After studying Appendix 1, you should be able to:
•
1 Explain how compound interest works.•
2 Use future value and present value tables to apply compound interest to accounting transactions.T
ime Value of Money
1
Appendix
Doug
Norman
Time value of money is widely used in business to measure today’s value of future cash outflows 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 usually paysinterest. In effect, interest is the time value of money. The amount of interest paid is 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, investInvest-ments in debt securities and savings accounts), installment sales, 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 in the accounting records of the other. All such assets and liabilities increase as interest is earned by the asset holder and decrease as payments are made by the liability holder.
COMPOUND INTEREST CALCULATIONS
Compound interest is a method of calculating the time value of money in which inter-est is earned on the previous periods’ interinter-est. That is, interinter-est for the period is added to the account balance and interest is earned on this new balance in the next period. In computing compound interest, it’s important to understand the difference between the interest period and the interest rate:
• Theinterest period is the time interval between interest calculations.
• Theinterest 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 period is 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 interest period, the rate must be adjusted before interest can be calculated. For example, suppose that a bank advertises interest at a rate of 12% per year compounded monthly. Here, the interest period would be one month. Since there are 12 interest periods in one year, the interest rate for one month is one-twelfth the annual rate, or 1%. In other words, if the rate statement period differs from the interest period, the stated rate must be divided by the number of interest periods included in the rate statement period. A few examples of adjusted 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 an interest 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 plus undistributed 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 the end of each period.CORNERSTONEA1-1(p. 700) shows how to compute future values using compound interest.
O B J E C T I V E
•
1As you can see in Cornerstone A1-1, the balance in the account continues to grow each month by an increasing amount of interest. The amount of monthly interest increases because interest iscompounded. 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 January interest ($100 3 0.005¼ 50¢).
In Cornerstone A1-1, the compound interest only amounts to $1.50. That might seem relatively insignificant, but if the investment period is sufficiently long, the amount of 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 the objective of giving you a university graduation present at age 21. How much would that investment be worth after 21 years? The answer is $3,514. In 21 years, the compound interest 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 the future 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 onpast and future cash flows and interest payments. In contrast, calculations of present values are determinations of present amounts based onexpected future cash flows.
C O R N E R S T O N E
A 1 - 1
Computing Future Values Using
Compound Interest
Information:
An investor deposits $20,000 in a savings account on January 1, 2015. The bank pays interest of 6% per year compounded 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 money will 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 $20,000.00 January interest ($20,000.00 3 ½%) 100.00 Account balance, 1/31/15 20,100.00 February interest ($20,100.00 3 ½%) 100.50 Account balance, 2/28/15 20,200.50 March interest ($20,200.50 3 ½%) 101.00 Account 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 complex situations, 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 deposits have the opposite effect, increasing the balance and the amount of interest earned.
PRESENT VALUE OF FUTURE CASH FLOWS
Whenever a contract establishes a relationship between an initial amount borrowed or loaned and one or more future cash flows, the initial amount borrowed or loaned is the present value of those future cash flows. The present value can be interpreted in two ways:
• From the borrower’s viewpoint, it is the liability that will be exactly paid by the future payments.
• From the lender’s viewpoint, it is the receivable balance that will be exactly satisfied by the future receipts.
For understanding cash flows, cash flow diagrams that display both the amounts and the 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 pointing upward and outflows as arrows pointing downward. For example, suppose that Hilliard Corporation 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 are shown in Exhibit A1-1.
The calculation that follows shows, from the borrower’s perspective, the relationship between the amount borrowed (the present value) and the future payments (future cash flows) required by Hilliard’s note.
Amount borrowed, 1/1/15 $100,000.00 Add: 2015 interest ($100,000.00 3 0.08) 8,000.00 Subtract payment on 12/31/15 (38,803.35) Liability at 12/31/15 69,196.65 Add: 2016 interest ($69,196.65 3 0.08) 5,535.73 Subtract payment on 12/31/16 (38,803.35) Liability at 12/31/16 35,929.03 Add: 2017 interest ($35,929.03 3 0.08) 2,874.32 Subtract 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, the three payments of $38,803.35 exactly pay off the liability created by the note. Because the reversal of future value calculations can present a burdensome and sometimes diffi-cult algebraic problem, shortcut methods using tables have been developed (see Exhibits A1-7, A1-8, A1-9, and A1-10, pp. 717–720, discussed later in this appendix).
Exhibit A1-1
Cash Flow Diagram
$38,803.35 $38,803.35 $38,803.35 $100,000
Interest and the Frequency of Compounding
The number of interest periods into which a compound interest problem is divided can make a significant difference in the amount of compound interest. For example, assume that 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 they have 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, which offers only annual compounding.
