time value of money - higher ed ebooks & digital … studying appendix 1, †you should be able...
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After studying Appendix 1,you should be able to:
•1 Explain how compoundinterest works.
•2 Use future value andpresent value tables toapply compound interest toaccounting transactions.
Time Value of Money1Appendix
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Time value of money is widely used in business to measure today’s value of future cashoutflows or inflows and the amount to which liabilities (or assets) will grow when com-pound interest accumulates.
In transactions involving the borrowing and lending of money, the borrower usuallypays interest. In effect, interest is the time value of money. The amount of interest paidis determined by the length of the loan and the interest rate.
However, interest is not restricted to loans made to borrowers by banks. Invest-ments (particularly, investments in debt securities and savings accounts), installmentsales, and a variety of other contractual arrangements all include interest. In all cases,the arrangement between the two parties—the note, security, or purchase agreement—creates an asset in the accounting records of one party and a corresponding liability inthe accounting records of the other. All such assets and liabilities increase as interest isearned by the asset holder and decrease as payments are made by the liability holder.
COMPOUND INTEREST CALCULATIONSCompound interest is a method of calculating the time value of money in which inter-est is earned on the previous periods’ interest. That is, interest for the period is added tothe account balance and interest is earned on this new balance in the next period. Incomputing compound interest, it’s important to understand the difference between theinterest period and the interest rate:
• The interest period is the time interval between interest calculations.• The interest rate is the percentage that is multiplied by the beginning-of-period
balance to yield the amount of interest for that period.
The interest rate must agree with the interest period. For example, if the interest periodis one month, then the interest rate used to calculate interest must be stated as a percent-age ‘‘per month.’’
When an interest rate is stated in terms of a time period that differs from the interestperiod, the rate must be adjusted before interest can be calculated. For example, supposethat a bank advertises interest at a rate of 12% per year compounded monthly. Here, theinterest period would be one month. Since there are 12 interest periods in one year, theinterest rate for one month is one-twelfth the annual rate, or 1%. In other words, if therate statement period differs from the interest period, the stated rate must be divided bythe number of interest periods included in the rate statement period. A few examples ofadjusted rates follow:
Stated Rate Adjusted Rate for Computations
12% per year compounded semiannually 6% per six-month period (12%/2)12% per year compounded quarterly 3% per quarter (12%/4)12% per year compounded monthly 1% per month (12%/12)
If an interest rate is stated without reference to a rate statement period or aninterest period, assume that the period is one year. For example, both ‘‘12%’’ and‘‘12% per year’’ should be interpreted as 12% per year compounded annually.
Compound interest means that interest is computed on the original amount plusundistributed interest earned in previous periods. The simplest compound interest calcu-lation involves putting a single amount into an account and adding interest to it at theend of each period. CORNERSTONE A1-1 (p. 700) shows how to compute future valuesusing compound interest.
O B J E C T I V E•1Explain how compound interest works.
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As you can see in Cornerstone A1-1, the balance in the account continues to groweach month by an increasing amount of interest. The amount of monthly interestincreases because interest is compounded. In other words, interest is computed on accu-mulated interest as well as on principal. For example, February interest of $100.50 con-sists of $100 interest on the $20,000 principal and 50¢ interest on the $100 Januaryinterest ($100 3 0.005 ¼ 50¢).
In Cornerstone A1-1, the compound interest only amounts to $1.50. That mightseem relatively insignificant, but if the investment period is sufficiently long, the amountof compound interest grows large even at relatively small interest rates. For example,suppose your parents invested $1,000 at ½% per month when you were born with theobjective of giving you a university graduation present at age 21. How much would thatinvestment be worth after 21 years? The answer is $3,514. In 21 years, the compoundinterest is $2,514—more than 2½ times the original principal. Without compounding,interest over the same period would have been only $1,260.
The amount to which an account will grow when interest is compounded is thefuture value of the account. Compound interest calculations can assume two fundamen-tally different forms:
• calculations of future values• calculations of present values
As shown, calculations of future values are projections of future balances based on pastand future cash flows and interest payments. In contrast, calculations of present valuesare determinations of present amounts based on expected future cash flows.
C O R N E R S T O N EA 1 - 1
Computing Future Values UsingCompound Interest
Information:
An investor deposits $20,000 in a savings account on January 1, 2015. The bank pays interest of 6% per yearcompounded monthly.
Why:
When deposits earn compound interest, interest is earned on the interest.
Required:
Assuming that the only activity in the account is the deposit of interest at the end of each month, how much moneywill be in the account after the interest payment on March 31, 2015?
Solution:
Monthly interest will be ½% (6% per year/12 months).
Account balance, 1/1/15 $20,000.00January interest ($20,000.00 3 ½%) 100.00Account balance, 1/31/15 20,100.00February interest ($20,100.00 3 ½%) 100.50Account balance, 2/28/15 20,200.50March interest ($20,200.50 3 ½%) 101.00Account balance, 3/31/15 $20,301.50
Note: Here, interest was the only factor that altered the account balance after the initial deposit. In more complexsituations, the account balance is changed by subsequent deposits and withdrawals as well as by interest.Withdrawals reduce the balance and, therefore, the amount of interest in subsequent periods. Additional depositshave the opposite effect, increasing the balance and the amount of interest earned.
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PRESENT VALUE OF FUTURE CASH FLOWSWhenever a contract establishes a relationship between an initial amount borrowed orloaned and one or more future cash flows, the initial amount borrowed or loaned is thepresent value of those future cash flows. The present value can be interpreted in twoways:
• From the borrower’s viewpoint, it is the liability that will be exactly paid by thefuture payments.
• From the lender’s viewpoint, it is the receivable balance that will be exactly satisfiedby the future receipts.
For understanding cash flows, cash flow diagrams that display both the amounts andthe times of the cash flows specified by a contract can be quite helpful. In these dia-grams, a time line runs from left to right. Inflows are represented as arrows pointingupward and outflows as arrows pointing downward. For example, suppose that HilliardCorporation borrows $100,000 from Citizens Bank of New Liskeard on January 1,2015. The note requires three $38,803.35 payments, one each at the end of 2015,2016, and 2017, and includes interest at 8% per year. The cash flows for Hilliard areshown in Exhibit A1-1.
The calculation that follows shows, from the borrower’s perspective, the relationshipbetween the amount borrowed (the present value) and the future payments (future cashflows) required by Hilliard’s note.
Amount borrowed, 1/1/15 $100,000.00Add: 2015 interest ($100,000.00 3 0.08) 8,000.00Subtract payment on 12/31/15 (38,803.35)Liability at 12/31/15 69,196.65Add: 2016 interest ($69,196.65 3 0.08) 5,535.73Subtract payment on 12/31/16 (38,803.35)Liability at 12/31/16 35,929.03Add: 2017 interest ($35,929.03 3 0.08) 2,874.32Subtract payment on 12/31/17 (38,803.35)Liability at 12/31/17 $ 0.00
Present value calculations like this one are future value calculations in reverse. Here, thethree payments of $38,803.35 exactly pay off the liability created by the note. Becausethe reversal of future value calculations can present a burdensome and sometimes diffi-cult algebraic problem, shortcut methods using tables have been developed (see ExhibitsA1-7, A1-8, A1-9, and A1-10, pp. 717–720, discussed later in this appendix).
Exhibit A1-1Cash Flow Diagram
$38,803.35 $38,803.35 $38,803.35
$100,000
1/1/15 12/31/15 12/31/16 12/31/17
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Interest and the Frequency of CompoundingThe number of interest periods into which a compound interest problem is divided canmake a significant difference in the amount of compound interest. For example, assumethat you are evaluating four 1-year investments, each of which requires an initial$10,000 deposit. All four investments earn interest at a rate of 12% per year, but theyhave different compounding periods. The data in Exhibit A1-2 show the impact of com-pounding frequency on future value. Investment D, which offers monthly compound-ing, accumulates $68 more interest by the end of the year than investment A, whichoffers only annual compounding.
