Measures of Return

Financial market assets generate two different streams of return: income through cash dividends or interest payments and capital growth through asset price appreciation. Headline stock market indices typically report on price appreciation only. They do not include the dividend income unless the index specifies it is a “total return” series. The ability to compute and compare different measures of return is critical in the proper evaluation of portfolio performance.

Holding Period Return

A holding period return is earned from holding an asset for a specified period. The time period may be as short as a day. Alternatively, it can run for many years. It is expressed as the total return. This means we look at the return as a composite of the price appreciation and the income stream.

The formula for the holding period return computation is as follows:

$$ \text{Holding Period Return (HPR)} = \frac {P_t – P_{t-1} + D_t} {P_{t-1}} $$

Where:

\(P_t\) is the price of the asset at time t when the asset is sold.

\(P_{t-1}\) is the price of the asset at time t-1 when the asset was bought.

\(D_t\) is the dividend per share paid between t and t-1.

Arithmetic or Mean Return

When we have assets for multiple holding periods, it is necessary to aggregate the returns into one overall return. An arithmetic mean is a simple process of finding the average of the holding period returns. For example, if a share has returned 15%, 10%, 12%, and 3% over the last four years, then the arithmetic mean is computed as follows:

$$ \text{Arithmetic mean} = \frac {15\% + 10\% + 12\% + 3\%} {4} = 10\% $$

Geometric Mean Return

Computing a geometric mean follows a principle similar to the one used in the computation of compound interest. Returns of the previous year are compounded to the initial value of the investment at the start of the new period in order to earn returns on your returns. A geometric return provides a more accurate representation of the portfolio value growth than an arithmetic return. Using the same annual returns of 15%, 10%, 12%, and 3% as shown above, we compute the geometric mean as follows:

$$ \text{Geometric mean} = [(1+15\%) × (1+10\%) × (1+12\%) × (1+3\%)]^{1/4} – 1 = 9.9\% $$

Note that the geometric return is slightly less than the arithmetic return. Arithmetic returns tend to be biased upwards unless the holding period returns are all equal.

Money-weighted or Internal Rate of Return

Arithmetic and geometric returns do not take the money invested in a portfolio at different periods into account. The money-weighted return computation methodology is similar to the one used in the calculation of an internal rate of return (IRR) or a yield-to-maturity. We examine the cash flows from the perspective of the investor. In this case, amounts invested in the portfolio are seen as cash outflows. On the other hand, the amounts the investor withdraws from the portfolio are cash inflows.

The IRR is the discount rate applied to determine the present value of the cash flows such that the cumulative present value of all the cash flows is zero. The IRR provides the investor with an accurate measure of the earnings the money invested attracted. Nonetheless, it does not allow for easy comparison between individuals.

Annualized Return

If the period during which the return is earned is not exactly one year, we can annualize the return to enable an easy comparative return. To annualize a return earned for a period shorter than one year, the return must be compounded by the number of periods in the year. A monthly return must be compounded 12 times, a weekly return 52 times, and a daily return 365 times. A weekly return of 2%, when annualized, is as follows:

$$ \text{Annualized return} = (1+2\%)^{52} – 1 = 180\% $$

When the holding period is longer than one year, we need to express the year as a fraction of the holding period and compound using this fractional number. For example, a year relative to a 20-month holding period is a fraction of 12/20. If we had a return of 12% for 20 months, then the annualized return is as follows:

$$ \text{Annualized return} = (1+12\%)^{12/20} – 1 = 7\% $$

Portfolio Return

When a portfolio comprises several assets, we may want to find the aggregate return of the portfolio as a whole. To compute this, we weight the returns of the underlying assets by the amounts allocated to them. A portfolio that consists of 70% equities which return 10%, 20% bonds which return 4%, and 10% cash which returns 1%, would have a portfolio return as follows:

$$ \text{Portfolio return} = (70\% × 10\%) + (20\% × 4\%) + (10\% × 1\%) = 7.9\% $$

Other Major Return Measures

The following are the other measures of returns that need to be taken into account when evaluating performance:

Gross and Net Return

A gross return is earned prior to the deduction of fees (management fees, custodial fees, and other administrative expenses). A net return is the return post-deduction of fees.

Pre-tax and After-tax Nominal Returns

In general, returns are presented pre-tax and with no adjustment for the effects of inflation. Tax considerations such as capital gains tax and tax on interest or dividend income will need to be deducted from the investment to determine post-tax returns.

Real Returns

Returns are typically presented in nominal terms, which consist of three components: the real risk-free return as compensation for postponing consumption, inflation as compensation for the loss of purchasing power, and a risk premium. Real returns are useful in comparing returns over different periods, given that inflation rates vary over time.

Leverage Returns

If an investor uses derivative instruments within a portfolio or borrows money to invest, then leverage is introduced into the portfolio. The leverage amplifies the returns on the investor’s capital, both upwards and downwards.

Question

What are the arithmetic mean and geometric mean, respectively, of an investment that returns 8%, -2%, and 6% each year for three years?

A. Arithmetic mean = 5.3%; Geometric mean = 5.2%.

B. Arithmetic mean = 4.0%; Geometric mean = 3.6%.

C. Arithmentic mean = 4.0%; Geometric mean = 3.9%.

Solution

The correct answer is C.

$$ \text{Arithmetic mean} = \frac {8\% + (-2\%) + 6\%} {3} = 4\% $$

$$ \text{Geometric mean} = [(1+8\%) × (1+(-2\%)) × (1+6\%)]^{1/3} – 1 = 3.9\% $$