I got confused the other day, when playing around with an Rscript for portfolio longevity. I've always used discrete returns (1+r(t)) in simple sims because 1) it's easy, 2) I don't know continuous time(CT), and 3) I am working inside the sim with very discrete annual steps. In the script I got from Professor Milevsky he uses e^(vt), a continuous form of return, but told me later that either approach is fine as long as the rates are matched to the compounding intervals.
So, why either? What difference does it make?
I simulate in 1 year steps in R and in real life I spend periodically and my bonds pay both discretely and annually or semi-annually. That means the discrete mode is more comfortable for me but if one is steeped in calculus then maybe "e^--" makes more sense.
How much of a difference is there, all else equal? Figure 1 should speak for itself (if I got the math and sim code right...always a good question to ask myself). Almost half of portfolios using the selected parameters (.04, .12) eventually go infinite (with 100 years as a proxy here) using CT, while with the linear it's more like a third. Otherwise the frequency distributions of the "finite portfolios" in years looks vaguely similar. An aside: using central tendency stats probably doesn't make sense with a defective distribution that looks like this so I don't use them here.
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| Figure 1 |
Conclusion: it's evidently a free policy choice to go either way -- with maybe small differences in inputs and methods -- but there is a big non-quantified impact on outcomes to my naïve eye. So, I'll stick with discrete since it is: 1) easier for me to think about, 2) it makes design sense, 3) the oft-critiqued normal distribution of discrete time returns is useful to me in easier math and coding, and 4) maybe it might be a little more conservative in a planning context.
I got turned around on this 5 times so feel free to comment. I'm still thinking about this a bit and might change something again.
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