Time Value of Money Questions and Answers [With PDF]

Would you rather have 1,000 today or 1,000 five years from now? Most people choose today, and the time value of money explains why: money you have now can be invested and earn interest, so it is worth more than the same amount later. This idea sits behind loans, savings, bonds, project evaluation and almost every financial decision. This chapter covers present value, future value, annuities, interest rates, simple and compound interest and the rules of 69 and 72. Here are 30 important questions with clear answers and worked examples. Every calculation has been rechecked.

This is Chapter 2 of our Finance course. It follows Introduction to Finance and comes before Short-Term and Mid-Term Financing.

Basics of the Time Value of Money

Q1. What is the time value of money?

The time value of money is the idea that a sum of money today is worth more than the same sum received in the future, because money today can be invested to earn a return. So 100 now and 100 four years from now are not equal in value.

Q2. Why is the time value of money important?

1. Uncertainty: the future is unsure, so firms prefer cash today to cash later.
2. Project evaluation: to judge a long-term project, a business must compare the project’s cost today with the money it will bring in later.
3. Decisions need one time frame: present and future values are not the same, so expected future income must be converted into present value before comparing.
4. Opportunity cost: money put into one project cannot be used in another. Time value of money equations measure this cost.
5. Inflation: prices rise over time, so the purchasing power of money falls. 1 today buys more than 1 tomorrow.

Q3. What is a timeline?

A timeline is a line that shows when cash flows happen: money paid or received today (time 0) and at each later period. It helps you see the pattern before choosing a formula.

Present Value

Q4. What is present value?

Present value (PV) is the worth today of a sum to be received in the future. It is the amount that, invested today at a given interest rate, would grow to that future sum. The process of finding it is called discounting.

Q5. What are the features of present value?

1. Discounting is the method of calculating present value.
2. Present value is always lower than the future amount.
3. The higher the interest (discount) rate, the lower the present value.
4. The longer the wait for the money, the lower its present value.
5. PVIF (Present Value Interest Factor) is the present value of 1 received after a certain period at a given rate.

Q6. How do you calculate present value?

1. Present value of a lump sum: PV = FV / (1 + i)ⁿ, where FV is the future value, i is the interest rate per year and n is the number of years.

Example: At 10%, what is the present value of 1,000 received after 5 years? PV = 1,000 / (1.10)⁵ = 1,000 / 1.6105 = 620.92. Receiving 1,000 in 5 years is worth about 621 today, so waiting costs 1,000 – 621 = 379 in present-value terms.

2. Discounting more than once a year: PV = FV / (1 + i/m)^{n x m}, where m is the number of compounding periods in a year.

Example: You want 50,000 in 5 years. A bank offers 9.5% a year compounded monthly. How much must you deposit today? Monthly rate = 0.095 / 12 = 0.0079167 and periods = 5 x 12 = 60. PV = 50,000 / (1.0079167)⁶⁰ = 50,000 / 1.6050 = 31,152.64.

3. Present value of unequal cash flows: discount each cash flow separately and add them: PV = CF1/(1+i) + CF2/(1+i)² + CF3/(1+i)³ + …

Example: You receive 1,000 at the end of year 1, 2,000 at the end of year 2 and 3,000 at the end of year 3. The interest rate is 10%.

Future Value

Q7. What is future value?

Future value (FV), also called compound value, is the amount a sum of money will grow to after a certain period if it earns interest. Interest is earned on the original sum and on the interest already added. The process of finding it is called compounding.

Q8. What are the features of future value?

1. It is found by adding interest to the present amount.
2. Future value is always higher than the present amount.
3. The more times interest is compounded in a year, the higher the future value.
4. The longer the period, the higher the future value.
5. The higher the interest rate, the higher the future value.
6. FVIF (Future Value Interest Factor) is the future value of 1 at a given rate and time.

Q9. How do you calculate future value?

1. Future value of a lump sum: FV = PV x (1 + i)ⁿ.

Example: What is 1,000 worth after 5 years at 5%? FV = 1,000 x (1.05)⁵ = 1,000 x 1.2763 = 1,276.28. At 10%, the same 1,000 would grow to 1,000 x 1.6105 = 1,610.51, which shows how a higher rate raises future value.

2. Compounding more than once a year: FV = PV x (1 + i/m)^{n x m}.

Example: You deposit 50,000 at 13.5% compounded monthly for 10 years. Monthly rate = 0.135 / 12 = 0.01125 and periods = 10 x 12 = 120. FV = 50,000 x (1.01125)¹²⁰ = 50,000 x 3.8285 = 191,423 (approximately).

3. Future value of unequal cash flows: compound each deposit for the number of periods it stays in the account.

Deposits at the end of each year (100 in year 1, 200 in year 2 and 250 in year 3, at 10%, valued at the end of year 3):

Deposits at the beginning of each year (same amounts, but each is deposited a year earlier, so it earns one more year of interest):

Q10. What is the difference between present value and future value?

Annuities

Q11. What is an annuity?

An annuity is a series of equal payments or receipts made at equal intervals of time for a specified number of periods. Both conditions must hold: equal amounts and equal gaps. Examples: loan instalments, rent and regular savings deposits.

Q12. What are the features of an annuity?

