So where did we mess up?
In the calculation of returns for the market cap weighted portfolio and
the portfolio optimization portfolio, we simply took the starting weights (W_{0})
and multiplied them by the relevant series of returns.
resEqual = as.matrix(returns) %*% t(ret)
and
subRes = as.matrix(subRes) %*% t(ret)
To correct this, we have 2 options.
 Recalculate the weight at each time point assuming a starting weight. Multiply those weights to each day’s returns to produce the series
 Assume a starting monetary value of the portfolio (1 is convenient), and apply the return series to the position values. At each period, calculate the daily portfolio return.
1 does not equal 2. Why?
Remember that we are using log returns.
#1 is a weighted arithmetic average.
#2 is the log of a sum of functions.
If we say that prices are final and weights are initial –
that is that we observe the price at the end of time t and the weight at the
beginning (as a function of prior period prices), then we can rewrite #2 as
The literature uses #1 because it makes the math easier. The numbers are approximately equal because log(1+r)
~= r (or exp(r) ~= 1+r, if you prefer).
However, they are different and the difference can compound with
time. Your brokerage account works
according to #2.
We will use #1 as that is the convention. If I was analyzing something for real, and I
was given a log returns data set to use, I would use #2. #1 may be more compact and make the math
nice. #2 will more closely reflect your
account balance.
How do we calculate w_{i,,t }given w_{i,t1}? We apply the returns to the weights and
restandardize the numbers.
Because we want to reuse
this methodology for cap weighted and optimized portfolios, we should just
create a function that will take a series of returns and a vector of weights,
and give back the return series.
reweight = function(returns,
startWeight){
n
= nrow(returns)
lastWeight
= as.vector(startWeight)
outReturn
= data.frame()
for(i in seq(1,n)){
rts
= as.vector(exp(returns[i,]))
w
= lastWeight * rts
sumW
= sum(w)
w
= w/sumW
r
= as.matrix(returns[i,]) %*% w
lastWeight
= w
outReturn
= rbind(outReturn,r)
}
return
(outReturn)
}
Substituting the function call in place
of the lines quoted above, the new Annualized Returns table looks like this:
CapWeight
Portfolio

EqualWeight
Portfolio

Portfolio
Optimization


Annualized
Return

0.0258

0.0128

0.0282

Annualized
Std Dev

0.2199

0.2242

0.1775

Annualized
Sharpe (Rf=0%)

0.1174

0.0570

0.1590

Here we can see the capitalization
weighted portfolio performs much better than previously calculated. This is expected. The Portfolio Optimization portfolio also
does better and has a slight advantage over the cap weight portfolio in the
Sharpe Ratio.
The correlation chart is:
The cap weight portfolio and the equal
weight portfolio are nearly perfectly correlated. The optimized portfolio is less correlated,
but still has a high degree of correlation.
NOTE: we can put a Beta on the optimized
portfolio of 0.70 (.87 * .1775 / .2199) using the calculations above.
The cumulative return graph is:
This really just shows us what we already know. The cap and equal weights are highly
correlated. The cap weight pulls away
after the 2009 bottom. The optimized
portfolio shows less volatility than the cap weight portfolio after the bottom. It has a steady march up while the cap weighted
portfolio whips up and down.
No comments:
Post a Comment