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1 February 2010 CFA Institute Journal Review

Return–Risk Ratios under Taxation (Digest Summary)

  1. Derek W. Johnson, CFA

Most asset allocation studies are framed in a tax-free context. The authors examine the return–risk ratio as defined by the Sharpe ratio to determine how asset allocations and their probability of achieving a desired rate of return differ under various taxable assumptions. The authors acknowledge that taxes detract from investment returns but find that taxable investors can accept a higher return−risk ratio than their tax-free counterparts because current U.S. tax structures for long-term capital gains differ from those for income received on cash. The authors also find that the ability to use losses to offset gains alters the return−risk ratio in favor of the taxable investor.

The authors use the Sharpe ratio (SR) to examine allocations between cash and equities and the return characteristics on these allocations under taxable and tax-free scenarios. They start with a simple two-asset model allocation composed of cash and a single equity asset. The equity is assumed to be subject to an advantageous capital gains tax rate. The authors find that most U.S. portfolios, from both individual (taxed) portfolios to highly diversified institutional (tax-free) funds, have nearly the same level of equity risk at a 60 percent allocation to equities. The authors interpret this result as representing both the maximum amount of volatility that investors are prepared to accept as well as the minimum risk required to achieve an acceptable return over cash. They assume that the base expected return for the equity investment will be 7 percent pretax, which represents a 4 percent rate for cash plus a 3 percent equity risk premium. The pretax assumption for the volatility of an all-equity portfolio is set at 16.5 percent. The authors also assume that interest on cash will be taxed at 40 percent and that all equity returns will be taxed at the long-term capital gains tax rate of 20 percent. Any losses to a taxable investor will be used to offset immediate or future gains.

The return−risk ratio measures the return that investors receive for moving toward different risk allocations; the more risk an investor is willing to assume, the more return the investor should expect. To translate a tax-free return−risk ratio into its taxable counterpart, both the return premium in the numerator and the volatility in the denominator must be restated in after-tax terms. Once these calculations are made, the authors find the slope of the return−risk ratio for the taxable investor to be greater than that for the tax-free investor. So, for a given increase in risk, the taxable investor is compensated more than the tax-free investor is. The main factors are the lower taxation of long-term capital gains versus that of cash and the benefit of tax loss offsets.

The authors also consider the probability of achieving a return that is higher than cash with different equity/cash allocations. A tax-free investor can always achieve a risk-free rate of 4 percent with 100 percent cash. Once equity is added to a portfolio, however, the probability of beating the 4 percent rate drops to less than 100 percent. They find that with an SR of 0.18, the probability of beating the 4 percent rate drops to 57 percent regardless of whether the equity exposure is 20 percent or 60 percent. The authors then show the probability of achieving a 1 percent and 2 percent net spread over the cash rate. To achieve a spread of 1 percent over the risk-free rate with an SR of 0.18, an investor would need 20 percent in equities for a 45 percent chance and 60 percent in equities for a 53 percent chance. These probabilities are applied to one-year returns. When the authors look at longer-term horizons, such as five years, the odds of achieving a positive spread over cash improve. The odds of achieving a 1 percent spread over cash, if one assumes a 7 percent risk premium, grows from 63 percent in one year to 77 percent in five years. Additionally, five-year after-tax portfolios have a 79 percent probability of achieving the spread because of the aforementioned tax loss offsets and lower taxation on capital gains compared with taxation on cash.

The authors conclude that although taxes lower prospective returns, the return−risk ratio and the net equity premium may actually increase. They find that, as a result, the incentive for risk taking is somewhat greater for taxed investors than for their tax-free counterparts.

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