Accounting and the Time Value of Money

Compound interest, annuity and present value techniques can be applied to many of the items found in financial statements.

In accounting, these techniques can be used to measure the relative values of cash inflows and outflows, evaluate alternative investment opportunities, and determine periodic payments necessary to meet future obligations. Some of the accounting items to which these techniques may be applied are notes receivables and payable, leases, amortization of premium and discounts etc.

 

In general business terms, interest is defined as the cost of using money overtime. This definition is in close agreement with the definition used by economists, who prefer to say that interest represents the time value of money.

Ignoring the effects of inflation, a dollar to day is worth more than a dollar to be received a year from now. In other words, we would all prefer to receive a specific amount of money now rather than on some future date. This preference rests on the time value of money. When payments for the time value of money are made or accrued interest expense is incurred, when payments for the time value of money are received or accrued, interest revenue is realized.

Inflows of dollars on various future dates should not be added together as if they were of equal value. These future cash inflows must be restated at their present values before they are aggregated. The concept of the time value of money tells us that more distant cash inflows have a smaller present value than cash inflows to be received within a shorter time span.

Similar reasoning applies to cash outflows. Before we add together cash outflows on various future dates, we must restate these outflows at their present values. The more distant the date of a cash outflow, the smaller is its present values.

As a simple example of this concept of present value, assume that you are trying to sell your car and you receive offers from three prospective buyers.

Buyers A offers you Br. 8000 to be paid immediately. Buyer B offers you Br. 8,200 to be paid one year from now. Buyer C offers the highest price, Br. 9,200 but this offer provides that payment will be made in five years. Assuming that the offers by B and C involves no credit risk and that money may be invested at 5% interest compounded annually, which offer would you accept?

You should accept the offer of Br. 8000 to be received immediately, because the present value of the other two offers is less than Br. 8000. if you were to invest Br. 8000 today, even at the modest rate of interest of 5%, your investment would be more than Br. 8200 in one year and considerably more than Br. 9,200 in five years.

This example suggests that the timing of cash receipts and payments has an important effect on the economic worth and the accounting values of both assets and liabilities. Consequently, investment and borrowing decisions should be made only after a careful analysis of the relative present values of the prospective cash inflows and outflows.

 

Accountants find many situations in which a reliable measurement of a transaction depends on the present value of future cash inflows and outflows. Some of the more prominent applications of the present and future value concepts are:

• Receivables and payables
• Asset valuation
• Bonds
• Leasing
• Pension and other post retirement benefits

 

Interest is the excess of resources (usually cash) received or paid over the amount of resources loaned or borrowed at an earlier date. Business transactions subject to interest state whether simple or compound interest is to be calculated.

Simple interest is the return on a principal amount for one time period. We may also think of simple interest as a return for more than one time period if we assume that the interest itself does not earn a return, but this kind of situation occurs rarely in the business world. Simple interest usually is applicable only to short-term investment and borrowing transactions involving a time span of less than one year.

Interest generally is expressed in terms of an annual rate. The formula for simple interest is:

I = p.r.t (interest = principal x annual rate of interest x number of years or fraction of a year that interest accrues). For example, interest on Br. 10,000 at 8% for one year is expressed as follows:

I = p.r.t

I = Br. 10,000 x 0.08 x 1

I = Br. 800

Compound interest is the return on a principal amount for two or more time periods, assuming that the interest for each time period is added to the principal amount at the end of each period, and earns interest in all subsequent periods. Because most investment and borrowing transactions involve more than one time period, business executives evaluate proposed transactions in terms of periodic returns, each of which is assumed to be reinvested to yield additional returns.

