Limited Time Offer: Save 10% on all 2021 and 2022 Premium Study Packages with promo code: BLOG10    Select your Premium Package »

Measures of Return

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 and 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 a return earned from holding an asset for a specified period of time. The time period may be as short as a day or many years and 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 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 and amounts withdrawn from the portfolio by the investor 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 what was actually earned on the money invested. 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 is made up of several assets, we may want to find the aggregate return of the portfolio as a whole. In order 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 for comparing returns over different periods given that inflation rates vary over time.

Leverage Returns

If an investor makes use of 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 which 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.

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

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

Featured Study with Us
CFA® Exam and FRM® Exam Prep Platform offered by AnalystPrep

Study Platform

Learn with Us

    Subscribe to our newsletter and keep up with the latest and greatest tips for success
    Online Tutoring
    Our videos feature professional educators presenting in-depth explanations of all topics introduced in the curriculum.

    Video Lessons



    Sergio Torrico
    Sergio Torrico
    2021-07-23
    Excelente para el FRM 2 Escribo esta revisión en español para los hispanohablantes, soy de Bolivia, y utilicé AnalystPrep para dudas y consultas sobre mi preparación para el FRM nivel 2 (lo tomé una sola vez y aprobé muy bien), siempre tuve un soporte claro, directo y rápido, el material sale rápido cuando hay cambios en el temario de GARP, y los ejercicios y exámenes son muy útiles para practicar.
    diana
    diana
    2021-07-17
    So helpful. I have been using the videos to prepare for the CFA Level II exam. The videos signpost the reading contents, explain the concepts and provide additional context for specific concepts. The fun light-hearted analogies are also a welcome break to some very dry content. I usually watch the videos before going into more in-depth reading and they are a good way to avoid being overwhelmed by the sheer volume of content when you look at the readings.
    Kriti Dhawan
    Kriti Dhawan
    2021-07-16
    A great curriculum provider. James sir explains the concept so well that rather than memorising it, you tend to intuitively understand and absorb them. Thank you ! Grateful I saw this at the right time for my CFA prep.
    nikhil kumar
    nikhil kumar
    2021-06-28
    Very well explained and gives a great insight about topics in a very short time. Glad to have found Professor Forjan's lectures.
    Marwan
    Marwan
    2021-06-22
    Great support throughout the course by the team, did not feel neglected
    Benjamin anonymous
    Benjamin anonymous
    2021-05-10
    I loved using AnalystPrep for FRM. QBank is huge, videos are great. Would recommend to a friend
    Daniel Glyn
    Daniel Glyn
    2021-03-24
    I have finished my FRM1 thanks to AnalystPrep. And now using AnalystPrep for my FRM2 preparation. Professor Forjan is brilliant. He gives such good explanations and analogies. And more than anything makes learning fun. A big thank you to Analystprep and Professor Forjan. 5 stars all the way!
    michael walshe
    michael walshe
    2021-03-18
    Professor James' videos are excellent for understanding the underlying theories behind financial engineering / financial analysis. The AnalystPrep videos were better than any of the others that I searched through on YouTube for providing a clear explanation of some concepts, such as Portfolio theory, CAPM, and Arbitrage Pricing theory. Watching these cleared up many of the unclarities I had in my head. Highly recommended.