The Mathematics of Asset Location
In a follow-up to his overview of the asset location decision, Kevin Grogan takes a more comprehensive look at how a security’s expected return and risk characteristics can change based on the tax treatment of the savings vehicle in which it’s held.
Our recommendation on asset location is to prefer holding tax-inefficient assets in tax-advantaged accounts. In our view, the expected return of an asset is close to irrelevant when determining where to locate it. This guidance applies to Roth as well as to traditional IRAs. For many, this can seem counterintuitive, given that much of what investors find in the financial media generally discusses locating the highest-expected-return asset classes in the Roth account.
This article is broken into two sections. First, we will discuss how a security’s expected return and risk characteristics can change based on where the asset is located. Second, we will walk through a Roth-versus-taxable-account asset location decision using after-tax asset allocation.
Section One: Asset Location’s Effect on Expected Return and Risk
The following table indicates the principal effectively owned by, return received by, and risk borne by individual investors in each savings vehicle. As shown, in contrast to bonds and stocks held in a Roth IRA, for bonds and stocks held in a tax-deferred account, the investor effectively owns each dollar of principal multiplied by one minus the tax rate on it but receives 100 percent of the returns and bears 100 percent of the risk.
| Savings Vehicle | Principal | Return | Risk |
| Roth IRA | 100% | 100% | 100% |
| Tax-deferred account | 1 – tax rate | 100% | 100% |
| Taxable account | |||
| Bonds | 100% | 1 – tax rate | 1 – tax rate |
| Stocks | 100% | 1 – tax drag | 1 – tax drag |
To illustrate the risk and return sharing of bonds held in taxable accounts, we assume bonds have a 3 percent expected return and a 4 percent standard deviation (a measure of risk), as well as that the investor is in the 25 percent tax bracket. We also assume bonds earn returns of -1 percent, 3 percent and 7 percent in three years; that is, they earn the mean return and one standard deviation below and above it. The standard deviation of these returns is 4 percent. Assuming the 1 percent loss is used to offset that year’s taxable income, the investor’s after-tax returns are -0.75 percent, 2.25 percent and 5.25 percent for a standard deviation of 3 percent. In this case, the investor receives each dollar of pretax return minus the tax rate on it and bears each unit of pretax risk less the tax rate.
When stocks are held in taxable accounts (with cost bases equal to market values), the investor owns 100 percent of the principal, but their after-tax return and risk is reduced by a tax drag, specifically the taxes due on dividends and realized capital gains. The tax drag number depends upon how actively the stocks are managed, but for purposes of this example, we will assume that 15 percent of the return on stocks is lost to taxes each year. (This effectively assumes that all capital gains are realized 366 days after the stocks are purchased.)
To once more illustrate risk and return sharing properties of taxable accounts, we assume stocks have a 7 percent expected return and a 19 percent standard deviation. Stocks, we further assume, earn pretax returns of -12 percent, 7 percent and 26 percent in three years; that is, they earn the mean return and one standard deviation below and above it. The standard deviation of these returns is 19 percent. If the loss is used that year to offset long-term capital gains, the after-tax returns are -10.2 percent, 5.95 percent and 22.1 percent for a standard deviation of 16.15 percent (or 85 percent of the pretax standard deviation).
The key message from these examples is that the same underlying asset can have different expected return and risk characteristics depending upon the type of savings vehicle in which it is located.
Section Two: Asset Location and Roth IRAs
Using the framework from the first section, we can evaluate asset location decisions when it comes to Roth IRAs versus taxable accounts. This section assumes the investor will use all assets in his or her lifetime and is not designating Roth accounts for future heirs. If the investor is designating the Roth for heirs, then this account is effectively being managed for a purpose other than retirement and should have its own Investment Policy Statement.
The following are our capital market and tax assumptions:
| Expected return on stocks | 7.0% |
| Expected return on bonds | 3.0% |
| Standard deviation on stocks | 19.0% |
| Standard deviation on bonds | 4.0% |
| Tax rate on stocks | 15.0% |
| Tax rate on bonds | 35.0% |
Assume that our hypothetical investor has $200 split evenly between a taxable account and a Roth. Let us also express the investor’s risk tolerance in terms of acceptable volatility, and, in this case, the client is comfortable with a portfolio after-tax standard deviation about 9.7 percent.