For example, if interest at 8% is compounded quarterly for one year on a principal amount of Br. 10,000 the total interest (compound interest) would be Br. 824.32, as computed below:

 

Period                Principal x Rate x Time =   Compound Interest     Accumulated Amounts

1^st quarter--------- Br. 10,000 x 0.08 x ¼          Br. 200.00                  Br. 10,200.00

2^nd quarter -------------10,200 x 0.08 x ¼                204.00                         10,404.00

3^rd quarter -------------10,404 x 0.08 x ¼                208.08                         10,612.08

4^th quarter ---------10,612.08 x 0.08 x ¼                 212.24                        10,824.32

     Interest                                                                824.32

 

N.B, In the computation of compound interest, the accumulated amount at the end of each period becomes the principal amount for purposes of computing interest for the following period.

 

Future value involves a current amount that is increased in the future as a result of compound interest accumulation. Present value, in contrast, involves a future amount that is decreased to the present as a result of compound interest discounting. Discounting, in effect, extracts the interest from a future value thereby returning to the principal amount.

The fact that investments have starting points and ending points makes it easier to understand present and future values. Present value in general refers to dollar (birr) values at the starting point of an investment, and future value refers to end-point dollar (birr) values.

If the dollar (birr) amount to be invested at the start is known, the future value of that amount at the end can be projected, provided the interest rate and number of interest compounding periods are also specified. Similarly, if the dollar (birr) amount available at the end of an investment period (future value) is known, the amount of money needed at the start of the investment period (present value) can be determined, again if the interest rate and number of interest compounding periods are known.

Present value and future value apply to interest calculations on both single payment amounts and periodic equal payment amounts (annuities)

 

   8.5.1 Future Value of a Single Sum

The accumulated amount (small a) of a single amount invested at compound interest may be computed period by period by a series of multiplication, as illustrated above for Br. 10,000 invested for one year at 8% compounded quarterly.

If n is used to represent the number of periods that interest is to be compounded, I is used to represent the interest per period, and p is the principal amount invested, the series of multiplications to compute the accumulated amount a in the example above may be determined as flows:

a = p (1 + i)ⁿ

a = Br. 10,000 (1.02)⁴

a = Br. 10,000 (1.02) (1.02) (1.02) (1.02)

a = Br. 10,824.32

 

The symbol a nù i is the amount to which 1 will accumulate at i rate of interest per period for n periods.

This symbol is read as “small a single n at i”.

^anù i = (1 + i)ⁿ or ^a 4 ù 2% = (1 + 0.02)⁴

 

Tables are available that give the value of a nù i

Use of these tables involves reference to a line showing the number of periods and a column showing the rate of interest per period.

 

Illustration of computation of future amount

1. If on the day her daughter was born, Bethel deposited Br. 10,000 in a savings account that guarantees to accumulate interest quarterly at 10% a year. What will be the amount in the savings account on her daughter’s 18^th birthday?

Solution: The amount in the savings account on the daughter’s 18^th birthday will be Br. 10,000 (1 + 0.025)⁷². Because Table 1 in the Appendix at the end of the chapter does not go beyond 50 periods, the amount in the savings account on the daughter’s 18^th birthday may be computed as follows:

Br. 10,000 (1 + 0.025) 50 x (1 + 0.025) 22

Br. 10,000 (3.437109) x (1.721571) = Br. 59,172

 

Determining the interest rate and number of periods

In some situations either the interest rate (i) or the number of periods (n) is not known, but sufficient data are available for their determination.

Example 1. If Br. 1000 is deposited at compound interest on January 1, 1990, and the amount on deposit on December 31, 1999 is Br. 1806.11, what is the semiannual interest rate accruing on the deposit?

Solution: The amount of 1 for 20 periods at the unstated rate of interest is 1.80611 (Br. 1806.11 ¸ Br. 1000 = 1.80611). Reference to table 1 in the Appendix at the end of this chapter indicates that 1806111 is the amount of 1 for 20 periods at 3%. Therefore, the semiannual interest rate is 3%.

 

Example 2. A family can invest Br. 150,000 today to provide for the college education of their child. The family believes that Br. 285,000 will be necessary for four years of college by the time the student matriculates. If the family can invest at 6% how many years will it take to accumulate Br. 285,000?

