CH 5; Time Value of Money Concepts

the knotworth company purchased a machine for $15,000 down and $500 a month payable at the end of each of the next 36 months. how would the company calculate the cash price of the machine, assuming the annual interest rate is known?

$15,000 plus the present value of an ordinary annuity of $500 for 36 periods; The cash price is equal to the present value of the future cash outflows. This includes the $15,000 today plus the value today (present value) of the $500 payments made at the endof each month (ordinary annuity).

concept check: annual installments; The Omagosh Company purchased office furniture for $25,800 and agreed to pay for the purchase by making five annual installment payments beginning one year from today. The installment payments include interest at 8%. The present value of an ordinary annuity for five periods at 8% is 3.99271. The present value of an annuity due for five periods at 8% is 4.31213. What is the required annual installment payment?

$25,800/3.99271*= $6,462 *present value of an ordinary annuity for five periods at 8%

(compound interest example) jacob lee invested $600 in a savings account paying 8% interest compounded twice a year. what will be his investment balance at the end of the year? rounded to nearest dollar

$649

determining the annuity amount when other variables are known; Assume that you borrow $700 from a friend and intend to repay the amount in four equal instalments beginning one year from today.Your friend wishes to be reimbursed for the time value of money at an 8% annual rate. What is the required annual payment that must be made (the annuity amount), to repay the loan in four years?

$700 PV = 3.31213 x annuity amount and we solve for annuity amount to get =$211.34 also we can use the present value of an ordinary annuity of $1 n=4, i=8%

Turp and Tyne Distillery is considering investing in a two-year project. The company's required rate of return is 10%. The present value of $1 for one period at 10% is .909 and .826 for two periods at 10%. The project is expected to create cash flows, net of taxes, of $240,000 in the first year, and $300,000 in the second year. The distillery should invest in the project if the project's cost is less than or equal to:

($240,00 x .909) = $218,160 ($300,000 x .826) = $247,800 $218,160+$247,800= $465,960

valuation of pension obligations; On January 1, 2021, the Stridewell Wholesale Shoe Company hired Terry Elliott. Terry is expected to work for 25 years before retirement on December 31, 2045. Annual retirement payments will be paid at the end of each year during his retirement period, expected to be 20 years The first payment will be on December 31, 2046. During 2021 Terry earned an annual retirement benefit estimated to be $2,000 per year. The company plans to contribute cash to a pension fund that will accumulate to an amount sufficient to pay Terry this benefit. Assuming thatStridewell anticipates earning 6% on all funds invested in the pension plan, how much would the company have to contribute at the end of 2021 to pay for pension benefits earned in 2021?

*this is a deferred annuity present value of an ordinary annuity of $1, n=20, i=6%; = 11.46992 PVA=$2,000 (annuity amount) x 11.46992=$22,940 PV=$22,940 (future amount) x .24698 (present value of $1, n=24,i=6%) =$5,666 1. To determine the required contribution, we calculate the present value on December 31, 2021, of the deferred annuity of $2,000 that begins on December 31, 2046, and is expected to end on December 31, 2065. 2. We can calculate the present value of the annuity using a two-step process. The first step computes present value of the annuity as of December 31, 2045, by multiplying the annuity amount by the 20-period ordinary annuity factor, which is 11.46992. 3. By doing so, we will get the PVA of $22,940. This is the present value as of December 31, 2045. This single amount is then reduced to present value as of December 31, 2021, by a second calculation, by multiplying the annuity amount by the 24-period present value factor, which is .24698. 4. Stridewell would have to contribute $5,666 at the end of 2021 to fund the estimated pension benefits earned by its employee in 2021. Viewed in reverse, $5,666 invested now at 6% will accumulate a fund balance of $22,940 at December 31, 2045. If the fund balance remains invested at 6%, $2,000 can be withdrawn each year for 20 years before the fund is depleted.

given $740 now and $1,000 three years from now would depend on your time value of money

- at a rate of 10% you would choose the $1,000 in 3 years, because the $740 invested at 10% for 3 years would grow to only $984.94 ($740 x 1.331 (FV of $1, i=10%, n=3) - at a rate of 11% or higher, you would choose the $740 now. you would invest the $740 now and have it grow to $1,012.05 in 3 years.

relation between the present value and the future value

- future value requires the ADDITION of compound interest - present value requires the REMOVAL of compound interest

concept check: present value concepts Harry Byrd's Chicken Shack agrees to pay an employee $50,000 a year for six years beginning two years from today and decides to fund the payments by depositing one lump sum in a savings account today. The company should use which present value concept to determine the required deposit?

