mathematics

Holding period return

Holding period return is a mathematics topic covered in the lgStudy science library. This page brings together a partial reference excerpt, illustrations, worked examples, real-world applications and a short study plan, so you can understand Holding period return rather than just read about it. In short: In finance, holding period return (HPR) is the return on an asset or portfolio over the whole period during which it was held. It is one of the simplest and most important measures of investment performance.

Key takeaways

Holding period return belongs to mathematics; place it in that map before memorising details.
Learn the definition first, then one example that makes the definition concrete.
Connect Holding period return to a quantity you can measure, compute or draw — that is where exam questions come from.
Reproduce the core statement of Holding period return from memory before moving on to harder problems.

Reference excerpt

In finance, holding period return (HPR) is the return on an asset or portfolio over the whole period during which it was held. It is one of the simplest and most important measures of investment performance. HPR is the change in value of an investment, asset or portfolio over a particular period. It is the entire gain or loss, which is the sum income and capital gains, divided by the value at the beginning of the period.

HPR = (End Value - Initial Value) / Initial Value where the End Value includes income, such as dividends, earned on the investment:

H P R n = I n c o m e + P n + 1 − P n P n {\displaystyle HPR_{n}\ =\ {\frac {Income+P_{n+1}-P_{n}}{P_{n}}}}

where P n {\displaystyle P_{n}} is the value at the start of the holding period and I n c o m e + P n + 1 {\displaystyle Income+P_{n+1}} is the total value at the end of the holding period.

Annualizing the holding period return

Over multiple years To annualize a holding period return means to find the equivalent rate of return per year. Assuming income and capital gains and losses are reinvested, i.e. retained in the portfolio, then:

Annualized rate of return = ( end value initial value ) 1 t − 1 {\displaystyle {\text{Annualized rate of return}}=\left({\frac {\text{end value}}{\text{initial value}}}\right)^{\frac {1}{t}}-1}

= ( holding-period return + 1 ) 1 t − 1 {\displaystyle =\left({\text{holding-period return}}+1\right)^{\frac {1}{t}}-1}

t being the length of the holding period, measured in years. For example, if you have held the item for half a year, t would equal 1/2, so 1/t would equal 2. (However, investment performance professionals generally advise against quoting annualized return over a holding period of less than a year).

From quarterly holding period returns To calculate an annual HPR from four quarterly HPRs, it is necessary to know whether income is reinvested within each quarter or not. If HPR1 through HPR4 are the holding period returns for four consecutive periods, assuming that income is reinvested, the annual HPR obeys the relation:

1 + H P R = ( 1 + H P R 1 ) ( 1 + H P R 2 ) ( 1 + H P R 3 ) ( 1 + H P R 4 ) {\displaystyle 1+HPR=\left(1+HPR_{1}\right)\left(1+HPR_{2}\right)\left(1+HPR_{3}\right)\left(1+HPR_{4}\right)}

Example with income not reinvested

To the right is an example of a stock investment of one share purchased at the beginning of the year for $100. Assume dividends are not reinvested. At the end of the first quarter the stock price is $98. The stock share bought for $100 can only be sold for $98, which is the value of the investment at the end of the first quarter. This is less than the purchase price, so the investment has suffered a capital loss. The first quarter holding period return is: ($98 – $100 + $1) / $100 = -1% Since the final stock price at the end of the year is $99, the annual holding period return is: ($99 ending price - $100 beginning price + $4 dividends) / $100 beginning price = 3% If the final stock price had been $95, the annual HPR would be: ($95 ending price - $100 beginning price + $4 dividends) / $100 beginning price = -1%.

Worked examples

Example 1 — a first encounter with Holding period return

Start with the simplest possible case. Write down what Holding period return claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In mathematics, the smallest case is usually a single object, a single equation or a single measurement. Check that every symbol or term in your sentence has a meaning in that case.

Example 2 — changing one variable

Take the situation from Example 1 and change exactly one quantity: double it, halve it, or set it to zero. Predict what should happen to Holding period return before you calculate. Comparing your prediction with the result is the fastest way to find out whether you understand the idea or only the words.

Example 3 — an exam-style question

Typical questions about Holding period return ask you to (a) state it precisely, (b) apply it to given data, and (c) explain a limitation. Practise writing all three answers in under five minutes; the third part is what separates a full-mark answer from an average one.

Applications of Holding period return

In research: Holding period return appears in mathematics research whenever the underlying quantities have to be modelled precisely. Papers usually cite it as a starting assumption and then explore where it breaks down.
In technology and industry: Engineering practice reuses Holding period return in design rules, simulations and safety margins. Knowing the idea lets you read a specification sheet and understand why the numbers look the way they do.
In the classroom: Holding period return is common in secondary-school and first-year university syllabi. It links to neighbouring topics Investment, Mathematical finance, so understanding it makes those chapters shorter.
In everyday life: Look for Holding period return outside the textbook — in sport, cooking, traffic, electronics or the sky above you. An example you found yourself is remembered far longer than one you were given.

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How to study Holding period return in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Holding period return means in your own words.
Compare your version with the excerpt and mark what you missed.
Work through the three examples above with pen and paper.
Explain Holding period return out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Holding period return in simple terms?

Why does Holding period return matter?

Because it connects several mathematics ideas at once: it gives you a definition you can apply, a quantity you can calculate, and a way to check whether a result is plausible.

How should I study Holding period return?

Read the excerpt, restate it from memory, then work through the examples and applications listed on this page. The five-step study plan above takes about twenty minutes.

What does this page cover?

It gives you a compact reference excerpt plus original lgStudy explanations, examples, applications and study material on Holding period return.