# Estimating realised gains given growth rate and churn

If one can estimate that the value of an investment portfolio will grow at $$g$$% per annum, and can estimate that approximately $$c$$% of that portfolio will be churned each year (sold and reinvested), how can the expected gain that would be realised each year on that churn be calculated?

Obviously, even if every investment in the portfolio were to grow at exactly $$g$$% each year, the actual gain realised in any given year will depend on exactly which $$c$$% of the portfolio is liquidated (and, in particular, for how long it has been held/growing).

Is there a statistically sound way of approaching this problem, or are further assumptions required? Would it help to add (or perhaps even replace the churn assumption with) the further assumption that investments are on average expected to be held for $$y$$ years, perhaps with some standard deviation $$\sigma$$?

• For clarification since this appears to me a somewhat unusual problem: 1) You always buy at par and you account for any gains only when selling? 2) You select the assets (or parts of your portfolio) to be churned randomly? – g g Oct 4 '18 at 9:12
• @gg: If I buy a portfolio of (say) equities with a view to long-term investing, I am unlikely to hold the entire portfolio and then divest it all at the same time. Instead, as conditions change, I will sell some of those equities and buy others. However, whenever I sell, the government want to know what capital gain (or loss) I have made in order to levy their taxes. How can I model the effect of this taxation with very broad assumptions about growth rate and churn? – eggyal Oct 4 '18 at 9:23
• What's the context? I ask because what's a reasonable model may depend greatly on your objective, and you'll find fantastic heterogeneity in the data. An individual investor's trading strategy may vary based on their individual situation, objectives, and sophistication. For example, a financially unconstrained investor could match gains and losses and not incur any capital gains tax liability for many years. Trading strategies may be significantly different in taxable and tax-deferred retirement accounts. – Matthew Gunn Oct 5 '18 at 3:12
• The most tax-efficient strategy is to never sell anything with a gain. Pushing in the other direction are an investor's desire for consumption or to rebalance a portfolio. Perhaps of interest: Constantinides writing on optimal trading strategies with capital gains taxes, psychological barriers to tax efficient investing, life-cycle effects, ... – Matthew Gunn Oct 5 '18 at 3:26

Assume we start at $$t=0$$ with $$P_0$$, there are $$t=1...N$$ subsequent periods, and at each period-end $$t$$ an (entirely arbitrary) portion $$c$$ of our portfolio $$P_t$$ is churned and $$(1-c)$$ remains untouched. $$P$$ grows over each period by a factor $$(1+g)$$: $$P_t = P_{t-1}(1+g)$$.

We can partition $$P_t$$ into sub-portfolios, each with its own churn history, as in:

$$P_1 = P_0(1+g) \equiv P_0(1+g)(c+(1-c))$$ $$P_2=P_1(1+g)\equiv P_0(1+g)^2(c(1-c)+(1-c)^2+c^2+(1-c)c)$$

At period $$t=N$$, we will have $$2^N$$ such sub-portfolios together forming $$P_N$$, each representing a particular history of churn/no-churn over $$t=1...N$$.

$$P_N \equiv P_0(1+g)^N((1-c)^N+(1-c)^{N-1}c+(1-c)^{N-2}c^2+\cdots+c^N)$$

Now let's place ourselves at period-end $$T=N+1$$ and let's consider the third term:

1. $$c^N$$ is the portion of $$P$$ that has been churned at each preceding period. This particular sub-portfolio has been held for 1 period only at this stage (since it was last churned at $$t=N$$), with cumulative growth $$g$$.
2. Similarly $$(1-c)^N$$ is the portion that has never been churned, and has been held for N+1 periods, with cumulative growth $$(1+g)^{N+1}-1$$.
3. Finally, there are far more in-between "churn histories" than the few $$(1-c)^{N-k}c^k$$ I've written here. For example, a large number of sub-portfolios have been churned $$k$$ times, some in the first $$k$$ periods, some in the last $$k$$ periods, and many more over any $$k$$ periods. There are in fact $$\binom{N}{k}$$ such sub-portfolios, for each $$k=0...N$$ (this covers $$c^N$$ and $$(1-c)^N$$ above).

Let's then rewrite $$P_N \equiv P_0(1+g)^N(\sum_{k=0}^{N}\binom{N}{k}c^k(1-c)^{N-k})$$.

Let's look at $$\binom{N}{k}c^k(1-c)^{N-k}$$, all the sub-portfolios that have been churned $$k$$ times: clearly some have been last churned at $$t=N$$ and held 1 period since, and some have been churned before that, but the earliest such sub-portfolio could have been last churned is $$t=k$$. If we call $$N-h$$ the last churn date of a sub-portfolio churned $$k$$ times, then $$h\in[0,N-k]$$.