FOUR BASIC COMPOUND INTERESTPROBLEMSAny present value or future value problems can be broken down into one or more of thefollowing four basic problems:
• computing the future value of a single amount• computing the present value of a single amount• computing the future value of an annuity• computing the present value of an annuity
Computing the Future Value of a Single AmountIn computing the future value of a single amount, the following elements are used:
• f: the cash flow• FV: the future value• n: the number of periods between the cash flow and the future value• i: the interest rate per period
To find the future value of a single amount, establish an account for f dollars and addcompound interest at i percent to that account for n periods:
FV ¼ (f )(1 þ i)n
The balance of the account after n periods is the future value.Because people frequently need to compute the future value of a single amount,
tables have been developed to make it easier. Therefore, instead of using the formulaabove, you could use the future value table in Exhibit A1-7 (p. 717), where M1 is themultiple that corresponds to the appropriate values of n and i:
FV ¼ (f )(M1)
For example, suppose Allied Financial loans $200,000 at a rate of 6% per year com-pounded annually to an auto dealership dealer for four years. Exhibit A1-3 shows how
Exhibit A1-2Effect of Interest Periods on Compound Interest
Investment Interest Period I N Calculation of Future Amount in One Year*
A 1 year 12% 1 ($10,000 3 1.12000) ¼$11,200B 6 months 6% 2 ($10,000 3 1.12360) ¼ 11,236C 1 quarter 3% 4 ($10,000 3 1.12551) ¼ 11,255D 1 month 1% 12 ($10,000 3 1.12683) ¼ 11,268
*The multipliers (1.12 for Investment A, 1.12360 for investment B, etc.) are taken from the future value table in Exhibit A1-7(p. 717).
O B J E C T I V E•2Use future value and present value tablesto apply compound interest to accountingtransactions.
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to compute the future value (FV) at the end of the four years—the amount that will berepaid. Assuming Allied’s viewpoint (the lender’s), using a compound interest calcula-tion, the unknown future value (FV) would be found as follows:
Amount loaned $200,000.00First year’s interest ($200,000.00 3 0.06) 12,000.00Loan receivable at end of first year 212,000.00Second year’s interest ($212,000.00 3 0.06) 12,720.00Loan receivable at end of second year 224,720.00Third year’s interest ($224,720.00 3 0.06) 13,483.20Loan receivable at end of third year 238,203.20Fourth year’s interest ($238,203.20 3 0.06) 14,292.19Loan receivable at end of the fourth year $252,495.39
As you can see, the amount of interest increases each year. This growth is the effect ofcomputing interest for each year based on an amount that includes the interest earned inprior years.
The shortcut calculation, using the future value table (Exhibit A1-7, p. 717), wouldbe as follows:
FV ¼ (f )(M1)¼ ($200,000)(1:26248)¼ $252,496
You can find M1 at the intersection of the 6% column (i ¼ 6%) and the fourth row (n ¼4) or by calculating 1.064. This multiple is the future value of the single amount afterhaving been borrowed (or invested) for four years at 6% interest. The future value of$200,000 is 200,000 times the multiple.
Note that there is a difference between the answer ($252,495.39) developed in thecompound interest calculation and the answer ($252,496) determined using the futurevalue table. This is because the numbers in the table have been rounded to five decimal pla-ces. If they were taken to eight digits (1.064 ¼ 1.26247696), the two answers would beequal. CORNERSTONE A1-2 shows how to compute the future value of a single amount.
Exhibit A1-3Future Value of a Single Amount: An Example
C O R N E R S T O N EA 1 - 2
Computing Future Value of a Single Amount
Information:
Kitchener Company sells an unneeded factory site for $200,000 on July 1, 2015. Kitchener expects to purchasea different site in 18 months so that it can expand into a new market. Meanwhile, Kitchener decides to invest the$200,000 in a money market fund that is guaranteed to earn 6%per year compounded semiannually (3% per six-month period).
(Continued)
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Computing the Present Value of a Single AmountIn computing the present value of a single amount, the following elements are used:
• f: the future cash flow• PV: the present value• n: the number of periods between the present time and the future cash flow• i: the interest rate per period
In present value problems, the interest rate is sometimes called the discount rate.
Why:
The future value of a single amount is the original cash flow pluscompound interest as of a specific future date.
Required:
1. Draw a cash flow diagram for this investment from Kitchener’s perspective.2. Calculate the amount of money in the money market fund on December 31, 2015, and prepare the journal
entry necessary to recognize interest income.3. Calculate the amount of money in the money market fund on December 31, 2016, and prepare the journal
entry necessary to recognize interest income.
Solution:
1.
15 15 16 16
2. Because we are calculating the value at 12/31/15, there is only one period:
FV ¼ (f )(FV of a Single Amount, 1 period, 3%)¼ ($200,000)(1:03)¼ $206,000
The excess of the amount of money over the original deposit is the interest earned from July 1 throughDecember 31, 2015.
Dec. 31, 2015 Cash 6,000Interest Income 6,000
(Record interest income)
3. FV ¼ (f )(FV of a Single Amount, 2 periods, 3%)¼ ($206,000)(1:032)¼ $218,545:40
The interest income for the year is the increase in the amount of money during 2016, which is $12,545.40($218,545.40 � $206,000). The journal entry to record interest income would be as follows:
Dec. 31, 2016 Cash 12,545.40Interest income 12,545.40
(Record interest income)
C O R N E R S T O N EA 1 - 2
(continued)
Assets 5 Liabilities 1
Shareholders’Equity
(Interest Income)þ6,000 þ6,000
Assets 5Liabilities1
Shareholders’Equity
(Interest Income)þ12,545.40 þ12,545.40
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To find the present value of a single amount, use the following equation:
PV ¼ f
(1 þ i)n
You could use the present value table in Exhibit A1-8 (p. 718), where M2 is the multiplefrom Exhibit A1-8 that corresponds to the appropriate values of n and i:
PV ¼ (f )(M2)
Suppose Marathon Oil has purchased property on which it plans to develop oilwells. The seller has agreed to accept a single $150,000,000 payment three years fromnow, when Marathon expects to be selling oil from the field. Assuming an interest rateof 7% per year, the present value of the amount to be received in three years from theborrower’s perspective can be calculated as shown in Exhibit A1-4.
The shortcut calculation, using the present value table (Exhibit A1-8, p. 718), wouldbe as follows:
PV ¼ (f )(M2)¼ ($150,000,000)(0:81630)¼ $122,445,000
You can find M2 at the intersection of the 7% column (i ¼ 7%) and the third row(n ¼ 3) in Exhibit A1-8 (p. 718) or by calculating [1/(1.07)3]. This multiple is the pres-ent value of a $1 cash inflow or outflow in three years at 7%. Thus, the present value of$150,000,000 is $150,000,000 times the multiple.
Although the future value calculation cannot be used to determine the presentvalue, it can be used to verify that the present value calculated by using the table is cor-rect. The following calculation is proof for the present value problem:
Calculated present value (PV) $122,445,000First year’s interest ($122,445,000 3 0.07) 8,571,150Loan payable at end of first year 131,016,150Second year’s interest ($131,016,150 3 0.07) 9,171,131Loan payable at end of second year 140,187,281Third year’s interest ($140,187,281 3 0.07) 9,813,110Loan payable at end of the third year (f ) $150,000,391
Again, the $391 difference between the amount here and the assumed $150,000,000cash flow is due to rounding.
When interest is compounded on the calculated present value of $122,445,000,then the present value calculation is reversed and we return to the future cash flowof $150,000,000. This reversal proves that $122,445,000 is the correct present value.CORNERSTONE A1-3 (p. 706) shows how to compute the present value of a single amount.
Exhibit A1-4Present Value of a Single Amount: An Example
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Computing the Future Value of an AnnuitySo far, we have been discussing problems that involve a single cash flow. However, thereare also instances of multiple cash flows one period apart. An annuity is a number ofequal cash flows: one to each interest period. For example, an investment in a securitythat pays $1,000 to an investor every December 31 for 10 consecutive years is an annu-ity. A loan repayment schedule that calls for a payment of $367.29 on the first day ofeach month can also be considered an annuity. (Although the number of days in amonth varies from 28 to 31, the interest period is defined as one month without regardto the number of days in each month.)