Solution:

1. Br. 285,000 = Br. 150,000 (1 + 6%)ⁿ

      Br. 285,000 ¸ 150,000 = 1.9000, which is the value in Table 1 of Br. 1 at 6%

       interest.

1. In Table 1, read down the 6% column to find 1.9000.
2. The table value 1.89830 is found on the line for 11 years and the value 2.01220 is found on the line for 12 years.
3. The number of interest periods is just over 11 years

 

8.5.2 Present Value of a single sum

Many measurement and valuation problems in financial accounting require the computation of the discounted present value of a principal amount to be paid or received on a fixed date. The present value represents the discounted amount (interest excluded) that will accumulate to the future amount (interest included). The present value of a future amount is always less than that future amount.

The computation of the present value of a single future amount is a reversal of the process of finding the amount to which a present amount will accumulate. We know that a = p (1 + i)ⁿ, and when we solve for p by dividing both sides of the equation by (1 + i)ⁿ, we have p = a / (1+i)ⁿ

Therefore, the formula for the present value of a due in n periods at i rate of interest per period is

P nùi = a / (1+i)ⁿ

a = future amount P = present value i = interest rate per period n = number of compounding period

Example: If we want an amount of Br. 30,000 after 12 years by making a single deposit in a saving account which will pay 16% interest compounded quarterly, what should the amount of initial deposit be?

 

Solution: The present value is Br. 30,000 discounted at 4% for 48 periods. Using Table 2 in the Appendix at the end of this chapter, the present value is

Br. 30,000 x 0.152195 = Br. 4565.84

Or using the formula:

P =a / (1+i)ⁿ =30,000/ (1+0.04)⁴⁸ = Br. 4565.84

 

Many measurement situations in financial accounting involve periodic deposits, receipts, withdrawals, or payments (called rents), with interest at a stated rate compounded at the time that each rent is paid or received. These situations are considered annuities if all the following conditions are present:

1. The periodic rents are equal in amount
2. The time period between rents is constant, such as a year, a quarter of a year or a month
3. The interest rate per time period remains constant
4. The interest is compounded at the end of each time period

In general, an annuity is a series of uniform payments or receipts (sometimes called rents) occurring at uniform intervals over a specified investment time frame, with all amounts earning compound interest at the same rate.

When rents are paid or received at the end of each period and the total amount on deposit is determined at the time the final rent is made, the annuity is an ordinary annuity (or annuity in arrears). When rents are paid or received at the beginning of each period, and the total amount on deposit is determined one period after the final rent, the annuity is annuity due (or annuity in advance). When the amount of an ordinary annuity remains on deposit for a number of period beyond the final rent, the arrangement is known as a deferred annuity.

The difference between the three types of annuity is illustrated below:

 

Period (Example: years)                 0           1            2           3           4       5

                                                                    R           R            R          R      R

Ordinary annuity

                                                       P                                                              A

                                                       R           R           R           R           R

Annuity due

                                                      P                                                              A

Deferred annuity                                         R             R          R

                                                       P                                                             A

   8.6.1 Amount of Annuity

Amount of an annuity is future values of a series of equal receipts or payments (rents) made at regular time intervals and at the same rate of interest compounded each time the receipts or payments are made.

A typical accounting application of the future value of an annuity is the establishment of a fund by equal annual contribution perhaps for the future expansion of a facility or payment of a debt.

 

     8.6.1.1 Amount of Ordinary Annuity

The amount of an ordinary annuity (or annuity in arrears) consists of the sum of the equal periodic rents and compound interest on the rents immediately after the final rent. Unless otherwise stated, all annuities are assumed to be ordinary annuities, meaning that every payment occurs at the end of the interest period.

