Читать книгу Time Value of Money and Fair Value Accounting - Dr Jae K Shim - Страница 8
ОглавлениеCHAPTER 1
Time value of money and its applications
A dollar now is worth more than a dollar to be received later. This statement sums up an important principle: money has a time value. The truth of this principle is not that inflation might make the dollar received at a later time worth less in buying power. The reason is that you could invest the dollar now and have more than a dollar at the specified later date.
Money has value because with it one can acquire assets and services and discharge obligations. The holding, borrowing or lending of money can result in costs or earnings. And the longer the time period involved, the greater the costs or the earnings. The cost or earning of money as a function of time is the time value of money.
Accountants must have a working knowledge of compound interest, annuities, and present value concepts because of their application to numerous types of business events and transactions which require proper valuation and presentation. These concepts are applied in the following areas: (1) sinking funds, (2) installment contracts, (3) pensions, (4) long-term assets, (5) leases, (6) notes receivable and payable, (7) business combinations, and (8) amortization of premiums and discounts.
Time value of money is also a critical consideration in financial and investment decisions. For example, compound interest calculations are needed to determine future sums of money resulting from an investment. Discounting, or the calculation of present value, which is inversely related to compounding, is used to evaluate the future cash flow associated with capital budgeting projects. There are plenty of applications of time value of money in accounting and finance. The purpose of this book to present the tools and techniques that will help you measure the present value of future cash inflows and outflows.
Accounting Applications
Financial reporting uses different measurements in different situations—historical cost for equipment, net realizable value for inventories, or fair value for investments. The FASB increasingly is requiring the use of fair values in the measurement of assets and liabilities. According to the FASB’s recent guidance on fair value measurements (ASC 820-10-05, FAS-57, Fair Value Measurements), the most useful fair value measures are based on market prices in active markets. Within the fair value hierarchy these are referred to as Level 1. Level 1 fair value measures are the most reliable because they are based on quoted prices, such as a closing stock price in the financial dailies and online finance sites.
However, for many assets and liabilities, market-based fair value information is not readily available. In these cases, fair value can be estimated based on the expected future cash flows related to the asset or liability. Such fair value estimates are generally considered Level 3 (least reliable) in the fair value hierarchy because they are based on unobservable inputs, such as a company’s own data or assumptions related to the expected future cash flows associated with the asset or liability. As discussed in the fair value guidance, present value techniques are used to convert expected cash flows into present values, which represent an estimate of fair value. This issue is covered in depth in Chapter 2.
Because of the increased use of present values in this and other contexts, it is important to understand present value techniques. The timing of the returns on an investment has an important effect on the worth of the investment (asset). Similarly, the timing of debt repayment has an important effect on the value of the debt commitment (liability).
GAAP addresses present value as a measurement basis for a broad array of transactions, such as accounts and notes receivable and payable, leases, pensions and other postretirement benefits, and long-term asset impairments. More are listed below.
Accounts and notes receivable and payable—these involve single sums (the face amounts) and may involve annuities, if there are periodic interest payments.
Leases—involve measurement of assets and obligations, which are based on the present value of annuities (lease payments) and single sums (if there are residual values to be paid at the conclusion of the lease).
Pensions and other postretirement benefits —involve discounted future annuity payments that are estimated to be paid to employees upon retirement.
Long-term asset impairments—evaluating various long-term investments or assessing whether an asset is impaired requires determining the present value of the estimated cash flows (may be single sums and/or an annuity).
Stock-based compensation—determining the fair value of employee services in compensatory stock-option plans.
Business combinations--determining the value of receivables, payables, liabilities, accruals, and commitments acquired or assumed in a “purchase.”
Environmental liabilities—Measuring the fair value of future obligations for asset retirements.
Disclosures--measuring the value of future cash flows from oil and gas reserves for disclosure in supplementary information.