In computing the future value of an annuity, the following elements are used:
• f : the amount of each repeating cash flow• FV: the future value after the last (nth) cash flow• n: the number of cash flows• i: the interest rate per period
C O R N E R S T O N EA 1 - 3
Computing Present Value of a Single Amount
Information:
On October 1, 2015, Adelsman Manufacturing Company sold a new machine to Raul Inc. The machine repre-sented a new design that Raul was eager to place in service. Since Raul was unable to pay for the machine on thedate of purchase, Adelsman agreed to defer the $60,000 payment for 15 months. The appropriate rate of interestin such transactions is 8% per year compounded quarterly (2% per three-month period).
Why:
The present value of a single cash flow is the original cash flow that must be invested to produce a known value ata specific future date.
Required:
1. Draw the cash flow diagram for this deferred-payment purchase from Raul’s (the borrower’s) perspective.2. Calculate the present value of this deferred-payment purchase.3. Prepare the journal entry necessary to record the acquisition of the machine.
Solution:
1.
1515 16 16 16 16
2. FV ¼ (f )(FV of a Single Amount, 5 periods, 2%)¼ ($60,000)(0:90573)¼ $54,344
3.Oct. 1, 2015 Equipment 54,344
Note Payable 54,344(Record purchase of equipment)
Assets 5 Liabilities 1Shareholders’
Equityþ54,344 þ54,344
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To find the future value of an annuity, use the following equation:
FV ¼ (f )(1 þ i)n � 1
i
� �
Alternatively, you could use the future value table in Exhibit A1-9 (p. 719), whereM3 is the multiple from Exhibit A1-9 that corresponds to the appropriate values of nand i:
FV ¼ (f )(M3)
Assume that CIBC wants to advertise a new savings program to its customers.The savings program requires the customers to make four annual payments of $5,000each, with the first payment due three years before the program ends. CIBC advertisesa 6% interest rate compounded annually. The future value of this annuity immediatelyafter the fourth cash payment from the investor’s perspective is shown in Exhibit A1-5.
Note that the first period in Exhibit A1-5 is drawn with a dotted line. When usingannuities, the time-value-of-money model assumes that all cash flows occur at the end ofa period. Therefore, the first cash flow in the future value of an annuity occurs at the endof the first period. However, since interest cannot be earned until the first deposit hasbeen made, the first period is identified as a no-interest period.
The future value (FV) can be computed as follows:
Interest for first period ($0 3 6%) $ 0.00First deposit 5,000.00Investment balance at end of first year 5,000.00Second year’s interest ($5,000.00 3 0.06) 300.00Second deposit 5,000.00Investment balance at end of second year 10,300.00Third year’s interest ($10,300.00 3 0.06) 618.00Third deposit 5,000.00Investment balance at end of third year 15,918.00Fourth year’s interest ($15,918.00 3 0.06) 955.08Fourth deposit 5,000.00Investment at end of fourth year $21,873.08
This calculation shows that the lender has accumulated a future value (FV) of $21,873.08by the end of the fourth period, immediately after the fourth cash investment.
The shortcut calculation, using the future value table (Exhibit A1-9, p. 719), wouldbe as follows:
FV ¼ (f )(M3)¼ ($5,000)(4:37462)¼ $21,873
Exhibit A1-5Future Value of an Annuity: An Example
Appendix 1 Time Value of Money 707
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You can find M3 at the intersection of the 6% column (i ¼ 6%) and the fourth row (n ¼ 4)in Exhibit A1-9 (p. 719) or by calculating (1.064 � 1)/0.06. This multiple is the future valueof an annuity of four cash flows of $1 each at 6%. The future value of an annuity of $5,000cash flows is $5,000 times the multiple. Thus, the table allows us to calculate the future valueof an annuity by a single multiplication, no matter how many cash flows are involved. COR-NERSTONE A1-4 shows how to compute the future value of an annuity.
Present Value of an AnnuityIn computing the present value of an annuity, the following elements are used:
• f : the amount of each repeating cash flow• PV: the present value of the n future cash flows• n: the number of cash flows and periods• i: the interest (or discount) rate per period
C O R N E R S T O N EA 1 - 4
Computing Future Value of an Annuity
Information:
Greg Smith is a lawyer and CA specializing in retirement and estate planning. One of Greg’s clients, the owner ofa large farm, wants to retire in five years. To provide funds to purchase a retirement annuity from London Life atthe date of retirement, Greg asks the client to give him annual payments of $170,000, which Greg will deposit ina special fund that will earn 7% per year.
Why:
The future value of an annuity is the value of a series of equal cash flows made at regular intervals with compoundinterest at some specific future date.
Required:
1. Draw the cash flow diagram for the fund from Greg’s client’s perspective.2. Calculate the future value of the fund immediately after the fifth deposit.3. If Greg’s client needs $1,000,000 to purchase the annuity, how much must be deposited every year?
Solution:
1.
2. FV ¼ (f )(FV of an Annuity , 5 periods, 7%)¼ ($170,000)(5:75074)¼ $977,626
3. In this case, the future value is known, but the annuity amount (f ) is not:
1,000,000 ¼ (f )(FV of an Annuity , 5 periods, 7%)1,000,000 ¼ (f )(5:75074)
f ¼ 1,000,000=5:75074f ¼ $173,890:66
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To find the present value of an annuity, use the following equation:
PV ¼ (f )1 � 1
(1 þ i)n
i
You could also use the present value table in Exhibit A1-10 (p. 720), where M4 is themultiple from Exhibit A1-10 that corresponds to the appropriate values of n and i:
PV ¼ (f )(M4)
For example, assume that Xerox Corporation purchased a new machine for its man-ufacturing operations. The purchase agreement requires Xerox to make four equallyspaced payments of $24,154 each. The interest rate is 8% compounded annually and thefirst cash flow occurs one year after the purchase. Exhibit A1-6 shows how to determinethe present value of this annuity from Xerox’s (the borrower’s) perspective. Note that thesame concept applies to both the lender’s and borrower’s perspectives.
The shortcut calculation, using the present value table (Exhibit A1-10, p. 720), wouldbe as follows:
PV ¼ (f )(M4)¼ ($24,154)(3:31213)¼ $80,001:19
You can find M4 at the intersection of the 8% column (i ¼ 8%) and the fourth row (n ¼ 4)in Exhibit A1-10 or by solving for [1 � (1/1.084)]/0.08. This multiple is the presentvalue of an annuity of four cash flows of $1 each at 8%. The present value of an annuityof four $24,154 cash flows is $24,154 times the multiple.
Again, although the compound interest calculation is not used to determine thepresent value, it can be used to prove that the present value found using the table is cor-rect. The following calculation verifies the present value in the problem:
Calculated present value (PV) $ 80,001.19Interest for first year ($80,001.19 3 0.08) 6,400.10Less: First cash flow (24,154.00)Balance at end of first year 62,247.29Interest for second year ($62,247.29 3 0.08) 4,979.78Less: Second cash flow (24,154.00)Balance at end of second year 43,073.07Interest for third year ($43,073.07 3 0.08) 3,445.85Less: Third cash flow (24,154.00)Balance at end of third year 22,364.92Interest for fourth year ($22,364.92 3 0.08) 1,789.19Less: Fourth cash flow (24,154.00)Balance at end of fourth year $ 0.11
This proof uses a compound interest calculation that is the reverse of the present valueformula. If the present value (PV) calculated with the formula is correct, then the proof
Exhibit A1-6Present Value of An Annuity: An Example
Appendix 1 Time Value of Money 709
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should end with a balance of zero immediately after the last cash flow. This proofends with a balance of $0.11 because of rounding in the proof itself and in the table inExhibit A1-10 (p. 720).
CORNERSTONE A1-5 shows how to compute the present value of an annuity.
SUMMARY OF LEARNING OBJECTIVES
LO1. Explain how compound interest works.• In transactions involving the borrowing and lending of money, it is custom-
ary for the borrower to pay interest.• With compound interest, interest for the period is added to the account and
interest is earned on the total balance in the next period.• Compound interest calculations require careful specification of the interest
period and the interest rate.