FOUR BASIC COMPOUND INTEREST
PROBLEMS
Any present value or future value problems can be broken down into one or more of the following 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 Amount
In 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 forf dollars and add compound interest ati percent to that account for n periods:
FV ¼ (f )(1 þ i)n
The balance of the account aftern 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 formula above, you could use the future value table in Exhibit A1-7 (p. 717), whereM1is the
multiple that corresponds to the appropriate values ofn 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 A1-2
Effect of Interest Periods on Compound InterestInvestment Interest Period I N Calculation of Future Amount in One Year* A 1 year 12% 1 ($10,000 3 1.12000) ¼$11,200 B 6 months 6% 2 ($10,000 3 1.12360) ¼ 11,236 C 1 quarter 3% 4 ($10,000 3 1.12551) ¼ 11,255 D 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 tables to apply compound interest to accounting transactions.
to compute the future value (FV) at the end of the four years—the amount that will be repaid. 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 $200,000.00
First year’s interest ($200,000.00 3 0.06) 12,000.00 Loan receivable at end of first year 212,000.00 Second year’s interest ($212,000.00 3 0.06) 12,720.00 Loan receivable at end of second year 224,720.00 Third year’s interest ($224,720.00 3 0.06) 13,483.20 Loan receivable at end of third year 238,203.20 Fourth year’s interest ($238,203.20 3 0.06) 14,292.19 Loan 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 of computing interest for each year based on an amount that includes the interest earned in prior years.
The shortcut calculation, using the future value table (Exhibit A1-7, p. 717), would be as follows:
FV ¼ (f )(M1)
¼($200,000)(1:26248) ¼$252,496
You can findM1at the intersection of the 6% column (i¼ 6%) and the fourth row (n ¼
4) or by calculating 1.064. This multiple is the future value of the single amount after having 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 the compound interest calculation and the answer ($252,496) determined using the future value table. This is because the numbers in the table have been rounded to five decimal pla-ces. If they were taken to eight digits (1.064¼ 1.26247696), the two answers would be equal.CORNERSTONEA1-2shows how to compute the future value of a single amount.
Exhibit A1-3
Future Value of a Single Amount: An Example
C O R N E R S T O N E
A 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 purchase a 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).
Computing the Present Value of a Single Amount
In 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 thediscount rate. Why:
The future value of a single amount is the original cash flow plus compound 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 through December 31, 2015.
Dec. 31, 2015 Cash 6,000
Interest 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 12,545.40
Interest income 12,545.40
(Record interest income)
C O R N E R S T O N E
A 1 - 2
(continued)
Assets 5 Liabilities 1 Shareholders’ Equity (Interest Income) þ6,000 þ6,000 Assets 5Liabilities 1 Shareholders’ Equity (Interest Income) þ12,545.40 þ12,545.40To 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), whereM2is the multiple
from Exhibit A1-8 that corresponds to the appropriate values ofn and i:
PV ¼ (f )(M2)
Suppose Marathon Oil has purchased property on which it plans to develop oil wells. The seller has agreed to accept a single $150,000,000 payment three years from now, when Marathon expects to be selling oil from the field. Assuming an interest rate of 7% per year, the present value of the amount to be received in three years from the borrower’s perspective can be calculated as shown in Exhibit A1-4.
The shortcut calculation, using the present value table (Exhibit A1-8, p. 718), would be 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 (i¼ 7%) and the third row
(n¼ 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 prespres-ent value of $150,000,000 is $150,000,000 times the multiple.
Although the future value calculation cannot be used to determine the present value, 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) $122,445,000 First year’s interest ($122,445,000 3 0.07) 8,571,150 Loan payable at end of first year 131,016,150 Second year’s interest ($131,016,150 3 0.07) 9,171,131 Loan payable at end of second year 140,187,281 Third year’s interest ($140,187,281 3 0.07) 9,813,110 Loan payable at end of the third year (f ) $150,000,391
Again, the $391 difference between the amount here and the assumed $150,000,000 cash 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 flow of $150,000,000. This reversal proves that $122,445,000 is the correct present value. CORNERSTONEA1-3(p. 706) shows how to compute the present value of a single amount.
Exhibit A1-4
Computing the Future Value of an Annuity
So far, we have been discussing problems that involve a single cash flow. However, there are also instances of multiple cash flows one period apart. An annuity is a number of equal cash flows: one to each interest period. For example, an investment in a security that 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 of each month can also be considered an annuity. (Although the number of days in a month varies from 28 to 31, the interest period is defined as one month without regard to 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 E
A 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 the date of purchase, Adelsman agreed to defer the $60,000 payment for 15 months. The appropriate rate of interest in 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 at a 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.