Fair Value Measurements
ASC 820-10-05, FAS157 (Fair Value Measurements) states that a fair value measurement reflects current market participant assumptions about future inflows of the asset and future outflows of the liability. A fair value measurement incorporates the attributes of the particular asset or liability (e.g., location, condition). In formulating fair value, consideration is given to the exchange price, which refers to the market price at the measurement date in an orderly transaction between the parties to sell the asset or transfer the liability. The focus is on the price that would be received to sell the asset or paid to transfer the liability (exit price), not the price that would be paid to buy the asset or received to assume the liability (entry price).
The asset or liability may be by itself (e.g., financial security, operating asset) or a group of assets or liabilities (e.g., asset group, reporting unit).
The Fair Value Hierarchy
A hierarchy list of fair value distinguishes between (1) assumptions based on market data from independent outside sources (observable inputs) and (2) assumptions by the company itself (unobservable inputs). The use of unobservable inputs allows for situations in which there is minimal or no market activity for the asset or liability at the measurement date. Valuation methods used to measure fair value shall maximize the use of observable inputs and minimize the use of unobservable ones.
Risk and Restrictions
An adjustment for risk should be made in a fair value measurement when market participants would include risk in the pricing of the asset or liability. Nonperformance risk of the obligation and the entity’s credit risk should be noted. Further, consideration should be given to the effect of a restriction on the sale or use of an asset that impacts its price.
The Difference between the Principal Market and the Most Advantageous Market
In a fair value measurement, we assume that the transaction occurs in the principal (main) market for the asset or liability. This is the market in which the company would sell the asset or transfer the liability with the greatest volume. If a principal market is nonexistent, then the most advantageous market should be used. This is the market in which the business would sell the asset or transfer the liability with the price that maximizes the amount that would be received for the asset or minimizes the amount that would be paid to transfer the liability after taking into account any transaction costs. The fair value measurement should incorporate transportation costs for the asset or liability.
Valuation Approaches
In fair value measurement, valuation techniques based on the market, income, and cost approaches may be used. The market approach uses prices for market transactions for identical or comparable assets or liabilities. The income approach uses valuation techniques to discount future cash flows to a present value amount. The cost approach is based on the current replacement cost such as the cost to buy or build a substitute comparable asset after adjusting for obsolescence. Input availability and reliability related to the asset or liability may impact the choice of the most suitable valuation method.
A single or multiple valuation technique may be needed, depending on the situation. For example, a single valuation method would be used for an asset having quoted market prices in an active market for identical assets. A multiple valuation method would be used to value a reporting unit.
The Three Levels of Fair Value Hierarchy
The fair value hierarchy prioritizes the inputs to valuation techniques used to measure fair value into three broad levels. Level 1, the highest priority, assigns quoted prices (unadjusted) in active markets for identical assets or liabilities. Level 3, the lowest priority, is assigned for unobservable inputs for the assets or liabilities.
Level 2 inputs are those except quoted prices included within Level 1 that are observable for the asset or liability, either directly or indirectly. Level 2 inputs include:
Quoted prices for similar assets or liabilities in active markets
Quoted prices for similar or identical assets or liabilities in markets that are not active, such as markets with few transactions, noncurrent prices, limited public information, and where price quotations show substantial fluctuation
Inputs excluding quoted prices that are observable for the asset or liability, such as interest rates observable at often quoted intervals and credit risks
Inputs obtained primarily from observable market information by correlation or other means
In the case of Level 3, unobservable inputs are used to measure fair value to the degree that observable inputs are not available. Unobservable inputs reflect the reporting company’s own assumptions about what market participants consider (e.g., risk) in pricing the asset or liability.
Financial Applications
In addition to accounting and business applications, compound interest, annuity, and present value concepts apply to personal finance and investment decisions. In purchasing a home or car, planning for retirement, and evaluating alternative investments, you will need to understand time value of money concepts. Other financial applications include determination of sinking funds--the contributions necessary to accumulate a fund for debt retirements and installment contracts--periodic payments on long-term purchase contracts or leases, periodic payments of an amortized loan, and valuations of businesses, stocks, bonds, real estate, and other financial securities.