The amount A of an ordinary annuity of n rents at I interest rate per period is determined by

          A =R[((1+i)ⁿ-1)/i]

 where R = the amount of each periodic rent

                                                    I = the interest rate per period

                                                   N = the number of rents

                                                    A = the amount of the ordinary annuity

Example: Compute the amount of an ordinary annuity of 16 rents of Br. 100 at 2%

Solution: A = Br. 100 [(1+0.02)¹⁶-1)/0.02]= Br 1863.93

 

     8.6.1.2 Amount of an Annuity Due

The amount of an annuity due (or annuity in advance) is the total amount on deposit one period after the final rent. This is illustrated below for an annuity due of 16 rents.

This diagram suggests that there are two ways of computing the amount of an annuity due of 16 rents of 1 at, say, 2% interest per period, as follows:

1. Take the amount of an ordinary annuity of 16 rents of 1 at 2% from Table 3 in the Appendix at the end of this chapter and accrue interest at 2% for one additional period:

   18.639285 x 1.02 = 19.01207 [A nùI x (1 + I)]
    

2. Take the amount of an ordinary annuity of 17 rents of 1 at 2% from Table 3 in the Appendix at the end of this chapter and subtract 1, the rent not made at the end of time period 17:

                       20.01207 – 1 = 19.01207 (A n + 1ùI – 1)

Example: Green Company needs Br. 200,000 on March 31, year 5. This amount is to be accumulated by making 16 equal deposits in a fund at the end of each quarter, starting March 31, year 1,and ending on December 31, year 4. The fund will earn interest at 8% compounded quarterly. Compute the periodic rents (deposits) that Green Company must make.

Solution: The balance in the fund on March 31, year 5, represents the amount of an annuity due of 16 rents at 2% per period (19.01207 as determined above). Therefore, the periodic rents are: Br. 200,000 ¸ 19.01207 = Br. 10,519.63. This result may be verified as follows:

Amount of ordinary annuity of 16 rents of Br. 10,519.63 at 2% on December 31, year 4: Br. 10,519.63 x 18.639285 ------------------------------------------Br. 196,078

Add: Interest for first quarter of year 5 Br. 196,078 x 0.02 ---------------------3,922

Balance in fund on March 31, year 5 (amount of

an annuity due of 16 rents of Br. 10,579.63 at 2%) -----------------------Br. 200,000

 

    8.6.1.3 Amount of Deferred Annuity

When the amount of an ordinary annuity remains on deposit for a number of periods beyond the final rent, the arrangement is known as a deferred annuity. When the amount of an ordinary annuity continues to earn interest for one additional period, we have an annuity due situation, when the amount of an ordinary annuity continues to earn interest for more than one additional period, we have a deferred annuity situation.

The amount of a deferred annuity may be computed by multiplying the amount of the ordinary annuity by the amount of 1 for the period of deferral to accrue compound interest.

Alternatively, we may take the amount of an ordinary annuity for all periods (including the period of deferral) and subtract from this the amount of the ordinary annuity for the deferral period when rents were not made, but interest continued to accumulate.

 

Example: On April 1, 1996 Delta company decided to accumulate cash to pay a debt, that matures on March 31,2002. The company deposited Br. 10,000 cash on March 31, 1996, 1997 and 1998. The interest rate is 10 percent compounded annually. Determine the amount that will accumulate on March 31, 2002.

Solution: using the Appendix at the end of this chapter, the amount of an ordinary annuity of three rents deferred for 4 periods may be computed as: Br. 10,000 x 3.31 x 1.4641 = Br. 48,461.71

 

   8.6.2 Present Value of an annuity

The present value of an annuity is the value of a series of equal future receipts or payments (rents) made at regular time intervals and discounted at the same compound interest rate on each date rents are due.

Present value of annuities are used more frequently in financial accounting. For example, the computation of the proceeds of bond issues, the value of plant assets acquired in purchase type business combination or through capital leases, the amount of past service pension costs, the amount of debt or receivables under installment contracts, and the amount of mortgage debt or investments in mortgage notes all require the application of the present-value- of annuity concept.