15
15 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 1
Shareholders’ Equity
To find the future value of an annuity, use the following equation:
FV ¼ (f ) (1 þ i)
n1
i
Alternatively, you could use the future value table in Exhibit A1-9 (p. 719), where M3 is the multiple from Exhibit A1-9 that corresponds to the appropriate values ofn
andi:
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 $5,000 each, with the first payment due three years before the program ends. CIBC advertises a 6% interest rate compounded annually. The future value of this annuity immediately after 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 using annuities, the time-value-of-money model assumes that all cash flows occur at the end of a period. Therefore, the first cash flow in the future value of an annuity occurs at the end of the first period. However, since interest cannot be earned until the first deposit has been 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%) $ 0.00
First deposit 5,000.00
Investment balance at end of first year 5,000.00 Second year’s interest ($5,000.00 3 0.06) 300.00
Second deposit 5,000.00
Investment balance at end of second year 10,300.00 Third year’s interest ($10,300.00 3 0.06) 618.00
Third deposit 5,000.00
Investment balance at end of third year 15,918.00 Fourth year’s interest ($15,918.00 3 0.06) 955.08
Fourth deposit 5,000.00
Investment at end of fourth year $21,873.08
This calculation shows that the lender has accumulated a future value (FV) of $21,873.08 by 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), would be as follows:
FV ¼ (f )(M3)
¼($5,000)(4:37462) ¼$21,873
Exhibit A1-5
You can findM3at the intersection of the 6% column (i¼ 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 value of an annuity of four cash flows of $1 each at 6%. The future value of an annuity of $5,000 cash flows is $5,000 times the multiple. Thus, the table allows us to calculate the future value of an annuity by a single multiplication, no matter how many cash flows are involved. COR-NERSTONEA1-4shows how to compute the future value of an annuity.
Present Value of an Annuity
In 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 E
A 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 of a large farm, wants to retire in five years. To provide funds to purchase a retirement annuity from London Life at the date of retirement, Greg asks the client to give him annual payments of $170,000, which Greg will deposit in a 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 compound interest 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=5:75074 f ¼ $173,890:66
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), whereM4is the
multiple from Exhibit A1-10 that corresponds to the appropriate values ofn and i:
PV ¼ (f )(M4)
For example, assume thatXerox Corporationpurchased a new machine for its man-ufacturing operations. The purchase agreement requires Xerox to make four equally spaced payments of $24,154 each. The interest rate is 8% compounded annually and the first cash flow occurs one year after the purchase. Exhibit A1-6 shows how to determine the present value of this annuity from Xerox’s (the borrower’s) perspective. Note that the same concept applies to both the lender’s and borrower’s perspectives.
The shortcut calculation, using the present value table (Exhibit A1-10, p. 720), would be as follows:
PV ¼ (f )(M4)
¼($24,154)(3:31213) ¼$80,001:19
You can findM4at the intersection of the 8% column (i¼ 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 present value of an annuity of four cash flows of $1 each at 8%. The present value of an annuity of four $24,154 cash flows is $24,154 times the multiple.
Again, although the compound interest calculation is not used to determine the present 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) $ 80,001.19 Interest for first year ($80,001.19 3 0.08) 6,400.10 Less: First cash flow (24,154.00) Balance at end of first year 62,247.29 Interest for second year ($62,247.29 3 0.08) 4,979.78 Less: Second cash flow (24,154.00) Balance at end of second year 43,073.07 Interest for third year ($43,073.07 3 0.08) 3,445.85 Less: Third cash flow (24,154.00) Balance at end of third year 22,364.92 Interest for fourth year ($22,364.92 3 0.08) 1,789.19 Less: 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 value formula. If the present value (PV ) calculated with the formula is correct, then the proof
Exhibit A1-6
should end with a balance of zero immediately after the last cash flow. This proof ends with a balance of $0.11 because of rounding in the proof itself and in the table in Exhibit A1-10 (p. 720).
CORNERSTONEA1-5shows 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
C O R N E R S T O N E
A 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 bank an installment note. The note requires Windsor to make four annual payments of $600,000 each on December 31 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 with compound 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’Equity
LO2. Use future value and present value tables to apply compound interest to accounting transactions.
• 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 is compounded (i.e., the future value) or
• the amount a series of future cash flows is worth today after taking into account compound interest (i.e., the present value).
