While studying for the CFA (Chartered Financial Analyst) level 3 exam, there is a concept I came across which is new to me. In a taxable investment, volatility lowers risk.

Here’s the formula:

σ_{AT} = σ ( 1 – t )

Basically, the formula says that the after-tax standard deviation is the standard deviation multiplied by 1 minus the tax rate. In other words, the taxing authority shares in the upside as well as the downside of your investment.

But do taxes actually reduce risk?

My initial thought is no, they reduce returns and using them to reduce risk is similar to double counting the effect of taxes on investments. I need to conduct an experiment to see why I am probably wrong. The CFA Institute has been at this a bit longer than me, so it is probably a safe assumption that they know better than I do.

I am going to walk through a hypothetical situation to flesh this idea out better in my head (I am attaching a downloadable spreadsheet **here** to go along with my example). Suppose we have an investor (I will call him Tim) and he purchases several stocks. Tim is subject to a 25% capital gains tax. Tim invests in the following:

- XYZ stock for $10,000.
- ABC stock for $10,000.
- AAA stock for $10,000.
- AZQ stock for $10,000.

A year later Tim has sold all of his stocks listed above. Tim’s investment performance is the following:

- XYZ stock sold for $8,000.
- ABC stock sold for $11,000.
- AAA stock sold for $6,000.
- AZQ stock sold for $13,000.

In total Tim initially invested $40,000 and now has $38,000. Tim’s pretax standard deviation is 27% and his return is -5% . Assuming no other taxable sources of income, Tim’s tax is zero dollars. No capital gains income means no tax. Tim only realizes the loss.

However, what if Tim had other income such as a salary from employment, and paid income taxes greater than the capital gains losses illustrated above?

Now, his after-tax gains and losses are the following:

- -$1,500 for XYZ stock, up from -$2,000.
- $750 for ABC stock, down from $1,000.
- -$3,000 for AAA stock, up from -$4,000.
- $2,250 for AZQ stock, down from $3,000.

All the gains and losses are reduced. Volatility (risk) of returns is indeed reduced. The after-tax standard deviation has fallen from 27% to 20%.

Interesting.

I was thinking about risk (standard deviation) in the wrong context. The risk is the volatility of returns, so it is not double counting. The risk is based on the returns and taxes reduce the gains or losses from returns. It is only natural then that risk is impacted directly. Taxes do indeed reduce risk. The key insight for me is that there have to be taxes payable to realize the benefits. Gains and losses do not happen in a vacuum, and portfolios should be constructed and analyzed on an after-tax basis.

Lesson learned.

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