 

     8.6.2.1 Present value of ordinary annuity

Present value of ordinary annuity is the discounted value of a series of future rents on a date one period before the first rent.

The present value of an ordinary annuity of five rents depicted above is the value of the rents, discounted at compound interest, at a point in time one period before the first rent. The present value of an ordinary annuity is computed as the total of the present values of the individual rents, but the use of a table, such as Table 4 in the Appendix at the end of this chapter is more efficient. The present value of an ordinary annuity may be computed using the following formula.

                          P = R [(1-(1/(1+i)ⁿ) / i]

 

Example: ERA company has outstanding a Br. 500,000 non interest bearing debt, payable Br. 100,000 a year for five years starting on December 31, year 1. what is the present value of this debt on January 1, year 1, for financial accounting, if 8% compounded annually is considered a fair rate of interest?

Solution: The present value of the debt on January 1, year 1, is equal to the present value of an ordinary annuity of five rents reported at Br. 399,271 (Br. 100,000 x 3.99271) in the accounting records on January 1, year 1.

The repayment program (loan amortization table) for this debt is summarized below:

ERA Company

Repayment program for Debt of Br. 399,271 at 8% interest

                                   Interest Expense    Repayment at       Net reduction     Debt balance

Date                            at 8% a year           end of year            in debt

Jan. 1, year 1                                                                                                       Br. 399,271

Dec. 31, year 1           Br. 31,942            Br. 100,000             Br. 68,058                331, 213

Dec. 31, year 2                 26,497                  100,000                   73,503                 257,710

Dec. 31, year 3                 20,617                  100,000                    79,383                178,327

Dec. 31, year 4                 14,266                  100,000                    83,734                  92,593

Dec. 31, year 5                   7,407                  100,000                    92,593                      –0-

 

Example 2

NINI corporation issued Br. 5 million face value amount of 9% five year bonds on June 30, year 5. The bonds pay interest on June 30 and December 31 and were issued to yield 10% compounded semiannually. Compute the proceeds of this bond issue.

Solution:

The proceeds of the bond issue may be computed as the total of (1) the present value of the Br. 5 million to be paid at maturity, discounted at the 5% semiannual current rate of interest for 10 periods, plus (2) the present value of an ordinary annuity of 10 rents of Br. 225, 000 (Br. 5,000,000 x 0.045 = Br. 225,000) semiannual interest payments, also discounted at 5% per period. That is,

Present value of Br. 5 million discounted at 5% for 10 six

month periods: Br. 5,000,000 x 0.613913 -------------------------------Br. 3,069,565

Add: present value of ordinary annuity of 10 rents

      of Br. 225,000 discounted at 5% Br. 225,000 x 7.721735 -------------1,737,390

Proceeds of bond issue ------------------------------------------------------Br. 4,806,955

 

      Alternatively, it can be computed as

Face amount of bonds ------------------------------------------------Br. 5,000,000

Less: present value of ordinary annuity of

        10 rents of Br. 25,000 interest deficiency

       discounted at 5% per period:  Br. 25,000 x 7.721735 ---------------193,043

Proceeds of bond issue -------------------------------------------------Br. 4,806,957*

* Br. 2 discrepancy between this amount and the amount computed above is caused by rounding in present value tables

 

     8.6.2.2 Present value of Annuity Due

Present value of annuity due is the discounted value of a series of future rents on the date the first rent is received or paid. That is, the present value falls on the date the first rent is made.

For this reason, an annuity due often is referred to as an annuity in advance.

Example: On January 1, year 1, Sosa corporation acquired a plant asset for Br. 64,682. Sosa agreed to make five equal annual payments starting on January 1, year 1, and ending on January 1, year 5, at 8% compounded annually. Compute the annual payments on the debt.