1. The amount of each payment is the same.
2. The interval between payments is the same.
3. It runs for a fixed number of periods (an annuity of unlimited length is a perpetuity).
4. The present value of an annuity is lower when the period is longer or the rate is higher.
5. The future value of an annuity is higher when the period is longer or the rate is higher.

Q13. How are annuities classified?

Q14. What is an ordinary annuity?

An annuity in which each payment is made at the end of each period, such as year or month. It can be valued as a present value or a future value.

Q15. What are the formulas for an ordinary annuity?

• Present value: PVA = A x [1 – 1/(1 + i)ⁿ] / i
• Future value: FVA = A x [(1 + i)ⁿ – 1] / i

Here A is the annuity payment, i is the interest rate per period and n is the number of periods.

Example: A = 1,000 a year for 3 years at 10%. PVA = 1,000 x [1 – 1/1.331] / 0.10 = 1,000 x 2.4869 = 2,486.85. FVA = 1,000 x [1.331 – 1] / 0.10 = 3,310.

Q16. What is an annuity due?

An annuity in which each payment is made at the beginning of each period. Because each payment is made one period earlier, an annuity due is worth more than an ordinary annuity of the same size. It can also be valued as a present value or a future value.

Q17. What are the formulas for an annuity due?

• Present value: PVAD = A x [1 – 1/(1 + i)ⁿ] / i x (1 + i)
• Future value: FVAD = A x [(1 + i)ⁿ – 1] / i x (1 + i)

In other words, multiply the ordinary annuity value by (1 + i). Using the same example (A = 1,000, i = 10%, n = 3): PVAD = 2,486.85 x 1.10 = 2,735.54 and FVAD = 3,310 x 1.10 = 3,641.

Interest Rates

Q18. What is the key element in the time value of money?

The interest rate (also called the discount rate when finding present value). It is the rate at which money grows over time, and it decides the future and present value of money.

Q19. What is the nominal (annual) rate?

The nominal or stated annual rate is the yearly rate of interest quoted in a loan or deposit agreement, before any effect of compounding. Example: a company takes a 100,000 loan at 12% a year. The nominal rate is 12%.

Q20. What is the effective annual rate (EAR)?

The effective annual rate is the interest rate you really earn or pay in a year after compounding: EAR = (1 + i/m)^m – 1. Example: a nominal rate of 12% compounded monthly (m = 12): EAR = (1 + 0.12/12)¹² – 1 = (1.01)¹² – 1 = 1.1268 – 1 = 12.68%. If the same 12% is compounded only once a year, EAR = 12%.

Q21. What are the similarities and differences between the nominal rate and the EAR?

• Similarity: with annual compounding, the nominal rate and EAR are equal.
• Difference: when compounding happens more often than once a year, the EAR is higher than the nominal rate, and it keeps rising as the number of compounding periods increases.

Simple and Compound Interest

Q22. What is simple interest?

Interest charged only on the original principal, at the same rate for every year.

Q23. How is simple interest calculated?

SI = P x i x n, where P is the principal, i is the rate per year and n is the number of years. Example: 1,000 deposited for 3 years at 10% gives SI = 1,000 x 0.10 x 3 = 300.

Q24. What is compound interest?

Interest calculated on the principal and on all the interest already added to it, in other words, interest on interest. It results from reinvesting the interest instead of paying it out.

Q25. How is compound interest calculated?

CI = P x (1 + i)ⁿ – P. Example: 1,000 for 3 years at 10%: CI = 1,000 x 1.331 – 1,000 = 331. Compare this with the simple interest of 300 in Q23. The extra 31 is the interest earned on interest.

Q26. How do you find the annual compound interest rate?

Rearrange FV = PV x (1 + i)ⁿ to find i.

Example: Harry borrows 10,000 from a bank for 3 years and repays 13,310 at the end. What annual compound rate did he pay?

Q27. What is the difference between simple interest and compound interest?

Rules of 69 and 72, and Perpetuity

Q28. What is the Rule of 69?

A quick method to estimate how many years it takes for an investment to double with compound interest: N = 0.35 + 69 / i, where i is the interest rate in percent. Example: at 10%, N = 0.35 + 6.9 = about 7.25 years.

Q29. What is the Rule of 72?

A simpler shortcut for the same purpose: N = 72 / i, with i in percent. Example: at 8%, money doubles in about 72 / 8 = 9 years. (The exact answer is 9.01 years.) At 10%, 72 / 10 = 7.2 years.

Q30. What is a perpetuity?

An annuity that goes on forever, with no end date. Its present value is PV = A / i. Example: a payment of 500 every year forever, at 10%, has a present value of 500 / 0.10 = 5,000.

Quick Revision Points

• PV = FV / (1 + i)ⁿ. FV = PV x (1 + i)ⁿ. Discounting finds PV, compounding finds FV.
• For compounding m times a year, use i/m per period and n x m periods.
• Ordinary annuity: payments at the end. Annuity due: payments at the beginning, so multiply by (1 + i).
• EAR = (1 + i/m)^m – 1. It is higher than the nominal rate when m is more than 1.
• Perpetuity PV = A / i. Rule of 72: years to double = 72 / rate.

Test yourself with the chapter quiz below, then continue to Short-Term and Mid-Term Financing.