There is an old saying: Be more concerned about return of your investment than return on your investment. The term return on investment, or simply return, is used to refer to any of a number of metrics of the change in an asset's or portfolio's accumulated value over some period of time. Of course, accumulated value can be measured in different ways. In investment management, a distinction is drawn between total returns and net returns. The former are calculated from accumulate values reflecting only price appreciation and income from dividends or interest. The latter are calculated from accumulated values that also reflect items such as management fees, custody fees, transaction costs, taxes, and perhaps even inflation. Measure time in appropriate units—days, weeks, months,
years, etc. Let
where log denotes a natural logarithm. Suppose an asset has accumulated value of USD 100 at time t–1 and accumulated value USD 110 at time t. Then it had a
over the period t–1 to t. Gross return is increasingly falling out of use. Note that simple return is gross return minus 1. Simple returns are sometimes called net returns to distinguish them from gross returns. This can be confusing, given the alternative meaning of net return indicated earlier. As gross returns fall out of use, so is this alternative use of the term net return. For small returns, simple and log returns closely approximate one another. This is evident in out example above. Simple and log returns are sometimes called arithmetic and geometric returns, respectively. Returns are most often calculated over a one-year period, but shorter periods are increasingly being used. If a period of less than a year is used, a return may be reported on an annualized basis. The result is called an annualized return. This may also be called an annual rate of return or simply a rate of return. For example, if a portfolio experiences a .01 log return over a one month period, it might be said to have experienced a .12 log rate of return over that month. Let z be a (simple or log) return calculated over a
period of one
for simple returns, and
for log returns. A shortcoming of returns is the fact that their calculation entails division. If an instrument or portfolio's accumulated value can take on the value 0 or negative values, returns may be poorly behaved or undefined. This is especially a problem with derivatives such as swaps. An equity investor might use returns to describe the performance of his portfolio. A derivatives trader would not. Increasingly, in quantitative finance, the notion of return is being used in an entirely mathematical sense. For example, if an implied volatility rises one day from .050 to .055, it is reasonable to say that the implied volatility had a 1-day 10% simple return. In this context, return is no longer a metric of change in accumulated value, but a metric of change in any time series. Suppose the temperature at the summit of Mt. Washington rises over a day from 50° F to 55° F. Isn’t it reasonable to say that the temperature had a 1-day 10% return? Accordingly, we may treat returns as a purely mathematical notion. Simple or log returns may be used in this manner. Such usage tends to be limited to very technical discourse.
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denote a
portfolio or asset's accumulated value at time t (see the
gross return of 1.100,
of a
year (i.e. there are n such periods in a year). Then the formulas
for annualizing the return are