One way to accomplish this would be to locate $100 of bonds in the taxable account and $100 of stocks in the Roth account.* The following table has the results:
| Scenario 1: Incorrect Location | ||||||
| After-Tax Expected Return | After-Tax Expected SD | Start $ | End $ Pre Tax | End $ After Tax | ||
| Taxable account | Stock | 6.0% | 16.2% | $0.00 | $0.00 | $0.00 |
| Bond | 2.0% | 2.6% | $100.00 | $103.00 | $101.95 | |
| Roth IRA | Stock | 7.0% | 19.0% | $100.00 | $107.00 | $107.00 |
| Bond | 3.0% | 4.0% | $0.00 | $0.00 | $0.00 | |
| Total | 4.5% | 9.6% | $200.00 | $210.00 | $208.95 |
The portfolio has an after-tax expected return of 4.5 percent and an after-tax standard deviation of 9.6 percent. At the end of the one-year period, the investor wound up with $208.95.
An alternative way to get to the 9.7 percent after-tax standard deviation would be to locate the assets in the optimal manner, with $100 of stocks in the taxable account and a 15/85 mix in the Roth account.*
| Scenario 2: Correct Location | ||||||
| After-Tax Expected Return | After-Tax Expected SD | Start $ | End $ Pre Tax | End $ After Tax | ||
| Taxable account | Stock | 6.0% | 16.2% | $100.00 | $107.00 | $105.95 |
| Bond | 2.0% | 2.6% | $0.00 | $0.00 | $0.00 | |
| Roth IRA | Stock | 7.0% | 19.0% | $15.00 | $16.05 | $16.05 |
| Bond | 3.0% | 4.0% | $85.00 | $87.55 | $87.55 | |
| Total | 4.8% | 9.7% | $200.00 | $210.60 | $209.55 |
The after-tax standard deviation of Scenario 2 is roughly the same as Scenario 1, even though the portfolio contains a higher allocation to stocks. The reason for this is that the government shares in the volatility of the stocks held in the taxable account. The after-tax expected return is higher in Scenario 2 than in Scenario 1, and the investor winds up with $209.55 at the end of a year.
The key insight from Robert Dammon, Chester Spatt and Harold Zhang’s 2004 Journal of Finance paper, “Optimal Asset Location and Allocation with Taxable and Tax-Deferred Investing,” is that Scenario 2 will have a higher after-tax expected return than Scenario 1, regardless of what the stock and bond returns are, as long as the following two conditions are met:
1. The tax rate on bond interest is greater than the tax rate on stock gains and dividends.
2. The interest rate on bonds is positive.
This is true even if we assume all stocks are taxed every year. In reality, some of the stock returns are tax-deferred even if the stock holding is in a taxable account. The size of the improvement in after-tax returns is exactly equal to the interest rate on bonds multiplied by the difference in tax rates.
Let’s look at one more scenario where the investor’s risk tolerance is expressed in terms of actual dollars allocated to stocks and bonds. In this scenario, we will say our investor wants a 50/50 asset allocation.
Scenario 1 still applies, but Scenario 2 has too much invested in equities. Scenario 3, then, locates $100 of stocks in the taxable account and $100 of bonds in the Roth.*
| Scenario 3: Correct Location | ||||||
| After-Tax Expected Return | After-Tax Expected SD | Start $ | End $ Pre Tax | End $ After Tax | ||
| Taxable account | Stock | 6.0% | 16.2% | $100.00 | $107.00 | $105.95 |
| Bond | 2.0% | 2.6% | $0.00 | $0.00 | $0.00 | |
| Roth IRA | Stock | 7.0% | 19.0% | $0.00 | $0.00 | $0.00 |
| Bond | 3.0% | 4.0% | $100.00 | $103.00 | $103.00 | |
| Total | 4.5% | 8.3% | $200.00 | $210.00 | $208.95 |
Scenario 3 provides exactly the same after-tax expected return as Scenario 1, but with less volatility. The reason for this is that the government is partaking in the volatility of the portfolio’s stocks.
It is important to recognize that the government shares in the risk and return of assets located in taxable retirement savings vehicles. Therefore, a bond or a bond fund held inside a Roth is effectively a different asset than the same bond or bond fund held in a taxable account. Using the assumptions from our last example, the return and risk for a bond held in a Roth IRA are 3 percent and 4 percent, but only 2 percent and 2.6 percent for the same bond held in a taxable account.
By employing this framework, we learn that the best assets to hold in the taxable account are those that make the best use of the preferential long-term capital gains treatment. This will typically be stocks, as long as the investor is willing to avoid short-term capital gains.
* This scenario is hypothetical and being presented for illustrative purposes only. It relies on the capital market assumptions cited in this article.
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