Time Value of Money

Time value of money is a very basic concept which is the ground of many financial analysis and models. The goal of a business organization is to maximize the stockholder’s wealth. So the financial manager of an organization is aware of the compensation an investor requires (Jones, 2006). The compensation is required for two reasons: Risk factors and the time value of money. Money invested in any assets has an opportunity cost of investing it at a rate of return. This is the composition of time value cost and risk factor. Time value is attached to money because of sacrificing the present consumption for more future consumption.

Building blocks for time value of money

According to the time preference theory as a rational human being everyone prefers anything in present instead of future; this theory is grounded on the theory of time value of money (Lewin & Cachanosky, 2015). For example, anyone would prefer to get a gift now than a year from now. If anyone has $100 today he can deposit it in a bank at 6% interest rate and get $106 at the end of a year. So $100 today and $106 is the equivalent figure. Any rational person would prefer $106 or a higher amount in next year.

Lewin & Cachanosky (2015) found four main reasons that affect the time preference of rational investor- i) future is very uncertain. This uncertainty stems from the uncertainty of purchasing power of money, the economic condition of a country and uncertainty of life. ii) Every investment has an opportunity cost. If the investor consumes any amount of money he or she will lose the opportunity of investing that sum of money at a certain rate. iii) The purchasing power of money I affected by inflation. The value of the money today will not be the same tomorrow because of inflation or price level of the market. And the fourth reason was stated before: consumption preference which is the main theme of the time value of money.

Future value of an amount is the Present value or the amount today plus the product of percentage and the amount. The problem given below shows the future value of an investment. There are four types of investment varying in duration and interest rate. To solve the problem the simple mathematical form of calculating future value is given here,

Future value    =          PV (1+k) n

Problem 5-1: Compound interest

Types

Investment (PV)

Interest rate(k)

Years (n)

PV (1+k) n

Future Value

A

7000               

9%

8

70000 (1+.09)8

13947

B

11000

7%

10

11000 (1+.07) ^10

21638.66

C

20000

8%

6

20000 (1+.08)^6

31737.49

D

750

12%

12

750 (1+12)^12

2921.98

When interest is paid on interest it is called compounding. If someone invests in any asset $100 at 10% next year he would get $110. The second year the interest would be paid on $110, not on $100. So the amount he would get is $121. In the given problem compound interest is paid on the investment. The value $121 at the end of the 2nd year is equivalent to $100 in the first year with the presence of 10% interest rate. Equivalence doesn’t mean that $100 equals to $121. It means that both amounts have the same value after taking the time difference into account.

Present value

Present value shows how much amount it is today of any future sum while future value shows how many sums can become in any future date. For example, if anyone buys Treasury bond that can be sold back to the government for $1000. If the time value of money is 8% the amount he or she would want to pay is $1000/1.08 or $926. Any rational investor would be indifferent between the amount $926 today and $1000 one year later with the presence of 8% time value of money or interest rate. The process of reducing a future sum by moving back to present is called discounting and k or r is called discount rate (Gitman and Madura, 2001). It is opposite to the compounding and refers to the shrinking sum of the present.

The discount factor is calculated by k percent for n periods. The discount factor is opposite to the compounding factor. The variable that affects the present value interest factor also affects present value in the same direction. In the given problem 4 types of investment are given. Every investment type has different present value factor which discounts the future value.

Problem 5-2: Present value calculation

Present value = FV ÷ (1+k) ^n

Types

Investment (FV)

Interest rate(k)

Years (n)

Present Value

A

1200

11%

7

1200÷(1+.11)^7

577.99

B

400

6%

6

400÷(1+.06)^6

281.98

C

1500

4%

8

1500÷(1+.04)^8

1096.035

D

2500

20%

8

2500÷(1.20)^8

581.42

Future Value

When the future value is numerator and the present value factor is denominator the present value increases for the decrease of present value factor. The discount rate is considered an opportunity cost of not consuming the sum today. Northington & Gerard (2011) stated the higher the discount rate is the higher the future value of present sum and lower value of future sum today. The discount rate represents the required rate of return of an investor.

The concept of time value of money can be used for sales forecasting. If the seller expects yearly 15% increase in sales. The increase is considered as compounding rate. The problem gives here presents the application of future value in an increase of sales. It is calculated by simply multiplied by the sum of expected sales and increase in sales.

Problem 5-3: Future value

Sales               = 20000 copies. It is expected that the sales will increase by 15% in next three years.