Time Value Fundamentals
The following four variables are fundamental to all time value problems (Exhibit 1).
Exhibit 1: Fundamental Variables*
1. Rate of interest. This rate, unless otherwise stated, is an annual rate that must be adjusted to reflect the length of the compounding period if less than a year.
2. Number of time periods. This is the number of compounding periods. (A period may be equal to or less than a year.)
3. Future value. The value at a future date of a given sum or sums invested assuming compound interest.
4. Present value. The present worth of a future sum or sums discounted assuming compound interest.
* Given any two variables of 1,2,3, or 1,2,4, one can determine the third variable. Many situations will be illustrated later in the book.
Exhibit 2 depicts the relationship of these four fundamental variables in a time diagram.
Exhibit 2: The relationship of four fundamental variables
Exhibit 3 shows all the tables we will be using throughout this book.
Exhibit 3: Summary of time value tables*
* Future and present value table values are truncated to three decimal digits for simplicity. In practice, you are advised to use financial calculators or Excel to ensure maximum accuracy.
** Hereafter in this book, the terms payment and receipt will be used interchangeably. A payment by one party in a transaction becomes a receipt to the other and vice versa,
How Do You Calculate Future Values? - How Money Grows
Simple Interest
Simple interest is the interest calculated on the amount of the principal only. It is the return on (or growth of) the principal for one time period. The following equation expresses simple interest.
Interest = p x i x n
Where p = principal
i = rate of interest for a single period n = number of periods
Example 1
Barstow Electric Inc. borrows $10,000 for 3 years with a simple interest rate of 8% per year. It computes the total interest it will pay as follows.
Interest = p x i x n = $10,000 x .08 x 3 = $2,400
Compound Interest
Compounding interest means that interest earns interest. The future value of a dollar is its value at a time in the future given its present sum. The future value of a dollar is affected both by the interest rate and the time at which the dollar is received. For the discussion of the concepts of compounding and time value, let us define:
Then,
The future value of an investment compounded annually at rate i for n years is
where FVF(i,n)=T1(i,n) is the future value (compound amount) of $1 and can be found in Table 1.
Example 2
To illustrate the difference between simple and compound interest, assume that Nolan Company deposits $1,000 in the First Bank, where it will earn simple interest of 8% per year. It deposits another $1,000 in the Second Bank, where it will earn compound interest of 8% per year compounded annually. In both cases, Nolan will not withdraw any interest until 3 years from the date of deposit.
Simple interest:
$1,000 × .08 × 3 years = $240; the future value = $1,240
Compound interest:
Note: Simple interest uses the initial principal of $1,000 to compute the interest in all 3 years. Compound interest uses the accumulated balance (principal plus interest to date) at each year-end to compute interest in the succeeding year. This explains the larger balance in the compound interest account. Obviously, any rational investor would choose compound interest, if available, over simple interest. In the example above, compounding provides $20 of additional interest revenue. Simple interest usually applies only to short-term investments and debts that involve a time span of one year or less.
Example 3
You place $1,000 in a savings account earning 8 percent interest compounded annually. How much money will you have in the account at the end of 4 years?
From Table 1, the T1 for 4 years at 8 percent is 1.360. Therefore,
An excerpt from Table 1 is given over the page.
Table 1: The Future Value of $1.00 (Compound Amount of $1.00) (1 + i)n = FVF (i,n) = T1 (i,n)
Example 4
You invested a large sum of money in the stock of TLC Corporation. The company paid a $3 dividend per share. The dividend is expected to increase by 14 percent per year for the next 3 years. You wish to project the dividends for years 1 through 3.
