# Measuring Risk from US Tax Bracket Changes

Let’s take a trip down the Amazon River. Some of the dangers we might face are malaria, wild predators, and who knows, maybe even a batch of hostile natives. But what if we knew without a doubt that we would arrive back to civilization unharmed? Our special knowledge would remove the danger from all those threats and hence their risk. Risk, then, is a measure of the likelihood that something adverse will affect us and “likelihood” implies a degree of uncertainty.

Just as we might take on risk in the Amazon from uncertainties outside our control, we take on financial risk with uncertainties that jeopardize our money. If we *choose *to go down the river (versus being forced to go down it) we take on the added risk of making a wrong choice and incurring an opportunity cost.

Variance in tax rates, just like variance in our investments’ returns, invites uncertainty and both have a critical effect on how much money we wind up with. Tax risk is often understood, but not quantified. I’d like to suggest a way to measure tax risk so we, as investors, can make better decisions on the types of accounts we contribute to.

This is a back of the envelope measure and does not account for realized or unrealized gains.

**RISK AND STANDARD DEVIATION (A PRIMER)**

Let’s say you have a boss who has bipolar disorder. We’ll call him “Boss A.” When he’s walking on sunshine, it’s a great day! He showers his employees with hilarious jokes, and spontaneous free lunches! When he’s depressed, it’s pretty bad. Withdrawn and cynical, he spills out his personal woes, gets moody and demands you work late. On these bad days, you find yourself hiding in the shadows of large office furniture to avoid him. Now there’s also “Boss B.” Aside from a few mild exceptions, he’s not too spontaneous or funny, but he’s not a downer either. Every day is pretty much like the others, no big surprises.

What if you could measure the emotion in bosses A and B and pretty well predict what each would do according to how each felt that day? For example, you measure Boss A by assigning 1-9 to his behavior: a “5” when he’s normal, “1” on his worst days, and a “9” when he’s happiest.

Now you’ve learned from experience that on Boss A’s best days there’s a real good chance he’ll take everyone to lunch and give them the afternoon off. On his most depressed days there’s a real good chance everyone needs to work late. After seeing this pattern over and over, you can predict by late morning whether you’re getting a free lunch, will have to work late, or neither.

You apply a similar scale to Boss B. On Boss B’s best and worst days, you never expect anything as extreme as a free lunch or working late. His best and worst days are a smidgen north and south of an average day, so the spread of numbers you assign are not as wide. You give him a 3 for a worst day, a 5 for an average day, and a 7 for his best.

There’s an advantage boring Boss B has over A. Boss B’s behavior from one day to the next doesn’t vary much, so he’s pretty predictable. What that means is you can plan ahead with some confidence– you can afford to run five minutes late, because you don’t need to crouch in the shadows once you arrive. You’re also pretty sure you’ll make your son’s evening band recital because you never end up working late.

Financial investments work like Bosses A and B or somewhere between. Stocks are a little like Boss A. They can have really high swings (and greater unpredictability) than other investments, so on their best days, you feel the great benefit of those swings, like a free lunch. On their worst days, you really feel the cost, like having to work late. The difference of course, is that investments either make or lose you money. Bonds vary as well, but in general, they’re more like Boss B; most of the time we can expect modest swings and more predictability.

Remember that measuring system we used for our two bosses, statistics uses something like it called standard deviation. A bigger standard deviation indicates wider swings and that means greater unpredictability, signaling greater possible risk. Stocks, for example, generally have a higher standard deviation than bonds.

In our quest to measure the risk in changing tax rates, we want to keep in mind standard deviation, because we are going to apply a well-known measure to the risk that stocks, bonds, or funds take on, but apply it to tax changes. This measure is called the Sharpe Ratio and standard deviation is the denominator of that ratio.

**THE SHARPE RATIO**

In financial markets, funds that take on risk attempt to compensate for it by paying a higher return. The Sharpe ratio is one measure of whether the return you receive justifies the risk you take on. The ratio divides historical risk into historical return (Avg. return – risk free rate / standard deviation). The Sharpe is a relative ratio so you want to be sure you are measuring apples with apples, for example two funds composed of large, blue chip stocks like Coke and Pepsi between 2001 and 2010.

