This is a continuation of Lumpsum vs. Dollar Cost Averaging (SIP) that modeled different return series and concluded that a prudent investor would be better off with a SIP because of a smaller probability of incurring a large loss. However, we stopped short of comparing different indexes to see if the conclusion held.

### The ‘average’ return

What happens if we take the average weekly return of an index and create a synthetic index that just gives those average returns without any variance? We end up with a parabolic looking cumulative return profile below:

The small cap index was chosen on purpose to illustrate how ‘average’ returns relate to real returns on an extremely volatile index.

The average return series is, of course, a fantasy. What we are interested in is in the probability of getting those returns.

### Probabilities

Just like our first post, we start by modeling the returns of the NIFTY 50, MIDCAP and SMLCAP indexes as a Generalised Lambda Distribution and running a 10,000 path simulation to obtain a series of DCA vs lumpsum investment returns. We then feed that into a empirical cumulative distribution function so that we can query it for probabilities under different thresholds. To put that in a picture:

The vertical lines mark the different thresholds we are interested in.

- The grey line on the left is at zero. We have SIP showing a 4.44% probability of negative returns and lumpsum showing 3.49%. Yes, there is a non-trivial possibility that SIPs will give negative returns. However, looking at the shapes below zero, SIP losses may not be as large as lumpsum losses.
- The red line in the middle is the start to finish return of the index. Here, we have SIP showing a 22.69% probability of exceeding those point-to-point returns and lumpsum showing 57.50%.
- The orange line on the right is the compounded ‘average’ return. We have SIP showing a 7.82% probability of exceeding that and lumpsum showing 37.20%.

### What does all this mean?

- It is possible for SIP returns to be negative over large periods of time. Enough to cover your entire investing lifetime. So, if you are investing in small-caps, make sure you are not 100% allocated to it.
- Lumpsum investing gives you a higher probability of higher returns across all indexes. The probability of negative returns are on par with that of SIP’s.
- Lumpsums have fatter left tails. However, if you are looking only at NIFTY 50 and MIDCAP, those probabilities are tiny.
- Lumpsums have a higher probability of achieving ‘average’ returns compared to SIPs.
- Lumpsums seem to be benefiting from “time in the market” on indexes that rise over a period of time.

Code and charts on github.

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