-present value of a deferred annuity; the calculation is the amount to be deposited today, the present value, of six equal payments (an annuity), that doesn't start for two years (deferred annuity)

(compound interest example) investing $1,000 in a savings account paying 10% interest compounded twice a year. what will be her investment balance at the end of he year? what is the effective annual interest rate?

1,102.50 10.25%

present value of a deferred annuity; At January 1, 2021, you are considering acquiring an investment that will provide three equal payments of $10,000 each to be received at the end of three consecutive years. However, the first payment is not expected until December 31, 2023. The time value of money is 10%. How much would you be willing to pay for this investment?

2 steps: 1. calculate the PV of the annuity as of the beginning of the annuity period 2. reduce the single amount calculated in (1) to its present value as of today step 1: PVA=$10,000 (annuity amount) x 2.48685 = $24,868 using present value of an ordinary annuity of $1 n=3, i=10% step 2: PV=$24,868 (future amount) x .82645 = $20,552 using present value of $1 n=2, i=10%

determining i when other variables are known - unequal cash flows; Suppose that you borrowed $400 from a friend and promised to repay the loan by making three annual payments of $100 at the end of each of the next three years plus a final payment of $200 at the end of year four. What is the interest rate implicit in this agreement?

400 PV = $100 (annuity amount) x PVA + $200 (single payment) x PV n=3, i=?, annuity=$100 & n=4, i=?, annuity=$200 This equation involves two unknowns and is not as easily solved as the two previous examples. One way to solve the problem is to trial-and-error the answer. For example, if we assumed i to be 9%, the total PV of the payments would be calculated as follows: PV = $100 (2.53129) + $200 (.70843) = $395. Because the present value computed is less than the $400 borrowed, using 9% removes too much interest. Recalculating PV with i = 8% results in a PV of $405. This indicates that the interest rate implicit in the agreement is between 8% and 9%.

(future value) oliver kim invests $5,000 in a bank account earning 8% interest compounding annually. How much will he have in his account in four years? the future value of $1 at 8% for four years is 1.36049 per Table 1 (Future Value of $1)

FV= 1 x FV Factor FV= $5,000 x 1.36049 FV=$6,802

determine unknown number of periods - ordinary annuity; Assume that you borrow $700from a friend and intend to repay the amount in equal instalments of $100 per year over a period of years. The payments will be made at the end of the year beginning one year from now. Your friend wishes to be reimbursed for the time value of money at a 7% annual rate. How many years would it take before you repaid the loan?

PV $700 = $100 (annuity amount) x ? using present value of an ordinary annuity of $1 n=? i=7% $700/$100=7.0 7.0 is the PVA table factor. In the PVA table (Table 4), search the 7% column (i= 7%) for this value and find 7.02358 in row 10. So it would take about 10 years to repay the loan.

(present value of a single amount) today's equivalent to a particular amount in the future

PV= FV / (1+i)^n $1,331 / (1+.10)^3 =$1,000

determining (i) when other variables are known; Suppose that a friend asked to borrow $331 today (present value) and promised to repay you $100 (the annuity amount) at the end of each year for each of the next four years. What is the annual interest rate implicit in this agreement?

PV=$331 n=4 i=? $331 (PV) / $100 (annuity amount) = 3.31 3.31= PVA table factor In the PVA table search row four (n=4) for 3.31. We find it in the 8% column The effective interest rate is 8%.

(determining an unknown number of periods) you want to invest $10,000 today to accumulate $16,000 for graduate school. If you can invest at an interest rate of 10% compounded annually, how many years will it take to accumulate the required amount?