Intrayear Compounding
Interest is often compounded more frequently than once a year. Banks, for example, compound interest quarterly, daily, or even continuously. If interest is compounded m times a year, then the general formula for solving the future value becomes
The formula reflects more frequent compounding (n x m) at a smaller interest rate per period (i/m). For example, in the case of semiannual compounding (m = 2), the above formula becomes
Example 5
You deposit $10,000 in an account offering an annual interest rate of 16 percent. You will keep the money on deposit for five years. The interest rate is compounded quarterly. The accumulated amount at the end of the fifth year is calculated as follows:
Where P = $10,000
Therefore,
Example 6
As the example shows, the more frequently interest is compounded, the greater the amount accumulated. This is true for any interest for any period of time. How often interest is compounded can substantially affect the rate of return. For example, an 8% annual interest compounded daily provides an 8.33% yield, or a difference of 0.33%. The 8.33% is the effective yield, frequently called annual percentage rate (APR). The annual interest rate (8%) is the nominal, stated, coupon, or face rate. When the compounding frequency is greater than once a year, the effective interest rate will always exceed the nominal rate.
The formula for calculating the effective interest rate or annual percentage rate (APR), in situations where the compounding frequency (n) is greater than once a year, is as follows.
APR = (1 + )m. To illustrate, if the stated annual rate is 8% compounded quarterly (or 2% per quarter), the effective annual rate is: Effective rate = (1 + .02)4 - 1 = (1.02)4 - 1 = 1.0824 - 1 = .0824 = 8.24%
Exhibit 4 shows how compounding for five different time periods affects the effective yield and the amount earned by an investment for one year.
Exhibit 4: Nominal and Effective Interest Rates with Different Compounding Periods
Note: Federal law requires the disclosure of interest rates on an annual basis in all contracts. That is, instead of stating the rate as “1% per month,” contracts must state the rate as “12% per year” if it is simple interest or “12.68% per year” if it is compounded monthly.
The Power of Compounding
The power of compounding is evident in two typical cases:
1. Start early.
2. Equally significant, the fact that a small difference in the interest rate makes a big difference in the future value amount.
Start Early
The current debate on Social Security reform provides a great context to illustrate the power of compounding. One proposed idea is for the government to give $1,000 to every citizen at birth. This gift would be deposited in an account that would earn interest tax-free until the citizen retires. Assuming the account earns a modest 8% annual return until retirement at age 65, the $1,000 would grow to $137,759. Why start so early? If the government waited until age 18 to deposit the money, it would grow to only $34,474 with annual compounding. That is, reducing the time invested by a third results in more than almost 75% reduction in retirement money (see Exhibit 5). The example illustrates the importance of starting early when the power of compounding is involved.
Exhibit 5: Future Values of When to Start
Impact of Interest Rate Difference
A small difference in the interest rate makes a big difference in the future value amount. To illustrate this point, assume that you had $1,000 in a tax-free retirement account. All the money is in stocks returning 12 percent or all the money in bonds earning 10 percent. Assuming reinvested profits and annually compounding, your investment will grow to $51,875 from bonds after 10 years. But your stocks, returning 2 percent more, would be worth $62,117, which implies a 2% higher interest would result in some 20% increase in the future value after 10 years. Exhibit 6 illustrates this impact.
Exhibit 6: Impact of Interest Rate Difference
Future Value of an Annuity
Two types of annuities are discussed below: the ordinary annuity (annuity in arrears) with the payment at the end of the year, and the annuity due (or annuity in advance) when the payment is made at the beginning of the year. In addition, the discussion examines the future difference in value between these two annuities.
Ordinary Annuity
An ordinary annuity is defined as a series of payments (or receipts) of a fixed amount for a specified number of periods. Each payment is assumed to occur at the end of the period. The future value of an annuity is a compound annuity which involves depositing or investing an equal sum of money at the end of each year for a certain number of years and allowing it to grow.
Then we can write
where FVF-OA(i,n) = T2(i,n) represents the future value of an ordinary annuity of $1 for n years compounded at i percent and can be found in Table 2.