C O R N E R S T O N EA 1 - 5
Computing Present Value of an Annuity
Information:
Windsor Builders purchased a subdivision site from the Royal Bank on January 1, 2015. Windsor gave the bankan installment note. The note requires Windsor to make four annual payments of $600,000 each on December31 of each year, beginning in 2015. Interest is computed at 9%.
Why:
The present value of an annuity is the value of a series of equal future cash flows made at regular intervals withcompound interest discounted back to today.
Required:
1. Draw the cash flow diagram for this purchase from Windsor’s perspective.2. Calculate the cost of the land as recorded by Windsor on January 1, 2015.3. Prepare the journal entry that Windsor will make to record the purchase of the land.
Solution:
1.
4
15 15 16 17 18
2. PV ¼ (f )(PV of an Annuity , 4 periods, 9%)¼ ($600,000)(3:23972)¼ $1,943,832
3.Jan. 1, 2015 Land 1,943,832
Notes Payable 1,943,832(Record purchase of land)
Assets 5 Liabilities 1Shareholders’
Equityþ1,943,832 þ1,943,832
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LO2. Use future value and present value tables to apply compound interest to accountingtransactions.• Cash flows are described as either
• single cash flows, or• annuities.
• An annuity is a number of equal cash flows made at regular intervals.• All other cash flows are a series of one or more single cash flows.• Accounting for such cash flows may require
• calculation of the amount to which a series of cash flows will grow when interest iscompounded (i.e., the future value) or
• the amount a series of future cash flows is worth today after taking into accountcompound interest (i.e., the present value).
C O R N E R S T O N E SFOR APPENDIX 1
CORNERSTONE A1-1 Computing future values using compound interest (p. 700)
CORNERSTONE A1-2 Computing future value of a single amount (p. 703)
CORNERSTONE A1-3 Computing present value of a single amount (p. 706)
CORNERSTONE A1-4 Computing future value of an annuity (p. 708)
CORNERSTONE A1-5 Computing present value of an annuity (p. 710)
KEY TERMS
Annuity (p. 706)Compound interest (p. 699)Future value (p. 700)Interest period (p. 699)
Interest rate (p. 699)Present value (p. 701)Time value of money (p. 699)
DISCUSSION QUESTIONS
1. Why does money have a time value?2. Describe the four basic time-value-of-money problems.3. How is compound interest computed? What is a future value? What is a present value?4. Define an annuity in general terms. Describe the cash flows related to an annuity from the
viewpoint of the lender in terms of receipts and payments.5. Explain how to use time-value-of-money calculations to measure an installment note liability.
CORNERSTONE EXERCISES
Cornerstone Exercise A1-1 Explain How Compound Interest WorksJim Emig has $6,000.
Required:Calculate the future value of the $6,000 at 12% compounded quarterly for five years. (Note:Round answers to two decimal places.)
OBJECTIVE•1CORNERSTONE A1-1
Appendix 1 Time Value of Money 711
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Cornerstone Exercise A1-2 Use Future Value and Present Value Tables to ApplyCompound InterestCathy Lumbattis inherited $140,000 from an aunt.
Required:If Cathy decides not to spend her inheritance but to leave the money in her savings account untilshe retires in 15 years, how much money will she have, assuming an annual interest rate of 8%,compounded semiannually? (Note: Round answers to two decimal places.)
Cornerstone Exercise A1-3 Use Future Value and Present Value Tables to ApplyCompound InterestLuAnn Bean will receive $7,000 in seven years.
Required:What is the present value at 7% compounded annually? (Note: Round answers to two decimalplaces.)
Cornerstone Exercise A1-4 Use Future Value and Present Value Tables to ApplyCompound InterestA bank is willing to lend money at 6% interest, compounded annually.
Required:How much would the bank be willing to loan you in exchange for a payment of $600 four yearsfrom now? (Note: Round answers to two decimal places.)
Cornerstone Exercise A1-5 Use Future Value and Present Value Tables to ApplyCompound InterestEd Flores wants to save some money so that he can make a down payment of $3,000 on a carwhen he graduates from university in four years.
Required:If Ed opens a savings account and earns 3% on his money, compounded annually, how much willhe have to invest now? (Note: Round answers to two decimal places.)
Cornerstone Exercise A1-6 Use Future Value and Present Value Tables to ApplyCompound InterestKristen Lee makes equal deposits of $500 semiannually for four years.
Required:What is the future value at 8%? (Note: Round answers to two decimal places.)
Cornerstone Exercise A1-7 Use Future Value and Present Value Tables to ApplyCompound InterestChuck Russo, a high school math teacher, wants to set up a RRSP account into which he will de-posit $2,000 per year. He plans to teach for 20 more years and then retire.
Required:If the interest on his account is 7% compounded annually, how much will be in his account whenhe retires? (Note: Round answers to two decimal places.)
Cornerstone Exercise A1-8 Use Future Value Tables to Apply Compound InterestLarson Lumber makes annual deposits of $500 at 6% compounded annually for three years.
Required:What is the future value of these deposits? (Note: Round answers to two decimal places.)
OBJECTIVE•2CORNERSTONE A1-2
OBJECTIVE•2CORNERSTONE A1-3
OBJECTIVE•2CORNERSTONE A1-3
OBJECTIVE•2CORNERSTONE A1-4
OBJECTIVE•2CORNERSTONE A1-4
OBJECTIVE•2CORNERSTONE A1-4
OBJECTIVE•2CORNERSTONE A1-4
712 Appendix 1 Time Value of Money
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Cornerstone Exercise A1-9 Use Future Value and Present Value Tables to ApplyCompound InterestMichelle Legrand can earn 6%.
Required:How much would have to be deposited in a savings account today in order for Michelle to beable to make equal annual withdrawals of $200 at the end of each of the next 10 years? (Note:Round answers to two decimal places.) The balance at the end of the last year would be zero.
Cornerstone Exercise A1-10 Use Future Value and Present Value Tables to ApplyCompound InterestBarb Muller wins the lottery. She wins $20,000 per year to be paid for 10 years. The provinceoffers her the choice of a cash settlement now instead of the annual payments for 10 years.
Required:If the interest rate is 6%, what is the amount the province will offer for a settlement today? (Note:Round answers to two decimal places.)
EXERCISES
Exercise A1-11 Practice with TablesRefer to the appropriate tables in the text.
Required:Note: Round answers to two decimal places. Determine:
a. the future value of a single cash flow of $5,000 that earns 7% interest compounded annuallyfor 10 years.
b. the future value of an annual annuity of 10 cash flows of $500 each that earns 7% com-pounded annually.
c. the present value of $5,000 to be received 10 years from now, assuming that the interest(discount) rate is 7% per year.
d. the present value of an annuity of $500 per year for 10 years for which the interest (discount)rate is 7% per year and the first cash flow occurs one year from now.
Exercise A1-12 Practice with TablesRefer to the appropriate tables in the text.
Required:Note: Round answers to two decimal places. Determine:
a. the present value of $1,200 to be received in seven years, assuming that the interest (dis-count) rate is 8% per year.
b. the present value of an annuity of seven cash flows of $1,200 each (one at the end of each ofthe next seven years) for which the interest (discount) rate is 8% per year.
c. the future value of a single cash flow of $1,200 that earns 8% per year for seven years.d. the future value of an annuity of seven cash flows of $1,200 each (one at the end of each of
the next seven years), assuming that the interest rate is 8% per year.
Exercise A1-13 Future ValuesRefer to the appropriate tables in the text.
Required:Note: Round answers to two decimal places. Determine:
a. the future value of a single deposit of $15,000 that earns compound interest for four years atan interest rate of 10% per year.
b. the annual interest rate that will produce a future value of $13,416.80 in six years from a sin-gle deposit of $8,000.