PV=10,000 FV=16,000 n=? i=10% 10k/16k=.0625 so we use the present value table & conclude n=5

valuing a note: one payment, explicit interest: the stridewell wholesale shoe company manufactures athletic shoes for sale to retailers. the company recently sold a large order of shoes to harmon sporting goods for $50,000. stridewell agreed to accept a note in payment for the shoes requiring payment of $50,000 in one year plus interest at 10%

PV=? FV=$55,000 at the end of year 1 n=1 i=10% $55,000(FV) x .90909 = $50,000(PV) by using present value of $1, n=1, i=10%; we arrive at .90909

valuing a note: one payment, no interest stated; the stridewell wholesale shoe company recently sold a large order of shoes to harmon sporting goods. terms of the sale require harmon to sign a noninterest-bearing note of $60,500 with payment due in 2 years

PV=? FV=60,500 at the end of year 2 n=2 i=10% to find the PV of the note (price of the shoes), we need to know either the cash price of the shoes or the appropriate interest rate for a transaction like this one. let's say the market rate is 10% $60,500 FV x .82645 = $50,000 PV we use the present value table of $1

valuation of long-term leases; On January 1, 2021, the Stridewell Wholesale Shoe Company signed a 25-year lease agreement for an office building. Terms of the lease call for Stridewell to make annual lease payments of $10,000 at the beginning of each year, with the first payment due on January 1, 2021. Assuming an interest rate of 10% properly reflects the time value of money in this situation, how should Stridewell value the asset acquired and the corresponding lease liability?

PVAD = $10,000 (annuity amount) × 9.98474 (PVAD of $1, n=25, i=10%) = $99,847 journal entry right of use asset 99,847 (d) lease payable (c) 99,847

annuity due example (cash flows occur at the beginning of each period)

a 3 year lease of a building that begins on december 31, 2021, and ends on december 31, 2024, may require the first year's lease payment in advance on december 31, 2021. the 3rd and last payment would take place on december 31, 2023, the beginning of the 3rd year of the lease.

effective rate

actual rate at which money grows per year (example: compounded semiannually - 12%/2=6% quarterly - 12%/4=3% monthly - 12%/12=1%)

interest

amount of money paid or received in excess of the amount of money borrowed or lent

ordinary annuity example (cash flows occur at the end of each period)

an installment note payable dated december 31, 2021, might require the debtor to make 3 equal annual payments, with the first payment due on december 31, 2022, and the last one on december 31, 2024

preview of accounting applications of present value techniques (single cash amount): monetary liabilities

are obligations to pay amounts of cash in the future, the amount of which is fixed or determinable (ex: notes payable)

annuity due

cash flows occur at the beginning of each period

ordinary annuity

cash flows occur at the end of each period

deferred annuity

exists when the first cash flow occurs more than one period after the date the agreement begins

Rita Grant wants to accumulate a sum of money to pay for graduate school. Rather than investing a single amount today that will grow to a future value, she decides to invest $10,000 a year over the next three years in a savings account paying 10% interest compounded annually. She decides to make the first payment to the bank one year from today.

first payment $10,000 x 1.21 = $12,100 second payment $10,000 x 1.10 = $11,000 third payment $10,000 x 1.00 = $10,000 12,100+11,000+10,000 = $33,100 FV @ the end of year 3 or use (FV of an ordinary annuity of $1)

(future value of a single amount) cindy johnson invested $1,000 in a savings account for 3 years paying 10% interest compounded annually

future value $1,331 at the end of year 3 (1,000 x 1.331) I x FV Factor = FV

due to the concept of time value of money

given a choice between $1,000 now and $1,000 three years from now you would chose to have the money now so you could earn interest on it.

preview of accounting applications of present value techniques (single cash amount): monetary assets

include money and claims to receive money in the future, the amount of which is fixed or determinable (ex: cash and most receivables)

compound interest

includes interest not only on the initial investment but also on the accumulated interest earned in previous periods (occurs when money remains invested for multiple periods) example: invested $1000 in a savings account paying 10% interest compounded annually. How much interest will she earn in each of the next 3 years, and what will be her investment balance after 3 years? $1,331

simple interest

initial investment x annual interest rate x period of time investment 1,000 x (annual interest rate 10%) x (time period 1 year) = (simple interest $100)

time value of money

means that money can be invested today to earn interest and grow to a larger dollar amount in the future (useful in valuing a variety of assets and liabilities)

concept check: annual deposits; The Stinch Fertilizer Corporation wants to accumulate $8,000,000 for plant expansion. The funds are needed on January 1, 2026. Stinch intends to make five equal annual deposits in a fund that will earn interest at 7% compounded annually. The first deposit is to be made on January 1, 2021.