The Sharpe has its limitations, but the higher the ratio the better; because it suggests that the risk you’re taking is low compared to the return you’re expecting. For example, consider that over ten years fund A has annual average returns of 10% and a standard deviation of 8% and fund B has the same annual average returns of 10% but a lower standard deviation of 5%. Fund A .10/.08 = 1.25; Fund B .10/.05 = 2. Accordingly, fund B has a greater return to risk relationship, at least as the Sharpe ratio measures it. Just like boss B who is pretty average even on his high and low days, when standard deviation doesn’t swing too high or too low from its mean (or average) we have a smaller divisor, like .05 instead of .08 and this suggests lower risk.

The Sharpe ratio also subtracts a risk-free rate from a fund’s total return. The risk-free rate, the return often tied to 3 month t-bills, is what a fund’s return should beat if it is taking on risk. Since t-bills are “risk free,” our riskier fund should at least beat them. Don’t you agree? For simplicity and since this is a comparative exercise, I’ve left this risk free rate out of our example.

**APPLYING THE SHARPE RATIO TO TAXATION RISK**

Tax rate changes invite uncertainty because of unforeseen volatility just like the changes in returns of stocks and bonds invite uncertainty. Like choosing among different stocks with risk/return relationships, we have choices among how the accounts holding our investments are taxed. Generally these accounts either are taxable, tax deferred, and tax-free.

The variance within tax brackets, capital gains taxes and dividend rates over the life of an investor’s portfolio sheds light on account-related risks as affected by taxation. Just like the risk of a stock, bond, or fund is measured by the distance of high and low swings in its returns from its average return, the higher the swing away from the average tax rate, the more risk you take on. Like an investment’s returns, tax rate hikes vary and aren’t entirely predictable.

If we modified our risk and returns to show taxes on our portfolio, we could use the Sharpe ratio to see the tax-affects across our tax-free, tax-deferred, and taxable accounts, providing a new, quantifiable perspective on the risk of changes in taxation. We will apply the Sharpe ratio to hypothetical portfolios in which tax brackets change with each portfolio to represent variance in taxation over time. We will then compare the before-tax ratio to the after-tax ratio to quantify taxation’s risk in our accounts. First, we must find the numerator, or the return to our tax-adjusted Sharpe ratio.

To apply tax-adjusted risk and return to our accounts, we can begin by thinking of taxable, tax-deferred, and tax-free accounts as typical funds with varying returns and risks measurable by the Sharpe. For example, say a mutual fund has a return of 10%. We can treat our tax-free and tax-deferred accounts just like this mutual fund. If my tax-free account had a return of 10% like fund A, then I will use 10% – the risk free rate as the numerator in my Sharpe Ratio. I used Excel’s RAND function to generate a number of hypothetical portfolios. However, if we applied the Sharpe to a real portfolio we could determine the excess return from our portfolio, similar to any fund inside that portfolio (R – R0, where R is the average return of the portfolio and R0 is its risk-free return from money market and treasuries). For example, say our average annual return on our account’s portfolio from 1972 to 2011 was 12% and 3% of the return came “risk-free” from money market and treasuries in that portfolio; we would arrive at 12% – 3% = 9% excess return.

Adjusting the average retirement income for a college-educated retiree from 1972 to 2011 yields the tax brackets below. In 2011, such a retiree roughly has a retirement income of $45,000. The table discounts income each year by 3.1% to account for inflation to arrive at that year’s average retirement tax bracket. The mean (average) tax rate from 1972-2011 turns out to be 18% using this method.

**ESTIMATED RETIREMENT TAX BRACKETS (40 YEARS)**

**Year and Bracket**

2011 (0.15) 2010 (0.15) 2009 (0.15) 2008 (0.15) 2007 (0.15) 2006 (0.15) 2005 (0.15) 2004 (0.15)

2003 (0.15) 2002 (0.15) 2001 (0.15) 2000 (0.15) 1999 (0.15) 1998 (0.15) 1997 (0.15) 1996 (0.15)

1995 (0.15) 1994 (0.15) 1993 (0.15) 1992 (0.15 )1991 (0.15) 1990 (0.28) 1989 (0.15) 1988 (0.15)

1987 (0.15) 1986 (0.18) 1985 (0.18) 1984 (0.18) 1983 (0.19) 1982 (0.22) 1981 (0.24) 1980 (0.14)