OBJECTIVE•2CORNERSTONE A1-5
OBJECTIVE•2CORNERSTONE A1-5
OBJECTIVE•2
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Appendix 1 Time Value of Money 713
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c. the size of annual cash flows for an annuity of nine cash flows that will produce a future valueof $79,428.10 at an interest rate of 9% per year.
d. the number of periods required to produce a future value of $17,755.50 from an initial de-posit of $7,500 if the annual interest rate is 9%.
Exercise A1-14 Future Values and Long-Term InvestmentsFired Up Pottery Inc. engaged in the following transactions during 2015:
a. On January 1, 2015, Fired Up deposited $12,000 in a certificate of deposit paying 6% inter-est compounded semiannually (3% per six-month period). The certificate will mature on De-cember 31, 2018.
b. On January 1, 2015, Fired Up established an account with Rookwood Investment Manage-ment. Fired Up will make quarterly payments of $2,500 to Rookwood beginning on March31, 2015, and ending on December 31, 2016. Rookwood guarantees an interest rate of 8%compounded quarterly (2% per three-month period).
Required:1. Prepare the cash flow diagram for each of these two investments.2. Calculate the amount to which each of these investments will accumulate at maturity. (Note:
Round answers to two decimal places.)
Exercise A1-15 Future ValuesOn January 1, Beth Walid made a single deposit of $8,000 in an investment account that earns8% interest.
Required:Note: Round answers to two decimal places.
1. Calculate the balance in the account in five years assuming the interest is compounded annually.2. Determine how much interest will be earned on the account in seven years if interest is com-
pounded annually.3. Calculate the balance in the account in five years assuming the 8% interest is compounded
quarterly.
Exercise A1-16 Future ValuesKashmir Transit Company invested $70,000 in a corporate bond on June 30, 2015. The bond earns12% interest compounded monthly (1% per month) and matures on March 31, 2016.
Required:Note: Round answers to two decimal places.
1. Prepare the cash flow diagram for this investment.2. Determine the amount Kashmir will receive when the bond matures.3. Determine how much interest Kashmir will earn on this investment from June 30, 2015,
through December 31, 2015.
Exercise A1-17 Present ValuesRefer to the appropriate tables in the text.
Required:Note: Round answers to two decimal places. Determine:
a. the present value of a single $14,000 cash flow in seven years if the interest (discount) rate is8% per year.
b. the number of periods for which $5,820 must be invested at an annual interest (discount)rate of 7% to produce an investment balance of $10,000.
c. the size of the annual cash flow for a 25-year annuity with a present value of $49,113 and anannual interest rate of 9%. One payment is made at the end of each year.
d. the annual interest rate at which an investment of $2,542 will provide for a single $4,000cash flow in four years.
e. the annual interest rate earned by an annuity that costs $17,119 and provides 15 paymentsof $2,000 each, one at the end of each of the next 15 years.
OBJECTIVE•2
OBJECTIVE•2
OBJECTIVE•2
OBJECTIVE•2
714 Appendix 1 Time Value of Money
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Exercise A1-18 Present ValuesWeinstein Company signed notes to make the following two purchases on January 1, 2015:
a. a new piece of equipment for $60,000, with payment deferred until December 31, 2016.The appropriate interest rate is 9% compounded annually.
b. a small building from Johnston Builders. The terms of the purchase require a $75,000 pay-ment at the end of each quarter, beginning March 31, 2015, and ending June 30, 2017.The appropriate interest rate is 2% per quarter.
Required:Note: Round answers to two decimal places.
1. Prepare the cash flow diagrams for these two purchases.2. Prepare the entries to record these purchases in Weinstein’s journal.3. Prepare the cash payment and interest expense entries for purchase b at March 31, 2015, and
June 30, 2015.4. Prepare the adjusting entry for purchase a at December 31, 2015.
Exercise A1-19 Present ValuesKrista Kellman has an opportunity to purchase a government security that will pay $200,000 infive years.
Required:Note: Round answers to two decimal places.
1. Calculate what Krista would pay for the security if the appropriate interest (discount) rate is6% compounded annually.
2. Calculate what Krista would pay for the security if the appropriate interest (discount) rate is10% compounded annually.
3. Calculate what Krista would pay for the security if the appropriate interest (discount) rate is6% compounded semiannually.
Exercise A1-20 Future Values of an AnnuityOn December 31, 2015, Natalie Livingston signs a contract to make annual deposits of $4,200 inan investment account that earns 10%. The first deposit is made on December 31, 2015.
Required:Note: Round answers to two decimal places.
1. Calculate what the balance in this investment account will be just after the seventh deposithas been made if interest is compounded annually.
2. Determine how much interest will have been earned on this investment account just afterthe seventh deposit has been made if interest is compounded annually.
Exercise A1-21 Future Values of an AnnuityEssex Savings Bank pays 8% interest compounded weekly (0.154% per week) on savings accounts.The bank has asked your help in preparing a table to show potential customers the number ofdollars that will be available at the end of 10-, 20-, 30-, and 40-week periods during which thereare weekly deposits of $1, $5, $10, or $50. The following data are available:
Length of Annuity Future Value of Annuity at an Interest Rateof 0.154% per Week
10 weeks 10.069620 weeks 20.295330 weeks 30.679640 weeks 41.2250
OBJECTIVE•2
OBJECTIVE•2
OBJECTIVE•2
OBJECTIVE•2
Appendix 1 Time Value of Money 715
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Required:Complete a table similar to the one below. (Note: Round answers to two decimal places.)
Amount of Each Deposit
Number of Deposits $1 $5 $10 $50
10203040
Exercise A1-22 Future Value of a Single Cash FlowJimenez Products has just been paid $25,000 by Shirley Enterprises, which has owed Jimenezthis amount for 30 months but been unable to pay because of financial difficulties. Had it beenable to invest this cash, Jimenez assumes that it would have earned an interest rate of 12% com-pounded monthly (1% per month).
Required:Note: Round answers to two decimal places.
1. Prepare a cash flow diagram for the investment that could have been made if Shirley had paid30 months ago.
2. Determine how much Jimenez has lost by not receiving the $25,000 when it was due30 months ago.
3. Conceptual Connection: Indicate whether Jimenez would make an entry to account for thisloss. Why, or why not?
Exercise A1-23 Installment SaleWilke Properties owns land on which natural gas wells are located. Windsor Gas Company signsa note to buy this land from Wilke on January 1, 2015. The note requires Windsor to pay Wilke$775,000 per year for 25 years. The first payment is to be made on December 31, 2015. Theappropriate interest rate is 9% compounded annually.
Required:Note: Round answers to two decimal places.
1. Prepare a diagram of the appropriate cash flows from Windsor Gas’s perspective.2. Determine the present value of the payments.3. Indicate what entry Windsor Gas should make at January 1, 2015.
Exercise A1-24 Installment SaleBailey’s Billiards sold a pool table to Sheri Sipka on October 31, 2015. The terms of the sale areno money down and payments of $50 per month for 30 months, with the first payment due onNovember 30, 2015. The table they sold to Sipka cost Bailey’s $800, and Bailey uses a perpetualinventory system. Bailey’s uses an interest rate of 12% compounded monthly (1% per month).
Required:Note: Round answers to two decimal places.
1. Prepare the cash flow diagram for this sale.2. Calculate the amount of revenue Bailey’s should record on October 31, 2015.3. Prepare the journal entries to record the sale on October 31. Assume that Bailey’s records
cost of goods sold at the time of the sale (perpetual inventory accounting).4. Determine how much interest income Bailey’s will record from October 31, 2015, through
December 31, 2015.5. Determine how much Bailey’s 2015 income before taxes increased from this sale.
OBJECTIVE•2
OBJECTIVE•2
OBJECTIVE•2
716 Appendix 1 Time Value of Money
NEL
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64
Appendix 1 Time Value of Money 717
NEL
Exhi
bitA
1-8
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sentV
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leA
mount
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(1þ
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n/i
1%2%
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030
0.96
117
0.94
260
0.92
456
0.90
703
0.89
000
0.87
344
0.85
734
0.84
168
0.82
645
0.79
719
0.76
947
0.74
316
0.71
818
0.69
444
0.64
000
0.59
172
30.