Example 7
You wish to determine the sum of money you will have in a savings account at the end of 6 years by depositing $1,000 at the end of each year for the next 6 years. The annual interest rate is 8 percent. The T2(8%,6 years) is given in Table 2 as 7.336. Therefore,
Example 8
You deposit $30,000 semiannually into a fund for ten years. The annual interest rate is 8 percent. The amount accumulated at the end of the tenth year is calculated as follows:
where A = $30,000
Therefore,
An excerpt from Table 2 is given below.
Table 2: The Future Value of an ordinary Annuity of $1.00 (Compounded Amount of an ordinary Annuity of $1.00) = FVF-OA (i,n) = T2(i,n)
Example 9
Michael receives an annual annuity of $2,000 that is invested at 10 percent immediately upon receipt of the annual payment. At the end of the 5-year annuity, Michael will have accumulated the following sums.
Michael will have $12,210 at the end of the annual annuity if it is invested at 10 percent.
Annuity Due
The formula for computing the future value of an annuity due must take into consideration one additional year of compounding, since the payment comes at the beginning of the year. Therefore, the future-value formula must be modified to take this into consideration by compounding it for one more year:
Note: The future value of an annuity due of $1is found by subtracting 1 (one extra payment) from the future value of an ordinary annuity of $1 for one more period. That is:
Example 10
Using the same investment as Example 8, the future-value computations for the annuity due would be as follows:
Future-Value Difference between Ordinary Annuity and Annuity Due
The future-value difference between the ordinary annuity and an annuity due in our examples is $13,431 - $12,210 = $1,221. Generally speaking, an annuity due is preferable over an ordinary annuity, since an amount equivalent to an additional year of compounding is received.
Can a Computer Help?
Financial calculators marketed by several manufacturers (e.g., Hewlett-Packard, Sharp, Texas Instruments) have a “future (compound) value” function. Future value is also incorporated as a built-in function in spreadsheet programs. For example, Excel has a routine FV(rate,nper,pmt,pv,type), which calculates the future value of an investment based on periodic, constant payments and a constant interest rate. To calculate the future value of an annuity due, use the formula FV × (1 + interest).
What is Present Value - How Much Money is Worth Now?
Present value is the present worth of future sums of money. The process of calculating present values, or discounting, is actually the opposite of finding the compounded future value, compounding. In connection with present value calculations, the interest rate i is called the discount rate. The discount rate we use is more commonly called the cost of capital, which is the minimum rate of return required by the investor (to be discussed later in the book).
Recall that Fn = P (1+i)n
Therefore,
Where PVF(i,n) = T3(i,n) represents the present value of $1 and is given in Table 3 in the Appendix.
Example 11
You have been given an opportunity to receive $20,000 6 years from now. If you can earn 10 percent on your investments, what is the most you should pay for this opportunity? To answer this question, you must compute the present value of $20,000 to be received 6 years from now at a 10 percent rate of discount. F6 is $20,000, i is 10 percent, and n is 6 years. PVF(10%,6) = T3 (10%,6) from Table 3 is 0.565.
An excerpt from Table 3 is given below.
Table 3: The Present Value of $1.00 (Discounted Amount of $1.00) = PVF (i,n) = T3(i,n)
This means that you can earn 10 percent on your investment, and you would be indifferent to receiving $12,800 now or $20,000 6 years from today since the amounts are time equivalent. In other words, you could invest $12,800 today at 10 percent and have $20,000 in 6 years.
An Application
Suppose you purchase a building with a noninterest-bearing as a consideration. There is no established exchange price for the building, and the note had no ready market. The noninterest-bearing note should be recorded at its fair market value, which is the present value of the future cash flows discounted at the prevailing rate of interest.
Present Value of Mixed Streams of Cash Flows
The present value of a series of mixed payments (or receipts) is the sum of the present value of each individual payment. We know that the present value of each individual payment is the payment times the appropriate T3 value.