We can now rearrange these sub-portfolios by $$h$$: there are $$\binom{N-h-1}{k-1}$$ sub-portfolios churned $$k$$ times whose last churn date is $$t=N-h$$, and since $$h\in[0,N-k]$$, we can rewrite:

$$\binom{N}{k}\equiv \sum_{h=0}^{N-k}\binom{N-h-1}{k-1}$$

From which follows

$$P_N \equiv P_0(1+g)^N(\sum_{k=1}^{N}\sum_{h=0}^{N-k}\binom{N-h-1}{k-1}c^k(1-c)^{N-k}+(1-c)^N)$$

The last term for that portion that was never churned, and now we can rearrange the sum to show

$$P_N \equiv P_0(1+g)^N(\sum_{h=0}^{N-1}\sum_{k=1}^{N-h}\binom{N-h-1}{k-1}c^k(1-c)^{N-k}+(1-c)^N)$$

And so there is a fraction $$\sum_{k=1}^{N-h}\binom{N-h-1}{k-1}c^k(1-c)^{N-k}$$ of the portfolio which was last churned at $$t=N-h$$, and has a holding period of $$h+1$$ as seen from $$T=N+1$$ with cumulative growth $$G_h=(1+g)^{h+1}-1$$ (this is your capital gain).

There is also a fraction $$(1-c)^N$$ which has been held for $$T+1$$ periods, and never churned.

And we have $$\mathbb{E}(C_{N+1})=\sum_{h=0}^{N-1}G_h\sum_{k=1}^{N-h}\binom{N-h-1}{k-1}c^k(1-c)^{N-k}+G_N(1-c)^N$$.

$$\mathbb{E}(C_{N+1})$$ is the expected capital gain of any randomly sampled portion of our portfolio at $$T=N+1$$ and in particular of the portion $$c$$ we churn on that date.

Now for some numerical examples:

• $$N=10, g=3\%, c=0\%, \mathbb{E}(C_{N+1})= 38.42\% = (1+3\%)^{11}$$
• $$N=10, g=3\%, c=100\%, \mathbb{E}(C_{N+1})=3\%$$
• $$N=10, g=3\%, c=10\%, \mathbb{E}(C_{N+1})=23.24\%$$

Yes, there are sound ways to address this problem. And, depending on the level of realism required and your goals, you will need to think a lot more to devise an acceptable strategy.

## Bird's eye view

Let us first make the assumption that each asset indeed has exactly the same growth, each period. Even in this most simple case you can follow different strategies for profit realisation. Two possible (extreme) strategies are

1. Always sell the "young" assets first
2. Always sell the "old" assets first.

Strategy 1) will lead to early low taxation of gains, while strategy 2) will lead to high taxation early on. But note that in strategy 1) your untaxed yet accrued profits will increase much more than for strategy 2). And unless you want to run your fund for all eternity pushing forward higher and higher unrealised gains, all your accrued profits have to be realised at some point in time. This means whatever your strategy, it will mainly shape the emergence of tax over time.That said, this can make a big difference with progressive tax rates and non-zero time value of money. To decide even between those two simple strategies, both items (discount rate and tax brackets) need to be incorporated into your model.

## Details on the two strategies

It is not difficult to develop the tax payments for the two simple strategies above, when you assume that you start fresh, i.e. all assets in the portfolio have zero age. In the young-assets-first strategy, you sell exactly the same $$c$$ percent of assets each period. This means you realise each period exactly $$c g$$ while the rest $$(1 - c)$$ of the portfolio will go on accruing happily until the day of final reckoning.

Only slightly more complicated is the case of old-assets-first. Let us assume for simplicity that $$c=0.1$$. Then you will sell over the first ten periods assets of age $$1, 2, \ldots, 10$$ periods until all assets are sold which were initially in your portfolio. From then on you will only sell assets of age $$10$$ with an according taxable gain.

Between these two extreme approaches lies the random strategy, where you pick $$c$$ percent of assets at random. For each asset, this means that it is being sold in any period with probability $$c$$. Using Excel you can then calculate the probability weighted time the assets from a cohort stay in your portfolio. (Assuming 10% again, it is 1 period for the 10% assets sold in the first period, it is 2 for the 9% = 10% * (1 - 10%) assets sold in the second period and so on.) With this calculation you can verify that the expected survival time and hence the average age of an asset when randomly sold is $$\frac{1}{c}$$. This is exactly the same time as for the old-assets-first strategy, once it reaches the steady state after all initial assets are sold.

## Non-constant gains

The real fun and room for optimisation starts of course only when you leave unrealistic assumptions such as constant gains. If your gains are volatile, your opportunities are endless, since you can then net gains with losses and do all kinds of things. Of course it is necessary to specify the return (and all other) assumptions in detail. Such strategies can then be modelled and optimised by Monte-Carlo simulation.