9705
90.
9423
20.
9151
40.
8890
00.
8638
40.
8396
20.
8163
00.
7938
30.
7721
80.
7513
10.
7117
80.
6749
70.
6406
60.
6086
30.
5787
00.
5120
00.
4551
74
0.96
098
0.92
385
0.88
849
0.85
480
0.82
270
0.79
209
0.76
290
0.73
503
0.70
843
0.68
301
0.63
552
0.59
208
0.55
229
0.51
579
0.48
225
0.40
960
0.35
013
50.
9514
70.
9057
30.
8626
10.
8219
30.
7835
30.
7472
60.
7129
90.
6805
80.
6499
30.
6209
20.
5674
30.
5193
70.
4761
10.
4371
10.
4018
80.
3276
80.
2693
36
0.94
205
0.88
797
0.83
748
0.79
031
0.74
622
0.70
496
0.66
634
0.63
017
0.59
627
0.56
447
0.50
663
0.45
559
0.41
044
0.37
043
0.33
490
0.26
214
0.20
718
70.
9327
20.
8705
60.
8130
90.
7599
20.
7106
80.
6650
60.
6227
50.
5834
90.
5470
30.
5131
60.
4523
50.
3996
40.
3538
30.
3139
30.
2790
80.
2097
20.
1593
78
0.92
348
0.85
349
0.78
941
0.73
069
0.67
684
0.62
741
0.58
201
0.54
027
0.50
187
0.46
651
0.40
388
0.35
056
0.30
503
0.26
604
0.23
257
0.16
777
0.12
259
90.
9143
40.
8367
60.
7664
20.
7025
90.
6446
10.
5919
00.
5439
30.
5002
50.
4604
30.
4241
00.
3606
10.
3075
10.
2629
50.
2254
60.
1938
10.
1342
20.
0943
010
0.90
529
0.82
035
0.74
409
0.67
556
0.61
391
0.55
839
0.50
835
0.46
319
0.42
241
0.38
554
0.32
197
0.26
974
0.22
668
0.19
106
0.16
151
0.10
737
0.07
254
110.
8963
20.
8042
60.
7224
20.
6495
80.
5846
80.
5267
90.
4750
90.
4288
80.
3875
30.
3504
90.
2874
80.
2366
20.
1954
20.
1619
20.
1345
90.
0859
00.
0228
012
0.88
745
0.78
849
0.70
138
0.62
460
0.55
684
0.49
697
0.44
401
0.39
711
0.35
553
0.31
863
0.25
668
0.20
756
0.16
846
0.13
722
0.11
216
0.06
872
0.04
292
130.
8786
60.
7730
30.
6809
50.
6005
70.
5303
20.
4688
40.
4149
60.
3677
00.
3261
80.
2896
60.
2291
70.
1820
70.
1452
30.
1162
90.
0934
60.
0549
80.
0330
214
0.86
996
0.75
788
0.66
112
0.57
748
0.50
507
0.44
230
0.38
782
0.34
046
0.29
925
0.26
333
0.20
462
0.15
971
0.12
520
0.09
855
0.07
789
0.04
398
0.02
540
150.
8613
50.
7430
10.
6418
60.
5552
60.
4810
20.
4172
70.
3624
50.
3152
40.
2745
40.
2393
90.
1827
00.
1401
00.
1079
30.
0835
20.
0649
10.
0351
80.
0195
416
0.85
282
0.72
845
0.62
317
0.53
391
0.45
811
0.39
365
0.33
873
0.29
189
0.25
187
0.21
763
0.16
312
0.12
289
0.09
304
0.07
078
0.05
409
0.02
815
0.01
503
170.
8443
80.
7141
60.
6050
20.
5133
70.
4363
00.
3713
60.
3165
70.
2702
70.
2310
70.
1978
40.
1456
40.
1078
00.
0802
10.
0599
80.
0450
70.
0225
20.
0115
618
0.83
602
0.70
016
0.58
739
0.49
363
0.41
552
0.35
034
0.29
586
0.25
025
0.21
199
0.17
986
0.13
004
0.09
456
0.06
914
0.05
083
0.03
756
0.01
801
0.00
889
190.
8277
40.
6864
30.
5702
90.
4746
40.
3957
30.
3305
10.
2765
10.
2317
10.
1944
90.
1635
10.
1161
10.
0829
50.
0596
10.
0430
80.
0313
00.
0144
10.
0068
420
0.81
954
0.67
297
0.55
368
0.45
639
0.37
689
0.31
180
0.25
842
0.21
455
0.17
843
0.14
864
0.10
367
0.07
276
0.05
139
0.03
651
0.02
608
0.01
153
0.00
526
210.
8114
30.
6597
80.
5375
50.
4388
30.
3589
40.
2941
60.
2415
10.
1986
60.
1637
00.
1351
30.
0925
60.
0638
30.
0443
0.0.
0309
40.
0217
40.
0092
20.
0040
522
0.80
340
0.64
684
0.52
189
0.42
196
0.34
185
0.27
751
0.27
751
0.18
394
0.15
018
0.12
285
0.08
264
0.05
599
0.03
819
0.02
622
0.01
811
0.00
738
0.00
311
230.
7954
40.
6341
60.
5066
90.
4057
30.
3255
70.
2618
00.
2109
50.
1703
20.
1377
80.
1116
80.
0737
90.
0491
10.
0329
20.
0222
20.
0150
90.
0059
00.
0023
924
0.78
757
0.62
172
0.49
193
0.39
012
0.31
007
0.24
698
0.19
715
0.15
770
0.12
640
0.10
153
0.06
588
0.04
308
0.02
838
0.01
883
0.01
258
0.00
472
0.00
184
250.
7797
70.
6095
30.
4776
10.
3751
20.
2953
00.
2330
00.
1842
50.
1460
20.
1159
70.
2953
00.
0588
20.
0377
90.
0244
70.
0159
60.
0104
80.
0037
80.
0014
226
0.77
205
0.59
758
0.46
369
0.36
069
0.28
124
0.21
981
0.17
220
0.13
520
0.10
639
0.08
391
0.05
252
0.03
315
0.02
109
0.01
352
0.00
874
0.00
302
0.00
109
270.
7644
00.
5858
60.
4501
90.
3468
20.
2678
50.
2073
70.
1609
30.
1251
90.
0976
10.
0762
80.
0468
90.
0290
80.
0181
80.
0114
60.
0762
80.
0024
20.
0008
428
0.75
684
0.57
437
0.43
708
0.33
348
0.25
509
0.19
563
0.15
040
0.11
591
0.08
955
0.06
934
0.04
187
0.02
551
0.01
567
0.00
971
0.00
607
0.00
193
0.00
065
290.
7493
40.
5631
10.
4243
50.
3206
50.
2429
50.
1845
60.
1405
60.
1073
30.
0821
50.
0630
40.
0373
80.
0223
70.
0135
10.
0082
30.
0050
60.
0015
50.
0005
030
0.74
192
0.55
207
0.41
199
0.30
832
0.23
138
0.17
411
0.13
137
0.09
938
0.07
537
0.05
731
0.03
338
0.01
963
0.01
165
0.00
697
0.00
421
0.00
124
0.00
038
718 Appendix 1 Time Value of Money
NEL
Exhi
bitA
1-9
Futu
reV
alu
eofa
nA
nnuity
FV
A¼
(1þ
i)n
�1
i
hi
n/i
1%2%
3%4%
5%6%
7%8%
9%10
%12
%14
%16
%18
%20
%25
%30
%
11.
0000
01.
0000
01.
0000
01.
0000
01.
0000
01.
0000
01.
0000
01.
0000
01.
0000
01.
0000
01.
0000
01.
0000
01.
0000
01.
0000
01.
0000
01.
0000
01.
0000
02
2.01
000
2.02
000
2.03
000
2.04
000
2.05
000
2.06
000
2.07
000
2.08
000
2.09
000
2.10
000
2.12
000
2.14
000
2.16
000
2.18
000
2.20
000
2.25
000
2.30
000
33.
0301
03.
0604
03.