Example 12
You are thinking of starting a new product line that initially costs $32,000. Your annual projected cash inflows are:
| Year 1 | $10,000 |
| Year 2 | $20,000 |
| Year 3 | $5,000 |
If you must earn a minimum of 10 percent on your investment, should you undertake this new product line?
The present value of this series of mixed streams of cash inflows is calculated as follows:
Since the present value of your projected cash inflows is less than the initial investment, you should not undertake this project.
Present Value of an Annuity
Interest received from bonds, pension funds, and insurance obligations all involve annuities. To compare these financial instruments, we need to know the present value of each. The present value of an annuity is a method of discounting an annuity to determine its worth in present-day dollars. It shows the amount of the lump-sum payment that would have to be received today to equal the annuity.
This analysis accommodates two types of annuities – ordinary annuities in which the equal payment comes at the end of the year, and annuities due in which the equal payment is made at the beginning of the year.
Ordinary Annuity (OA)
The present value of an ordinary annuity (Pn) can be found by using the following equation:
where PVF-OA(i,n) = T4(i,n) represents the present value of an annuity of $1 discounted at i percent for n years and is found in Table 4.
Example 13
Assume that the cash inflows in Example 11 form an annuity of $10,000 for 3 years. Then the present value is
An excerpt from Table 4 is given opposite.
Table 4: The Present Value of an ordinary annuity of $1.00 (Discounted Amount of an ordinary annuity of $1.00) = PVF-OA (i.n) = T4(i,n)
Example 14
Judy has been offered a 5-year annuity of $2,000 a year or a lump sum payment today. Since Judy wants to invest the money in a security paying 10 percent interest, she decides to take the lump-sum payment today. How large should the lump-sum payment be to equal the 5-year, $2,000 annual annuity at 10 percent interest?
Using the present-value interest factor for an ordinary annuity (PVIFA) of 5 years paying 10 percent interest provides an easy solution:
The lump-sum payment today for Judy should be $7,598 to equal the 5-year, $2000 annual annuity at 10 percent interest.
Annuity Due (AD)
The formula for the computation of the present value of an annuity due must take into consideration one additional year of compounding since the payment occurs at the beginning of the year. Therefore, the future formula must be modified as follows:
using the present-value factor for an ordinary annuity T4 or present-value factor for an annuity due = PVF-AD(i,n) = T5(i,n).
Example 15
Using the same information as in Example 14 and the present-value actor for an annuity T4 of 5 years paying 10 percent interest modified for an annuity due:
Or alternatively,
An excerpt from Table 5 is given below.
Table 5: The Present Value of an annuity due of $1.00 (Discounted Amount of an annuity due of $1.00) = PVF-AD (i,n) = T5(i,n)
Note: The present value of an annuity due of $1can also be found by adding 1 (one extra payment) to the future value of an ordinary annuity of $1 for one less period. That is: Pn = A × [T4(i,n-1) + 1]. In this example, then, Pn = $2,000 × [T4(10%, 4 years) + 1] = $2,000(3.170 + 1) = $2,000(4.170) = $8,340.
Present-Value Difference between Ordinary Annuity and Annuity Due
The present-value difference between an ordinary annuity and an annuity due is substantial. In the above examples, the difference is $8,340 - $7,582 = $756. If an annuity due and an ordinary annuity have the same number of equal payments and the same interest rates. Then the present value of the annuity due is greater than the present value of the ordinary annuity, since an additional year of compounding is essentially received.
Can a Computer Help?
Computer software can be extremely helpful in making present-value calculations. For example, PV(rate,nper,pmt,fv,type) of Excel determines the present value of an investment, based on a series of equal payments, discounted at a periodic interest rate over the number of periods, To calculate the present value of an annuity due, use the following formula: PV x (1 + interest). Financial calculators can do this too.