Increase rate   = 15%

Increased sales in next 1st

year= 20000(1+0.15)

                                                = 23000

Increased sales in Next 2nd

year= 23000 (1+0.15) or 20000 (1+.15) ^2

                                                = 26450

Increased sales in next 3rd

year           = 26450 (1+.15) or 20000 (1+.15) ^3

Any rational investor will compare the present value of the investments to identify the most lucrative or profitable investment for him. To do this, an investor calculates the present value of all future cash flows. Different investment assets have different maturity periods and interest rates. The future cash flows are discounted at present interest rate.

Problem 5-4: Present Value Comparison

Present value = FV ÷ (1+k) ^n

Percent

9%

Years

Amount

PV

A

Today

1,200

1200

---------

B

12

12,500

4448.39

12500÷(1+.09)^12

C

25

30,000

3479

30000÷(1.09)^25

I should choose investment asset B. Offer B provides more present value than other offers. So it will be optimal decision for me to take offer B considering all equivalent cash flows. Present value mirrors the future value.

Compound Annuity and Annuity Factor

Any fixed periodical payments either inflow or outflow is regarded as the annuity. For example, the monthly installments of purchasing a flat or house can be called an annuity. Present value can be calculated from a future value and the other way around. FVIFA or PVIFA is called the annuity factor which is multiplied with constant C or annuity. When the annuity factor is multiplied with annuity it gives the future value of the annuity (Brigham & Ehrhardt, 2017). In opposite when the future sum is divided by annuity factor it gives present value of the annuity. In the problem, 5-5 future values of various annuity payments are calculated.

Problem 5-5: Compound Annuity

Compound Annuity    = FVA = C     PVA= C /       PVIFA or FVIFA=

Types

Payment (C)

Interest (k)

Years (n)

Future Value

(C× FVA)

A

1500

6%

8

(= 16.667

25000

B

10000

12%

6

4111.7

C

350

5%

5

1515.317

D

125

4%

4

453.74

An annuity can be paid yearly, semi-annually, quarterly, monthly, bi-monthly, weekly and daily. It depends on the terms or covenant of the investment. Compound interest rate differs when the payment frequency differs. Payment frequency is incorporated while calculating the annuity factor. The interest rate is divided by the periods and the time is also multiplied by the period. In the given problem it will be seen that how the change in interest rate, time and payment frequency change the terminal value.

Problem 5-6: Compound interest rate with non-annual period

A) Future value           =          PV (1+k) n

= 70000 (1+.06) ^5     = 9363.58

B)        1. Future value at semi-annual compounding           = 70000 (1+ .06/2) ^2×5

                                                                                                                                = 9407.42

            2. Future value at bi-monthly compounding              = 7000 (1+.06/24) ^5×24

                                                                                                = 9445.475

C) Future value at 12% interest rate                                       = 70000 (1+.12) ^5

                                                                                                = 12336.4

1. Future value at semi-annual compounding          = 70000 (1+.12/2)^5×2

= 12535.93

2. Future value at bi-monthly compounding            = 7000 (1+ .12/24)^ 5×24

= 12735.78

D) Future value after 12 years at 6%                                      = 7000 (1+ .06) ^12

                                                                                                                                                                = 14085.38

1. Future value at semi-annual compounding          = 7000 (1+.06/2)^12×2

= 14229.56

2. Future value at bi-monthly compounding            = 7000 (1+.06/24)^12×24

It is seen that bi-monthly compounding interest rate gives slightly more future value than semi-annually compounding rate. In bi-monthly compounding the interests are given at the end of 15 days on the other hand semi-annual compounding gives interest at the end of 6 months. It is also seen that doubling the term gives higher future value than doubling the interest rate. It means that term to maturity has a higher impact on annuity factor than the interest rate. The terminal value is more than double when term to maturity is 12 years instead of 6 years.

= 14368.12

E)

It is very clear from the concept of time value of money any investor or firm needs to be compensated for delayed consumption and savings. The rate of the compensation depends on prevailing interest in market, uncertainty or risk of future outcome and inflation of the economy. Time value of money is very basic concept for various financial management tools and techniques.

References

Brigham, E., & Ehrhardt, M. (2017). Financial management. Boston (MA): Cengage Learning.

Gitman, L., & Madura, J. (2001). Introduction to finance. Boston: Addison Wesley.

Jones, C. (2006). Introduction to Financial Management (12th ed.). Boston: Richard D.

Lewin, P., & Cachanosky, N. (2015). The Time-Value of Money and the Money-Value of Time: Duration, Roundaboutness, Productivity and Time-Preference in Finance and Economics. SSRN Electronic Journal. http://dx.doi.org/10.2139/ssrn.2613469

Northington, S., & Gerard, G. (2011). Finance. New York: Ferguson’s.