0909
03.
1216
03.
1525
03.
1836
03.
2149
03.
2464
03.
2781
03.
3100
03.
3744
03.
4396
03.
5056
03.
5724
03.
6400
03.
8125
03.
9900
04
4.06
040
4.12
161
4.18
363
4.24
646
4.31
013
4.37
462
4.43
994
4.50
611
4.57
313
4.64
100
4.77
933
4.92
114
5.06
650
5.21
543
5.36
800
5.76
563
6.18
700
55.
1010
15.
2040
45.
3091
45.
4163
25.
5256
35.
6370
95.
7507
45.
8666
05.
9847
16.
1051
06.
3528
56.
6101
06.
8771
47.
1542
17.
4416
08.
2070
39.
0431
06
6.15
202
6.30
812
6.46
841
6.63
298
6.80
191
6.97
532
7.15
329
7.33
593
7.52
000
7.71
561
8.11
519
8.53
552
8.97
748
9.44
197
9.92
992
11.2
5879
12.7
5603
77.
2135
47.
4342
87.
6624
67.
8982
98.
1420
18.
3938
48.
6540
28.
9228
09.
2004
39.
4871
710
.089
0110
.730
4911
.413
8712
.141
5212
.915
9015
.073
4917
.582
848
8.28
567
8.58
297
8.89
234
9.21
423
9.54
911
9.89
747
10.2
5980
10.6
3663
11.0
2847
11.4
3589
12.2
9969
13.2
3276
14.2
4009
15.3
2700
16.4
9908
19.8
4186
23.8
5769
99.
3685
39.
7546
310
.159
1110
.582
8011
.026
5611
.491
3211
.977
9912
.487
5613
.021
0413
.579
4814
.775
6616
.085
3517
.518
5119
.085
8520
.798
9025
.802
3232
.015
0010
10.4
6221
10.9
4972
11.4
6388
12.0
0611
12.5
7789
13.1
8079
13.8
1645
14.4
8656
15.1
9293
15.9
3742
17.5
4874
19.3
3730
21.3
2147
23.5
2131
25.9
5868
33.2
5290
42.6
1950
1111
.566
8312
.168
7212
.807
8013
.486
3514
.206
7914
.971
6415
.783
6016
.645
4917
.560
2918
.531
1720
.654
5823
.044
5225
.732
9028
.755
1432
.150
4242
.566
1356
.405
3512
12.6
8250
13.4
1209
14.1
9203
15.0
2581
15.9
1713
16.8
6994
17.8
8845
18.9
7713
20.1
4072
21.3
8428
24.1
3313
27.2
7075
30.8
5017
34.9
3107
39.5
8050
54.2
0766
74.3
2695
1313
.809
3314
.680
3315
.617
7916
.626
8417
.712
9818
.882
1420
.140
6421
.495
3022
.953
3824
.522
7128
.029
1132
.088
6536
.786
2042
.218
6648
.496
6068
.759
5897
.625
0414
14.9
4742
15.9
7394
17.0
8632
18.2
9191
19.5
9863
21.0
1507
22.5
5049
24.2
1492
26.0
1919
27.9
7498
32.3
9260
37.5
8107
43.6
7199
50.8
1802
59.1
9592
86.9
4947
127.
9125
515
16.0
9690
17.2
9342
18.5
9891
20.0
2359
21.5
7856
23.2
7597
25.1
2902
27.1
5211
29.3
6092
31.7
7248
37.2
7971
43.8
4241
51.6
5951
60.9
6527
72.0
3511
109.
6868
416
7.28
631
1617
.257
8618
.639
2920
.156
8821
.824
5323
.657
4925
.672
5327
.888
0530
.324
2833
.003
4035
.949
7342
.753
2850
.980
3560
.925
0372
.939
0187
.442
1313
8.10
855
218.
4722
017
18.4
3044
20.0
1207
21.7
6159
23.6
9751
25.8
4037
28.2
1288
30.8
4022
33.7
5023
36.9
7370
40.5
4470
48.8
8367
59.1
1760
71.6
7303
87.0
6804
105.
9305
617
3.63
568
285.
0138
618
19.6
1475
21.4
1231
23.4
1444
25.6
4541
28.1
3238
30.9
0565
33.9
9903
37.4
5024
41.3
0134
45.5
9917
55.7
4971
68.3
9407
84.1
4072
103.
7402
812
8.11
667
218.
0446
037
1.51
802
1920
.810
9022
.840
5625
.116
8727
.671
2330
.539
0033
.759
9937
.378
9641
.446
2646
.018
4651
.159
0963
.439
6878
.969
2398
.603
2312
3.41
353
154.
7400
027
3.55
576
483.
9734
320
22.0
1900
24.2
9737
26.8
7037
29.7
7808
33.0
6595
36.7
8559
40.9
9549
45.7
6196
51.1
6012
57.2
7500
72.0
5244
91.0
2493
115.
3797
514
6.62
797
186.
6880
034
2.94
470
630.
1654
621
23.2
3919
25.7
8332
28.6
7649
31.9
6920
35.7
1925
39.9
9273
44.8
6518
50.4
2292
56.7
6453
64.0
0250
81.6
9874
104.
7684
213
4.84
051
174.
0210
022
5.02
560
429.
6808
782
0.21
510
2224
.471
5927
.298
9830
.536
7834
.247
9738
.505
2143
.392
2949
.005
7455
.456
7662
.873
3471
.402
7592
.502
5812
0.43
600
157.
4149
920
6.34
479
271.
0307
253
8.10
109
1067
.279
6323
25.7
1630
28.8
4496
32.4
5288
36.6
1789
41.4
3048
46.9
9583
53.4
3614
60.8
9330
66.1
7894
79.5
4302
104.
6028
913
8.29
704
183.
6013
824
4.48
685
326.
2368
667
3.62
636
1388
.463
5124
26.9
7346
30.4
2186
34.4
2647
39.0
8260
44.5
0200
50.8
1558
58.1
7667
66.7
6476
76.7
8981
88.4
9733
118.
1552
415
8.65
862
213.
9776
128
9.49
448
392.
4842
484
3.03
295
1806
.002
5725
28.2
4320
32.0
3030
36.4
5926
41.6
4591
47.7
2710
54.8
6451
63.2
4904
73.1
0594
84.7
0090
98.3
4706
133.
3338
718
1.87
083
249.
2140
234
2.60
349
471.
9810
810
54.7
9118
2348
.803
3426
29.5
2563
33.6
7091
38.5
5304
44.3
1174
51.1
1345
59.1
5638
68.6
7647
79.9
5442
93.3
2398
109.
1817
715
0.33
393
208.
3327
429
0.08
827
405.
2721
156
7.37
730
1319
.488
9830
54.4
4434
2730
.820
8935
.344
3240
.709
6347
.084
2154
.669
1363
.705
7774
.483
8287
.350
7710
2.72
313
121.
0999
416
9.37
401
238.
4993
333
7.50
239
479.
2210
968
1.85
276
1650
.361
2339
71.7
7764
2832
.129
1037
.051
2142
.930
9249
.967
5858
.402
5868
.528
1180
.697
6995
.338
8311
2.96
822
134.
2099
419
0.69
889
272.
8892
339
2.50
277
566.
4808
981
9.22
331
2063
.951
5351
64.3
1093
2933
.450
3938
.792
2345
.218
8552
.966
2962
.322
7173
.639
8087
.346
5310
3.96
594
124.
1353
614
8.63
093
214.
5827
531
2.09
373
456.
3032
266
9.44
745
984.
0679
725
80.9
3941
6714
.604
2130
34.7
8489
40.5
6808
47.5
7542
56.0
8494
66.4
3885
79.0
5819
94.4
6079
113.
2832
113
6.30
754
164.
4940
224
1.33
268
356.
7868
553
0.31
173
790.
9479
911
81.8
8157
3227
.174
2787
29.9
8548
Appendix 1 Time Value of Money 719
NEL
Exhi
bitA
1-10
Pre
sentV
alu
eofa
nA
nnuity
PV
A¼
1�
1
(1þ
i)n
i
n/i
1%2%
3%4%
5%6%
7%8%
9%10
%12
%14
%16
%18
%20
%25
%30
%
10.