Perpetuities
Some annuities go on forever, called perpetuities. An example of a perpetuity is preferred stock which yields a constant dollar dividend indefinitely. The present value of a perpetuity is found as follows:
Example 16
Assume that a perpetual bond has an $80-per-year interest payment and that the discount rate is 10 percent. The present value of this perpetuity is:
Deferred Annuities
A deferred annuity is an annuity in which the payments begin after a specified number of periods. A deferred annuity does not begin to produce payments until two or more periods have expired. For example, “an ordinary annuity of six annual payments deferred
4 years” means that no payments will occur during the first 4 years, and that the first of the six payments will occur at the end of the fifth year. “An annuity due of six annual payments deferred 4 years” means that no payments will occur during the first 4 years, and that the first of six payments will occur at the beginning of the fifth year.
Future Value of a Deferred Annuity
Computing the future value of a deferred annuity is relatively straightforward. Because there is no accumulation or investment on which interest may accrue, the future value of a deferred annuity is the same as the future value of an annuity not deferred. That is, computing the future value simply ignores the deferred period.
Example 17
Assume that Allison Corporation plans to purchase a land site in 6 years for the construction of its new corporate headquarters. Because of cash flow problems, Allison budgets deposits of $80,000, on which it expects to earn 6% annually, only at the end of the fourth, fifth, and sixth periods. What future value will Allison have accumulated at the end of the sixth year? Allison determines the value accumulated by using the standard formula for the future value of an ordinary annuity:
Present Value of a Deferred Annuity
Computing the present value of a deferred annuity must recognize the interest that accrues on the original investment during the deferral period. Two options are available to compute the present value of a deferred annuity, which are illustrated below.
Example 18
Joy has developed and copyrighted an online CPE course for CPAs. He agrees to sell the copyright to a CPE provider for six annual payments of $5,000 each. The payments will begin 5 years from today. Given an annual interest rate of 8%, what is the present value of the six payments?
Option 1: First, compute the present value of an ordinary annuity of $1 as if the payments had occurred for the entire period. We then subtract the present value of payments that were not received during the deferral period. We are left with the present value of the payments actually received subsequent to the deferral period. Use only Table 4, as shown, step by step, below.
Step 1:
Obtain the present value of an ordinary annuity of $1 for total periods (10) [number of payments (6) plus number of deferred periods (4)] at 8%
Step 2:
Get the present value of an ordinary annuity of $1 for the number of deferred periods (4) at 8%
Step 3:
Take the difference.
Step 4:
Multiply an annual payment by this difference to yield the present value of six annual payments of $5,000 deferred 4 periods
The subtraction of the present value of an annuity of $1 for the deferred periods eliminates the nonexistent payments during the deferral period. It converts the present value of an ordinary annuity of $1.00 for 10 periods to the present value of 6 annual payments of $1.00, deferred 4 periods. Symbolically,
Option 2: Use both Table 4 and Table 3 (in this sequence) to compute the present value of the 6 payments. You can first discount the annuity 6 periods. However, because the annuity is deferred 4 periods, you must treat the present value of the annuity as a future amount to be discounted another 4 periods. Calculation using formulas would be done in two steps, as follows.
Step 1:
Present value of an ordinary annuity = $5,000 T4(8%, 6 years) = $5,000 (4.623) = $23,115
Step 2:
Present value of a single sum = $23,115 T3(8%, 4 years) = $23,115 (.735) = $16,990
The present value of $16,990 computed by using both options should be the same, except for rounding errors.
What are the Applications of Future Values and Present Values?
Future and present values have numerous applications in financial and investment decisions. Each of these applications is presented below.
Deposits to Accumulate a Future Sum (or Sinking Fund)
A financial manager might wish to find the annual deposit (or payment) that is necessary to accumulate a future sum. To find this future amount (or sinking fund) we can use the formula for finding the future value of an annuity.
Solving for A, we obtain:
Example 19
You wish to determine the equal annual end-of-year deposits required to accumulate $5,000 at the end of 5 years in a fund. The interest rate is 10 percent. The annual deposit is:
In other words, if you deposit $819 at the end of each year for 5 years at 10 percent interest, you will have accumulated $5,000 at the end of the fifth year.