C O R N E R S T O N E S
FOR APPENDIX 1 CORNERSTONEA1-1Computing future values using compound interest (p. 700)
CORNERSTONEA1-2Computing future value of a single amount (p. 703) CORNERSTONEA1-3Computing present value of a single amount (p. 706) CORNERSTONEA1-4Computing future value of an annuity (p. 708) CORNERSTONEA1-5Computing 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 Works Jim 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
•
1 CORNERSTONEA1-1Cornerstone Exercise A1-2 Use Future Value and Present Value Tables to Apply Compound Interest
Cathy 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 until she 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 Apply Compound Interest
LuAnn Bean will receive $7,000 in seven years. Required:
What is the present value at 7% compounded annually? (Note: Round answers to two decimal places.)
Cornerstone Exercise A1-4 Use Future Value and Present Value Tables to Apply Compound Interest
A 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 years from now? (Note: Round answers to two decimal places.)
Cornerstone Exercise A1-5 Use Future Value and Present Value Tables to Apply Compound Interest
Ed Flores wants to save some money so that he can make a down payment of $3,000 on a car when he graduates from university in four years.
Required:
If Ed opens a savings account and earns 3% on his money, compounded annually, how much will he have to invest now? (Note: Round answers to two decimal places.)
Cornerstone Exercise A1-6 Use Future Value and Present Value Tables to Apply Compound Interest
Kristen 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 Apply Compound Interest
Chuck 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 when he retires? (Note: Round answers to two decimal places.)
Cornerstone Exercise A1-8 Use Future Value Tables to Apply Compound Interest Larson 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.)
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2 CORNERSTONEA1-4Cornerstone Exercise A1-9 Use Future Value and Present Value Tables to Apply Compound Interest
Michelle Legrand can earn 6%. Required:
How much would have to be deposited in a savings account today in order for Michelle to be able 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 Apply Compound Interest
Barb Muller wins the lottery. She wins $20,000 per year to be paid for 10 years. The province offers 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 Tables Refer 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 annually for 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 Tables Refer 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 of the 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 Values
Refer 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 at an 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
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2c. the size of annual cash flows for an annuity of nine cash flows that will produce a future value of $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 Investments
Fired 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 March 31, 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 Values
On January 1, Beth Walid made a single deposit of $8,000 in an investment account that earns 8% 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 Values
Kashmir Transit Company invested $70,000 in a corporate bond on June 30, 2015. The bond earns 12% 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 Values Refer 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 is 8% 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 an annual 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 $4,000 cash flow in four years.
e. the annual interest rate earned by an annuity that costs $17,119 and provides 15 payments of $2,000 each, one at the end of each of the next 15 years.
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2Exercise A1-18 Present Values
Weinstein 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 purchaseb at March 31, 2015, and June 30, 2015.
4. Prepare the adjusting entry for purchasea at December 31, 2015. Exercise A1-19 Present Values
Krista Kellman has an opportunity to purchase a government security that will pay $200,000 in five years.
Required:
Note: Round answers to two decimal places.
1. Calculate what Krista would pay for the security if the appropriate interest (discount) rate is 6% compounded annually.
2. Calculate what Krista would pay for the security if the appropriate interest (discount) rate is 10% compounded annually.
3. Calculate what Krista would pay for the security if the appropriate interest (discount) rate is 6% compounded semiannually.
Exercise A1-20 Future Values of an Annuity
On December 31, 2015, Natalie Livingston signs a contract to make annual deposits of $4,200 in an 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 deposit has been made if interest is compounded annually.
2. Determine how much interest will have been earned on this investment account just after the seventh deposit has been made if interest is compounded annually.
Exercise A1-21 Future Values of an Annuity
Essex 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 of dollars that will be available at the end of 10-, 20-, 30-, and 40-week periods during which there are weekly deposits of $1, $5, $10, or $50. The following data are available:
Length of Annuity Future Value of Annuity at an Interest Rate of 0.154% per Week 10 weeks 10.0696 20 weeks 20.2953 30 weeks 30.6796 40 weeks 41.2250 OBJECTIVE
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2Required:
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
10 20 30 40
Exercise A1-22 Future Value of a Single Cash Flow
Jimenez Products has just been paid $25,000 by Shirley Enterprises, which has owed Jimenez this amount for 30 months but been unable to pay because of financial difficulties. Had it been able 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 paid 30 months ago.
2. Determine how much Jimenez has lost by not receiving the $25,000 when it was due 30 months ago.
3. Conceptual Connection:Indicate whether Jimenez would make an entry to account for this loss. Why, or why not?
Exercise A1-23 Installment Sale
Wilke Properties owns land on which natural gas wells are located. Windsor Gas Company signs a 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. The appropriate 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 Sale
Bailey’s Billiards sold a pool table to Sheri Sipka on October 31, 2015. The terms of the sale are no money down and payments of $50 per month for 30 months, with the first payment due on November 30, 2015. The table they sold to Sipka cost Bailey’s $800, and Bailey uses a perpetual inventory 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.
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