Solution: The annual payments on the debt are present value (Br. 64,682) divided by present value of annuity due of at 8% for 5 periods (4.312127) = Br. 15,000

 

     8.6.2.3 Present Value of Deferred Annuity

Present value of deferred annuity is the discounted value of a series of future rents on a date that is more than one period before the date that the first rent is received or paid. The present value of deferred annuity may be computed by using two different methods as follows:

1. discount the present value of the ordinary annuity portion at compound interest for the period the annuity is deferred, or
2. determine the present value of an ordinary annuity equal to the total number of period involved and subtract from this the present value of the “missing” ordinary annuity for rents equal in number to the number of periods the annuity is deferred.

Example: Daof Company wants to know the amount at time period 0 that would pay a debt of five payments of Br. 100,000 each, payments starting at time period 4, and interest compounded at 8% per time period.

Solution: Using Table 2 and 4 in the Appendix at the end of this chapter, we may compute the present value at time period 0 (today) of the ordinary annuity of five rents of 1 deferred for three periods as follows:

1. Present value of ordinary annuity of five rents of

      1 at 8% at time period 3, discounted at 8% for

      three periods: 3,992710 x 0.793832 --------------------------------------------3.169541

      or (2) present value of ordinary annuity of eight rents

     of 1 at 8% at time period 0, less the present value of

    ordinary annuity of three rents    of 1 (the rents note made)

   at 8% at time period 0.5.746639 – 2.577097 -------------------------------------3169542

Thus, cash in the amount of Br. 316,954 (Br. 100,000 x 3.169542) is needed at time period 0 to repay the debt.

The repayment of debt is summarized below:

 

Daof Company

                          Repayment program for debt of Br. 316,954 at 8% interest

Time period       Interest Expense                 Repayment        Net reduction          Debt

                           at 8% per period                                            in debt                   Balance

    0                    present value of                                                                        Br. 316,954

    1                    debt                                                                                                342,310

    2                    Br.25,356                                                                                       369,695                                                                                  

    3                        29,576                                                                                        399,271

   4                        31,942   --------------------Br. 100, 000-------Br. 68, 058            331,213

   5                       26,497-----------------------      100, 000-------      73, 503            257,710

   6                       20,617-----------------------      100, 000-------      79, 383            178,327

   7                       14,26617--------------------      100, 000-------      85, 734             92,593

   8                        7,407------------------------      100, 000-------      92, 593                  0

 

Interest is the cost of borrowing money. It is normally stated as a percentage of the amount borrowed (principal) calculated on a yearly basis.

The concepts of present value is described as the amount that must be invested now to produce a known future value. This is the opposite of the compound interest discussion in which the present value was known and the future value was determined.

An annuity is a series of equal periodic payments or receipts called rents. An annuity requires that the rents be paid or received at equal time intervals, and that compound interest be applied. The future amount of an annuity is the sum (future value) of all the rents (payments or receipts) plus the accumulated interest on them. If the rents occur at the end of each time period, the annuity is known as an ordinary annuity. If rents occur at the beginning of each time period, it is an annuity due. Thus, in determining the amount of an annuity for a given set of facts, there will be one less interest period for an ordinary annuity than for an annuity due.

A deferred annuity is an annuity in which two or more periods must pass, after it has been arranged, before the rents will begin. For example, an ordinary annuity of 10 annual rents deferred five years mean that no rents will occur during the first five years, and that the first of the 10 rents will occur at the end of the sixth year.

An annuity due of 10 annual rents deferred five years means that no rents will occur during the first five years, and that the first of the 10 rents will occur at the beginning of the sixth year.

In general, compound interest, annuity and present value techniques can be applied to many of the items found in financial statements. In accounting, these techniques can be used to measure the relative values of cash inflows and outflows, evaluate alternative investment opportunities, and determine periodic payments necessary to meet future obligations. Some of the accounting items to which these techniques may be applied are notes receivable and payable, leases, amortization of premium and discounts etc.