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

Present Values and Future Values of Investments

Present Values and Future Values of Investments

Some types of investments are known to accumulate interest more than once a year. This results from semi-annual, quarterly, monthly or daily compounding. This, in turn, leads to different present values (PV) or future values (FV) of an investment depending on the frequency of compounding employed.

We have previously seen that the effective annual rate of interest increases as the number of compounding periods per year increases. In calculating the present value or future value of an investment with multiple compounding periods per year, the most important thing is to ensure that the interest rate used corresponds to the number of compounding periods present per year.

Future Value

$$ FV= PV \left\{ \left( 1+ \frac {r_q}{m} \right) \right\}^{ m*n}$$

Where:

rq is the quoted annual rate;

m represents the number of compounding periods (per year); and

lastly, n is the number of years

Present Value

Suppose you make PV the subject of the above formula, you should find that:

$$ PV= FV \left\{ \left( 1+ \frac {r_q}{m} \right) \right\}^{ -m*n}$$

Example: Present Value With Monthly Compounding

You wish to have $10,000 in your savings account at the end of the next 3 years. Assume that the account offers a return of 9 percent per year, subject to monthly compounding. How much would you need to invest now so as to have the specified amount after the three years?

Solution

First, we write down the formula to use,

$$ PV= FV \left\{ \left( 1+ \frac {r_q}{m} \right) \right\}^{ -m*n}$$

Secondly, we establish the components that we already have:

rq = 0.09,   m = 12 since compounding is monthly, n = 3 years; and

then, we factor everything into the equation to find our PV.

$$ \begin{align*} PV & = 10,000 \left\{ \left(1+\frac {0.09}{12} \right) \right\}^{-12*3} \\ & = 10,000*1.0075^{-36} \\ & = $7,641.50 \\ \end{align*} $$

Therefore, you will need to invest at least $7,642 in your account to ensure that you have $10,000 after three years.

Question 1

Elizabeth Mary invests $2,000  in a project that pays a rate of return of 8% compounded quarterly. How much interest will Mary have earned after investing in the project for two years?

A. $2,300

B. $2,343.32

C. $343.32

Solution

The correct answer is C.

$$ \begin{align*} FV & = 2000 \left\{ \left(1+\frac {0.08}{4} \right) \right\}^{4*2} \\ & = 2,000*1.02^8 \\ & = $2,343.32 \\ \end{align*} $$

Therefore, interest gained = 2,343.32-2,000= $343.32

Question 2

What if the project paid a rate of return of 8% compounded daily? How much interest would Elizabeth Mary earn?

A. $2,347

B. $347

C. $2,340

Solution

The correct answer is B.

$$ \begin{align*} FV & = 2,000 \left\{ \left( 1+ \frac {0.08}{365} \right) \right\}^{365*2} \\ & = 2,000*1.00021918^{730} \\ & = $2,347 \\ \end{align*} $$

Similarly, the interest = 2,347-2,000 = $347

You should notice that with a higher compounding frequency, the corresponding profit is also higher. This confirms that interest earned increases as the number of compounding periods per year increases.

Note

We can convert our stated annual rates into the effective annual rate of interest, and arrive at similar answers. However, if we do that, we should ensure that we use years in the computation.

Reading 6 LOS 6d

Solve time value of money problems for different frequencies of compounding.

 

 

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.