# Radioactive Half-Life of Expenses

In the world of science, the term “half-life” is used to describe “the time required for a quantity to reduce to half its initial value”. One common application of this concept is for radioactive decay and the half-life is a useful measure in science to characterize how quickly something will decay over time. Probably the most common application of this concept is used in carbon dating which allows scientists to fairly accurately estimate the age of objects containing radiocarbon.

It occurred to me that portfolio expenses can have a similar effect as radioactive decay. Namely, the higher the expense, the faster your portfolio will reduce to half of the value it would have achieved without the expense. Let’s take a look at the math of compounding returns to see how quickly a portfolio’s value decays with expenses.

## Lump-Sum Investment

Consider the case of investing a lump-sum amount. The compounded return
of the investment with a rate of return `r`

, after `N`

years is

```
final_value = initial_value * (1 + r)^N
```

So, without any expenses eating into the return, the ratio of the final_value to the initial value represents how much your portfolio would grow without expenses

```
growth_without_expenses = final_value / initial_value
= (1 + r)^N
```

Now consider a expense or expense of `e`

being subtracted off of that return every year.

```
final_value = initial_value * [(1 + r)(1 - e)]^N
```

Now the growth is smaller,

```
growth_with_expenses = final_value / initial_value
= [(1 + r)(1 - e)]^N
```

And to compute the half-life of these expenses, we want to know when the the growth with expenses is half of the growth without those expenses.

```
growth_with_expenses / growth_without_expenses = 1/2
```

or,

```
[(1 + r)(1 - e)]^N / (1 + r)^N = 0.5
```

Solving for `N`

gives us the half-life as a function of expenses,

```
N = log(0.5) / log(1-e)
```

That’s it! We now have an equation that directly computes how many years it takes for a given expense to cut your total portfolio in half. Notice that for a lump-sum investment, it doesn’t matter how big your return is, it only matters what the expense is.

In the following plot I show the half-life as well as the 3/4-life and 7/8-life. Losing half of your portfolio is terrible, but even losing a quarter or eighth can be painful, especially when we’re talking about a portfolio that might need to grow to over $1 million to support a comfortable retirement. Do you want to pay $125,000 (1/8 of $1M) or more in expenses over the course of your investment life?

Let me walk you through some key points from the graph above.

- You want your half-life number to be very large which means you want to be as close to zero expenses as possible.
- According to Morningstar, in 2013 the “typical” managed fund of stocks has a expense of 1.2%. Such a expense would eat up half of your gains in 57 years which seems like a long time. But you would lose an eighth of your portfolio value in just 11 years.
- It gets even worse if you consider the “all-in” costs of actively managed funds. John Bogle suggests that there are many unaccounted costs in an actively managed fund that can increase the actual “all-in” expense to something more like 2.21%. The main unaccounted for costs are “transaction costs, cash drag, and sales loads”. These extra costs are hard to quantify but they effectively can reduce your expected return beyond the published expenses. In any case, if you believe Bogle’s hypothesis, then an actively managed fund with a 2.2% expense has a half-life of only 31 years and a 7/8-life of just 6 years! That means that you could be losing as much as an eighth of your portfolio every 6 years if you invest in actively-managed funds.
- Contrast this with a leading low-cost index fund from Vanguard like the Total Stock Market Index fund (VTSAX) which has an expense ratio of 0.05%. This expense would take 1385 years to lose half it’s value or 267 years to lose an eighth. To put it another way, after 40 years an expense of 0.05% would only take 2.0% off of your portfolio value which is less than some funds would cost in a single year.

By now, just about everybody knows that low-cost funds are a great way to invest and maximize returns. Putting these expenses in terms of how they eat away at your portfolio can be eye-opening.