑡𝑢𝑟𝑛𝑠)
𝑆𝐷(𝑃𝑜𝑟𝑡𝑓𝑜𝑙𝑖𝑜 𝑒𝑥𝑐𝑒𝑠𝑠 𝑟𝑒𝑡𝑢𝑟𝑛𝑠)
Portfolio optimization
Next is the minimum variance portfolio, whose weights are:
𝒘 𝒗 =
1
𝟏′∑−𝟏 𝟏
∑−𝟏 𝟏
Finally we consider the long-only minimum variance
portfolio. There are not closed form solutions, but we can
easily get the optimal weights using the “optimalPortfolio”
function from the “RiskPortfolios” package.
In this function we set “mu=NULL” and “semiDev=NULL”
because we do not set any target return or target
semideviation. “type='minvol’” indicates that we want to
minimize the variance, while “constraint='lo’” requires longonly
(i.e., positive) weigths.