Example 20
You need a sinking fund for the retirement of a bond 30 years from now. The interest rate is 10 percent. The annual year-end contribution needed to accumulate $1,000,000 is
Example 21
A company needs to create a $15 million sinking fund at the end of 8 years at 10 percent interest to retire $15 million in outstanding bonds. The amount that should be deposited in the account at the end of each year is:
Thus, if the company deposits $1,311,647.42 at the end of each year for the next 8 years in an account earning 10 percent interest, it will accumulate the $15 million needed to retire the bonds.
Amounts of periodic withdrawals
You might want to determine how much you can withdraw with present savings earning interests over time.
Example 22
Jack and Jill Smiths have saved $40,000 to finance their daughter’s college education. They deposited the money in the Downey Savings and Loan Association, where it earns 4% interest compounded semiannually. What equal amounts can their daughter withdraw at the end of every 6 months during her 4 college years, without exhausting the fund?
Determining the answer by simply dividing $40,000 by 8 withdrawals is wrong, since this ignores the interest earned on the money remaining on deposit. She must consider that interest is compounded semiannually at 2% (4%/2) for 8 periods (4 years x 2). Thus, using the same present value of an ordinary annuity formula, she determines the amount of each withdrawal that she can make as follows.
Determining the Number of Periods Required
You might want to know how long it will take your money to double or how long it will take to reach your monetary goal.
Case 1: Single-Deposit Investment. The number of years it will take to reach a certain future sum can be computed as follows:
Solving for T1 we obtain:
where Fn = future value in period n, and P = a present sum of money or base-period value.
Example 23
At an interest rate of 12 percent, you want to know how long it will take for your money to double. The value is computed as follows:
Therefore,
From Table 1 in the Appendix, a T1 of 2 at 12 percent gives n = almost 6 years. It will take almost 6 years. Alternatively, from Table 3 in the Appendix, a T3 of 0.5 at 12 percent also yields n = almost 6 years.
Example 24
You want to have $250,000. You have $30,000 to invest. The interest rate is 12 percent. The number of years it will take to reach your goal is computed below:
From Table 1, n is approximately 18.5 years.
Case 2: Equal Periodic Deposits. The number of years it will take for equal periodic deposits to reach a certain future sum can be computed as follows:
Solving for T2, we obtain:
where Fn = future value in period n, and P = a present sum of money or base-period value.
Example 25
You want $500,000 in the future. The interest rate is 10 percent. You can deposit $80,000 each year. The number of years it will take to accomplish this objective is computed below:
From Table 2, we see that n is approximately 5 years.
Can a Computer Help?
More exact answers can be obtained using computer software. For example, Excel has a routine NPER(rate,pmt,pv,fv,type), which calculates the number of periods for an investment based on periodic, constant payments and a constant interest rate. Furthermore, many financial calculators contain preprogrammed formulas and perform many present-value and future-value applications.
Computing Interest Rate
You might want to know about the interest rate you are charged on a loan or the interest rate you must earn on your annual deposits to reach your investment goal.
Case 1: Single-Deposit Loan. The interest rate you are charged on a loan can be computed as follows:
Solving for T1 we obtain
where Fn = future value in period n, and P = a present sum of money or base-period value.
Example 26
You agree to pay back $3,000 in 6 years on a $2,000 loan made today. You are being charged an interest rate of 7 percent, calculated as follows:
From Table 1 in the Appendix, a T1 of 1.5 at 6 years is at 7 percent.
Case 2: Equal Periodic Deposits. The interest rate you must earn on your annual deposits to reach your investment goal can be computed as follows:
Solving for T2(i,n), we obtain
where Fn = future value in period n, and P = a present sum of money or base-period value.
Example 27
You want to have $500,000 accumulated in a pension plan after 9 years. You deposit $30,000 per year. The interest rate you must earn is computed below.
From Table 2, the interest rate is approximately 15 percent.
