Quantitative thoughts

Bespoke blogged about average monthly returns of the DJI and emphasized April. Before jumping on that information, let’s check some weak points.
In that post, only average returns are presented. We need at least extreme points (min;max) and confidence ranges. Second problem – the normal market have upward trend and we need to get rid of that. To do so, either we have subtract the  rolling mean (a tough way) or use logarithmic prices (that’s the easy way!).

Instead of DJI, I took S&P500 from 1950 until now.

Sure, average return in April is above 0, but based on historical data, negative return is possible. I conducted t-test, where null hypothesis was, that average return is equal to zero. I got p-value of 0.0042, so null hypothesis can be rejected (return is above 0).

The graph below shows cumulative return of investment, investing only in April. Keep in mind, that this is log scale and real return would be higher.

Conlusion: based on this data, expect positive return in April.

R code

require(xts)
require(quantmod)
require(ggplot2)
getSymbols('^GSPC',from='1950-01-01')
return<-Delt(Cl(to.monthly(log(GSPC))))
return[1]<-0
temp<-as.double(format(index(return),'%m'))
 
temp<-data.frame(as.double(return),as.numeric(temp))
qplot(factor(as.numeric(temp[,2])),as.double(return),data=temp,geom = "boxplot",ylab='Returns',xlab='Months')
 
t.test(return[which(format(index(return),'%m')=='04')])
plot(cumprod(return[which(format(index(return),'%m')=='04')]+1),main='Cumulative return, 1950-present')

Latvia to divide Parex Banka into two
Regardless of this news my broker sold Parex bank’s bonds.
Let’s make simple calculation. This bond has 2 outstanding coupons, worth of 5.6%. Total payoff will be approximately 1115 Eur. Somebody paid 980 Eur and is happy with 13.7% premium. Good premium, but downside risk is -100%.
Somebody is very well informed

Inspired by CXO group report, I did a rerun of the same strategy on my data. Easter’s dates can be find at wikipedia. Overall, my results are similar to CXO group’s results.

In the graph below, I plotted daily returns on Easter week (Monday to Thursday) from 1982 to 2009. I prefer this way of showing things, where the range, minimum, maximum and mean of returns are presented.
It is clear, that only returns on Thursdays are above zero and t-test confirms that:
t = 2.235, df = 27, p-value = 0.03389. The rest is close to random.

The graph below shows daily returns one week after Easter holidays. Although Monday looks like negative day, but it lacks significance:
t = -1.1517, df = 27, p-value = 0.2595 (should be less that 0.1).
The rest is noise.

In summary, only returns on Thursdays have positive bias.

#R-Language code to repeat this test.
require(quantmod)
require(xts)
easter<-as.matrix(read.table(file='data/easter.csv'))
getSymbols('^GSPC',from='1980-01-01')
 
GSPC.delta<-Delt(log(Cl(GSPC)))
GSPC.delta<-Delt(Cl(GSPC))
 
#returns during the week before Easter
#returns on Thursdays
nasa<-data.frame(as.double(GSPC.delta[(as.Date(easter,'%m/%d/%y')-3)]),4)
colnames(nasa)<-c('ret','weekday')
 
#Monday to Wednesday
for(i in 1:3)
{
tmp<-data.frame(as.double(GSPC.delta[(as.Date(easter,'%m/%d/%y')-(3+i))]),(4-i))
colnames(tmp)<-c('ret','weekday')
rbind(nasa,tmp)
nasa<-rbind(nasa,tmp)
}
t.test(nasa$ret[nasa$weekday==4])
 
require(ggplot2)
qplot(factor(as.numeric(nasa$week)),as.double(nasa$ret),data=nasa,geom = "boxplot",ylab='Returns',xlab='Weekdays')
 
#returns during the week after Easter
 
GSPC.delta<-Delt(Cl(GSPC))
nasa<-data.frame(as.double(GSPC.delta[(as.Date(easter,'%m/%d/%y')+1)]),1)
colnames(nasa)<-c('ret','weekday')
for(i in 2:5)
{
print(i)
tmp<-data.frame(as.double(GSPC.delta[(as.Date(easter,'%m/%d/%y')+i)]),i)
colnames(tmp)<-c('ret','weekday')
rbind(nasa,tmp)
nasa<-rbind(nasa,tmp)
}
t.test(nasa$ret[nasa$weekday==1]);t.test(nasa$ret[nasa$weekday==2])
qplot(factor(as.numeric(nasa$week)),as.double(nasa$ret),data=nasa,geom = "boxplot",ylab='Returns',xlab='Weekdays')

Recently, I was busy testing the following strategy:

If SPY and VIX daily returns are positive, then short SPY at close and keep it for one day.

The strategy is dump simple and it has very good feature – short side. There are not so many successful short side strategies. For testing purpose I used daily Yahoo data from 1995 until present. For commissions and bid/ask spread I used 5 $ fee per 10 000 $ trade. Here we go:

Annualized Return                     0.0421
Annualized Std Dev                   0.0488
Annualized Sharpe (Rf=0%)        0.8621
t = 3.3787, df = 3811, p-value = 0.0007356

Up to this point is was relatively easy to make a test (the true is, that I spent some time cracking and hacking blotter package, but I will write about it in separate post). My second objective was the improvement of this strategy.

One of the way to understand the strategy is to look how the components are related to each other or correlated. To do that, I took daily returns of SPY and VIX at day 0 and plotted against SPY next day’s (day+1) returns.

What can I tell by looking at this plot? I couldn’t figure out any linear relation between returns of SPY and VIX at day+0 and returns of SPY at day+1. Should I try something like random forest method?

I tried to add some TA flavors, like RSI, but the improvements were not very significant. Simplicity is genius!

It is know, that the first day of the month provides bullish edge. According to Quantifiable edges not all the months are equal. So, I made a test on S&P500 index, from January, 1980 until February, 2010. It is true, March isn’t the best month to run this strategy.

Only 3 months have significant results based on p-values:
“month 5, p-value 0.0399233570186162″
“month 7, p-value 0.0466800163648646″
“month 11, p-value 0.0218919220125013″

p.s. if somebody is interested in R-Language code to repeat this test, then let me know.