9901
00.
9803
90.
9708
70.
9615
40.
9523
80.
9434
00.
9345
80.
9259
30.
9174
30.
9090
90.
8928
60.
8771
90.
8620
70.
8928
60.
8333
30.
8000
00.
7692
32
1.97
040
1.94
156
1.91
347
1.88
609
1.85
941
1.83
339
1.80
802
1.78
326
1.75
911
1.73
554
1.69
005
1.64
666
1.60
523
1.56
564
1.52
778
1.44
000
1.36
095
32.
9409
92.
8838
82.
8286
12.
7750
92.
7232
52.
6730
12.
6243
22.
5771
02.
5312
92.
4868
52.
4018
32 5
2.67
32.
2458
92.
1742
72.
1064
81.
9520
01.
8161
14
3.90
197
3.80
773
3.71
710
3.62
990
3.54
595
3.46
511
3.38
721
3.31
213
3.23
972
3.16
987
3.03
735
2.91
371
2.79
818
2.69
006
2.58
873
2.36
160
2.16
624
54.
8534
34.
7134
64.
5797
14.
4518
24.
3294
84.
2123
64.
1002
03.
9927
13.
8896
53.
7907
93.
6047
83.
4330
83.
2742
93.
1271
72.
9906
12.
6892
82.
4355
76
5.79
548
5.60
143
5.41
719
5.24
214
5.07
569
4.91
732
4.76
654
4.62
288
4.48
592
4.50
756
4.11
141
3.88
867
3.68
474
3.49
760
3.32
551
2.95
142
2.64
275
76.
7281
96.
4719
96.
2302
86.
0020
55.
7863
75.
5823
85.
3892
95.
2063
75.
0329
54.
8684
24.
5637
64.
2883
04.
0385
73.
8115
33.
6045
93.
1611
42.
8021
18
7.65
168
7.32
548
7.01
969
6.73
274
6.46
321
6.20
979
5.97
130
5.74
664
5.53
482
5.33
493
4.96
764
4.63
886
4.34
359
4.07
757
3.83
716
3.32
891
2.92
470
98.
5660
28.
1622
47.
7861
17.
4353
37.
1078
26.
8016
96.
5152
36.
8016
95.
9952
55.
7590
25.
3282
54.
9463
74.
6065
44.
3030
24.
0309
73.
4631
33.
0190
010
9.47
130
8.98
259
8.53
020
8.11
090
7.72
173
7.36
009
7.02
358
6.71
008
6.41
766
6.14
457
5.65
022
5.21
612
4.83
323
4.49
409
4.19
247
3.57
050
3.09
154
1110
.367
639.
7868
59.
2526
28.
7604
88.
3064
17.
8868
77.
4986
77.
1389
66.
8051
96.
4950
65.
9377
05.
4527
35.
0286
44.
6560
14.
3270
63.
6564
03.
1473
412
11.2
5508
10.5
7534
9.95
400
9.38
507
8.86
325
8.38
384
7.94
269
7.53
608
7.16
073
6.81
369
6.19
437
5.66
029
5.19
711
4.79
322
4.43
922
3.72
512
3.19
026
1312
.133
7411
.348
3710
.634
969.
9856
59.
3935
78.
8526
88.
3576
57.
9037
87.
4869
07.
1033
66.
4235
55.
8423
65.
8423
34.
9095
18.
8526
83.
7801
03.
2232
814
13.0
0370
12.1
0625
11.2
9607
10.5
6312
9.89
864
9.29
498
8.74
547
8.24
424
7.78
615
7.36
669
6.62
817
6.00
207
5.46
753
5.00
806
4.61
057
3.82
408
3.24
867
1513
.865
0512
.849
2611
.937
9411
.118
3910
.379
669.
7122
59.
1079
18.
5594
88.
0606
97.
6060
86.
8108
66.
1421
75.
5754
65.
0915
84.
6754
72.
8492
63.
2682
116
14.7
1787
13.5
7771
12.5
6110
11.6
5230
10.8
3777
10.1
0590
9.44
665
8.85
137
8.31
256
7.82
371
6.97
399
6.26
506
5.66
850
5.16
235
8.31
256
3.88
741
3.28
324
1715
.562
2514
.291
8713
.166
1212
.165
6711
.274
0710
.477
269.
7632
29.
1216
48.
5436
38.
0215
57.
1196
36.
3728
65.
7487
05.
2223
34.
7746
33.
9099
33.
2948
018
16.3
9827
14.9
9203
13.7
5351
12.6
5930
11.6
8959
10.8
2760
10.0
5909
9.37
189
8.75
563
8.20
141
7.24
967
6.46
742
5.81
785
5.27
316
4.81
219
3.92
794
3.30
369
1917
.226
0115
.678
4614
.323
8013
.133
9412
.085
3211
.158
1210
.335
609.
6036
08.
9501
18.
3649
27.
3657
86.
5503
75.
8774
65.
3162
44.
8435
03.
9423
53.
3105
320
18.0
4555
16.3
5143
14.8
7747
13.5
9033
12.4
6221
11.4
6992
10.5
9401
9.81
815
9.12
855
8.51
356
7.46
944
6.62
313
5.92
884
5.35
275
4.86
958
3.59
288
3.31
579
2118
.856
9817
.011
2115
.415
0214
.029
1612
.821
1511
.764
0810
.835
5310
.016
809.
2922
48.
6486
97.
5620
06.
6869
65.
9731
45.
3836
84.
8913
23.
9631
13.
3198
422
19.6
6038
17.6
5805
15.9
3692
14.4
5112
13.1
6300
12.0
4158
11.0
6124
10.2
0074
9.44
243
8.77
154
7.64
465
6.74
294
6.01
133
5.40
990
4.90
943
3.97
049
3.32
296
2320
.455
8218
.292
2016
.443
6114
.856
8413
.488
5712
.303
3811
.272
1910
.371
069.
5802
18.
8832
27.
7184
36.
7920
66.
0442
55.
4321
24.
9245
33.
9763
93.
3253
524
21.2
4339
18.9
1393
16.9
3554
15.2
4696
13.7
9864
12.5
5036
11.4
6933
10.5
2876
9.70
661
8.98
474
7.78
432
6.83
514
6.07
263
5.45
095
4.93
710
3.98
111
3.32
719
2522
.023
1619
.523
4617
.413
1515
.622
0814
.093
9412
.783
3611
.653
5810
.674
789.
8225
89.
0770
47.
8431
46.
8729
36.
0970
95.
4669
14.
9475
93.
9848
93.
3286
126
22.7
9520
20.1
2104
17.8
7684
15.9
8277
14.3
7519
13.0
0317
11.8
2578
10.8
0998
9.92
897
9.16
095
7.89
566
6.90
608
6.11
818
5.48
043
4.95
632
3.98
791
3.32
970
2723
.559
6120
.706
9018
.327
0316
.329
5914
.643
0313
.210
5311
.986
7110
.935
1610
.026
589.
2372
27.
9425
56.
9351
56.
1363
65.
4918
94.
9636
03.
9903
33.
3305
428
24.3
1644
21.2
8127
18.7
6411
16.6
6306
14.8
9813
13.4
0616
12.1
3711
11.0
5108
10.1
1613
9.30
657
7.98
442
6.96
066
6.15
204
5.50
160
4.96
967
3.99
226
3.33
118
2925
.065
7921
.844
3819
.188
4516
.983
7115
.141
0713
.590
7212
.277
6711
.158
4110
.198
289.
3696
18.
0218
16.
9830
46.
1655
55.
5098
33.
5907
23.
9938
13.
3316
830
25.8
0771
22.3
9646
19.6
0044
17.2
9203
15.3
7245
13.7
6483
12.4
0904
11.2
5778
10.2
7365
9.42
691
8.05
518
7.00
266
6.17
720
5.51
681
4.97
894
3.99
505
3.33
206
720 Appendix 1 Time Value of Money
NEL