1979 (0.24) 1978 (0.25) 1977 (0.25) 1976 (0.25) 1975 (0.25) 1974 (0.25) 1973 (0.25) 1972 (0.25)

Now that we have the average tax-bracket of 18%, we can begin to find the tax-adjusted Sharpe ratio. As mentioned, standard deviation is a distance from the average. If we own a tax-deferred account, like an IRA or 401k and our mean tax rate is 18%, then we’ll keep 1.00 – 0.18, or 82% of our portfolio’s returns after taxes. To find the tax-adjusted standard deviation in our account’s portfolio, we will adjust our returns to reflect the effects of taxation compared to the mean. We will adjust upward our return when the tax rate is below our mean of 18%, since a lower than average tax rate allows us to keep more of our money as though our returns were higher; add zero additional return when the rate is 18%, since it is the average, and adjust our return downward when the tax rate is greater than 18%, since a higher than average tax rate will take more money from our portfolio as though our returns were lower. Below are three examples to better explain this.

1. WE ARE TAXED AT 18%

• Annual return on my account’s portfolio = 9%.

• Tax rates have a mean of 18%, which is also the rate we are taxed at.

• We keep (after taxes) 1 – 0.18 or 82% of our portfolio’s returns.

• Our tax-adjusted return is 82% of 9%, or 7.3%

• The change from the mean of 82% in terms of extra return from taxes saved is 0%, since the mean is 82%

2. WE ARE TAXED AT 15%

• Annual return on our portfolio = 9%.

• Tax rates have a mean of 18%.

• We keep (after taxes) 1 – 0.15 or 85% of our portfolio’s returns.

• Our tax adjusted return is 85% of 9%, or 7.6%

• Our change from the mean of 82% in terms of extra return from taxes saved is 3.7% or (.85/0.82)-1.

3. WE ARE TAXED AT 22%

• Annual return on our portfolio = 9%.

• Tax rates have a mean of 18%.

• We keep (after taxes) 1 – 0.22 or 78% of our returns.

• Our tax-adjusted return is 78% of 9%, or 7.2%

• The change from the mean of 82% in terms of extra return from taxes saved is

– 4.9 % or (0.78/0.82)-1.

This can be expressed as:

*R _{p}−R_{pf} +(1-T_{p })/(1-T_{μ })-1*

σ

Using the RANDBETWEEN function in Excel and the tax table that I created above, you can generate your own hypothetical portfolios between 1972 and 2011 (or fewer years) to see the tax-adjusted effects on your returns for each year. You can then use the tax-adjusted returns to arrive at an average return for returns. Finally you can use Excel’s STDEV function to get the standard deviation for those tax-adjusted, annual returns and arrive at a tax-reflective Sharpe ratio. Compare this ratio to ratios derived from other accounts with different tax effects and you get an inkling of how much extra risk you take on due to taxes.

**CONCLUSION****
**When considering taxation for retirement we often consider two options; a tax-free or tax-deferred account. The Sharpe ratio on a tax-free account like a Roth IRA, is the same as a before tax Sharpe ratio, because the uncertainty, or standard deviation due to changing tax brackets is zero. I ran 300 simulations comparing the before tax Sharpe to the tax-adjusted Sharpe. The tax-adjusted Sharpe was on average 14 percentage points lower and as much as 34 percentage points lower than the before tax Sharpe, due to the risk of volatility in retirement tax rates.

Does that mean you take on less risk when you invest in a Roth versus a traditional IRA? Yes, as far as the Sharpe sees it. But there are other factors to consider. For example, as your income increases, so may your tax bracket and with it the amount your next Roth contribution is taxed. Thus, even though the money you’ve already invested in a tax-free account suffers zero future volatility from taxes, the money you plan to invest still may. The great thing is you have foreseeable control over where you put your money when you receive that next promotion according to how it affects your tax bracket. Why, because you can own both types of accounts, tax-free and tax-deferred. You don’t have to settle for merely erratic Boss A or Boring boss B. You can have a little of each. Multiple retirement accounts can help to hedge against the risk of wrong choice, or opportunity cost, giving you greater control on how you manage the risk of taxation so you can grow your accounts more efficiently.

*This information is not intended to be a substitute for specific individualized tax advice. We suggest that you discuss your specific tax issues with a qualified tax advisor.*