Portfolio Theory
Dr. Andrea Rigamonti
andrea.rigamonti@econ.muni.cz
Lecture 11
Content:
• Return distribution statistics
• Alpha
• Turnover and net returns
Return distribution statistics
We first may start the results evaluation by computing
some statistics that describe the return distribution:
• Mean: the higher the better
• Standard deviation: the lower the better
• Skewness: a positive value is preferable
• (Excess) Kurtosis: a lower value is preferable unless the
skewness is significantly positive
We then compute the Sharpe ratio to quantify the riskadjusted
return.
Return distribution statistics
To evaluate downside risk, instead or in addition to the
standard deviation and the Sharpe ratio we may compute
some or all of these risk measures:
• Downside deviation: the lower the better
• CVaR: the lower (in absolute value) the better
• (Maximum) drawdown: the lower the better
We then quantify the risk-adjusted return with an
appropriate measure, like the Sortino ratio.
Alpha
The alpha is used to check if the returns of the investment
are explained by a given asset pricing model. It is obtained
as the intercept in a regression of the portfolio returns
over the returns of the factors of the model considered.
Usually, the CAPM (in which case the alpha is called
“Jensen’s alpha”) or the Fama-French three-factor model
are used. In the first case the regression takes the form:
𝑅 = 𝛼 + 𝛽 𝑅 𝑀𝑘𝑡 − 𝑅𝑓
while in the second case it is:
𝑅 = 𝛼 + 𝑏1 𝑅 𝑀𝑘𝑡 − 𝑅𝑓 + 𝑏2 𝑆𝑀𝐵 + 𝑏3 𝐻𝑀𝐿
Alpha
If 𝛼 is significantly greater than zero, it means that our
strategy achieves returns higher than those predicted by
the model based on the portfolio exposure to the factors.
To check if we have a significant positive 𝛼, we compute its
standard errors and then perform a test of hypothesis. The
usual standard errors for the linear regression are generally
not appropriate, as they assume homoskedasticity (i.e.,
constant variance) and no autocorrelation (i.e., no time
dependency).
We may use instead the Newey-West standard errors,
which can be easily computed in R.
Turnover and net returns
The turnover quantifies how much trading the strategy
requires. The higher the turnover, the higher the
transaction costs, which translates into lower net returns.
The turnover at a certain period 𝑡 is given by:
𝑇𝑂𝑡 = ෍
𝑖=1
𝑁
𝑤𝑖,𝑡 − 𝑤𝑖,𝑡−1
For each stock we compute the absolute value of the
change in its weight compared to the previous period,
and we sum all these 𝑁 values. We do this for each of the
𝑇 periods in which we applied our strategy, and then we
compute the mean to get the average turnover.
Turnover and net returns
To correctly use this formula, we need to account for the
effect of realized returns during the previous period.
Suppose we have a portfolio with two assets updated
monthly, and the weights at time 𝑡 − 1 were 0.5 and 0.5,
while now at time 𝑡 we want to change them to 0.4 and 0.6.
If during that month the first asset experienced a +10%
return, and the second one a -20% return, when we update
the portfolio at time 𝑡 we no longer have the two original
weights, but 0.5 + 0.5 × 0.1 = 0.55 for the first asset and
0.5 − 0.5 × 0.2 = 0.4 for the second.
Turnover and net returns
These weights do not sum up to 1 because the total
value of the portfolio changed compared to period 𝑡 − 1.
We need to account for this by dividing both weights by
their sum.
So in this example where they sum to 0.55 + 0.4 = 0.95
we have Τ0.55 0.95 ≈ 0.58 for the first weight and
Τ0.4 0.95 ≈ 0.42 for the second. Therefore, the actual
turnover is 0.4 − 0.58 + 0.6 − 0.42 = 0.36.
Turnover and net returns
A more accurate evaluation can be obtained by considering
the portfolio returns net of transaction costs.
Transaction costs can be fixed or proportional to the
amount of trading. The latter are more appropriate, and
can be accounted for by multiplying the turnover of each
asset for the proportional cost.
Frazzini (2012) suggests using transaction costs equal to 10
basis points (bp). A basis point is equal to 0.01%.
Therefore, for example, if we need to buy or sell 5% of the
positions in a certain asset, transaction costs are equal to
0.05 × 0.001 = 0.00005, which means that 0.005% of the
money invested in that position is lost in transaction costs.
Turnover and net returns
The portfolio returns net of transaction costs can be used
to compute all the other statistics we listed before.
Finally, it is useful to visualize the value 𝑉 of the portfolio
over time, which can be computed as
𝑉𝑇 = 𝑉0 + ෍
𝑡=1
𝑇
( 𝑉𝑡−1 𝑅𝑡)
It is appropriate to compute the value both ignoring and
net of transaction costs, and to plot it.
Turnover and net returns
We might want to also compute the evolution of real
wealth, i.e., accounting for the inflation. We can do it by
dividing the value of the portfolio over time by the deflator.
We can compute the deflator 𝐷 using a formula analogous
to the one used to compute the value of the portfolio,
simply replacing the return with the inflation rate 𝐼:
𝐷 𝑇 = 𝐷0 + ෍
𝑡=1
𝑇
( 𝐷𝑡−1 𝐼𝑡)
The two series need to have the same starting value (e.g., 1
unit of wealth), and the same frequency (e.g., monthly).