Terminal Wealth Distribution for Dollar-Cost Averaging

Summary

The document considers the terminal wealth distribution when monthly investments rise at a fixed rate while asset returns are independent and identically distributed lognormal variables. It contrasts this regular-contribution plan with a lump-sum investment, whose terminal wealth is lognormal under the stated return assumption.

The accepted answer expresses final wealth as a sum of contributions, where each successive investment is multiplied by the returns earned from its investment date through the horizon. Because this sum combines differently weighted products of lognormal returns, the answer says there is no simple closed-form distribution or straightforward numerical-integral representation, citing a difficulty with the lognormal characteristic function. It suggests calculating initial moments and potentially approximating the distribution with a Cornish–Fisher expansion around a lognormal form. The document supplies no numerical example or validation of that approximation, and its conclusions depend on the assumed IID lognormal return model and regular contribution growth.

Key ideas

• A lump-sum investment has lognormal terminal wealth under the stated IID lognormal return assumption.
• Growing periodic contributions produce terminal wealth as a sum of compounded contributions.
• Each contribution earns returns only from the period in which it is invested onward.
• The answer reports no simple closed-form distribution for this sum.
• Moment calculations and a Cornish–Fisher approximation are proposed, without demonstrated results.

Tags

# Terminal wealth distribution from dollar cost averaging


# Terminal wealth distribution from dollar cost averaging












If monthly stock market returns follow an IID lognormal distribution, the terminal wealth distribution of investing a lump sum for many years is also lognormal. What is the terminal wealth distribution of monthly investments that grow x% a year, reflecting the experience of someone who invests a fraction of each paycheck?

## Answer by Kermittfrog (score 6, accepted)

https://quant.stackexchange.com/a/69420

I understand that you assume multiplicative gross returns, $W_t=W_{t-1}R_{t-1,t}=W_{t-2}R_{t-1,t}R_{t-2,t-1}$ and so on.

Let's assume that you are investing $I$ at the onset, and increase your investment by a factor of $1+\alpha$ each period, then your (random) final wealth will be:

$$ \begin{align} W_N&=I\left[\prod_{i=1}^NR_i+(1+\alpha)\prod_{i=2}^NR_i+\ldots+(1+\alpha)^{N-2}R_{N-1}R_{N}+(1+\alpha)^{N-1}R_N\right]\\ &=I\sum_{j=1}^ {N}(1+\alpha)^{j-1}\prod_{i=j}^NR_i \end{align} $$

As the lognormal distribution does not have a well defined characteristic function, there exists no sensible closed form / simple numerical-integral-like representation for the distribution. You can, of course, compute the first couple of moments of $W_N$ and try your luck with a Cornish-Fisher expansion around the lognormal distribution, or something the like.

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.