n=5 i=7% use future value of an annuity due of $1 at 7% for 5 periods and get = 6.15 $8 mil x / 6.15 = $1,300,813

the versa tile company purchased a delivery truck on feb 1 2021. the agreement required versa tile to pay the purchase price of $44,000 on feb 1, 2022. assuming an 8% rate of interest, to calculate the price of the truck versa tile would multiply $44,000 by the:

present value of $1; the calculation is for the present value TODAY of the $44,000 to be paid one year from now.

concept check: accounting applications of PV techniques; On May 31, 2021, the Gusto Beer Company leased a machine from B. A. Lush, Inc. The lease agreement requires Gusto to pay six annual payments of $16,000 on each May 31, with the first payment due on May 31, 2021. Assuming an interest rate of 6% and that this lease is treated as an installment sale (capital lease), Gusto will initially value the machine by multiplying $16,000 by which of the following?

present value of an annuity due of $1 at 6% for 6 periods. The calculation is how much is recorded today, the present value of equal payments that start today (annuity due).

valuation of installment notes; On January 1, 2021, the Stridewell Wholesale Shoe Company purchased a $35,000 machine, with a $5,000 down payment and a 5-year installment note for the remaining $30,000. Terms of the note call for Stridewell to make annual instalment payments at the beginning, with the first payment due on January 1, 2021, and at each December 31 thereafter. Assuming an interest rate of 4%, what is the amount of the annual installment payments?

present value of an annuity due of $1, n=5, i=4% = 4.62990 $30,000(note)/4.62990 = $6,480 is the amount of the annual installment payments journal entries: - Stridewell initially will record the machine at its $35,000 cost with credits to cash and notes payable. machine (d) 35,000 cash (c) 5,000 notes payable (c) 30k - Stridewell will record the first installment payment at the date of purchase. notes payable (d) 6,480 cash (c) 6,480 - No interest is needed for the first installment payment because no time had yet passed, so no interest had accrued. With the second payment, though, the payment represents part interest and part reduction of the loan interest expense (d) 941 notes payable (d) 5,539 cash (c) 6,480 4% x ($30,000 - 6,480) = $941.

justin investor wants to calculate how much money he needs to deposit today into a savings account that earns 4% in order to be able to withdraw $6,000 at the end of each of the next 5 years. he should use which present value concept?

present value of an ordinary annuity of $1 for 5 periods; because the calculation is how much needs to be deposited TODAY (present value) so that equal amounts can be withdrawn over the next 6 years at the end of the year (ordinary annuity)

basic annuities: annuity

series of cash flows of the same amount received or paid each period ex: - a loan on which periodic interest is paid in equal amounts - a lease paid in equal installments during a specified period of time

if you have a set of present value tables, an annual interest rate, the dollar amount of equal payments made, and the number of semiannual payments, what other information do you need to calculate the present value of the series of payments?

the timing of the payments (whether they are at the beginning or end of the period)

concept check: accumulated savings; I.R. Wright plans to make quarterly deposits of $200 for five years into a savings account. The first deposit will be made IMMEDIATELY. The savings account pays interest at an annual rate of 8%, compounded quarterly. How much will Wright have accumulated in the savings account at the end of the five-year period? Round to the nearest dollar.

use future value of an annuity due for 20 (5 years x quarterly deposits) periods at 2% to get = 24.2974 200 x 24.2974 = $4,859

concept check: accumulated savings ordinary annuity; U. B. Wong plans to make quarterly deposits of $200 for five years into a savings account. The deposits will be made at the END of each quarter. The savings account pays interest at an annual rate of 8%, compounded quarterly. How much will Wong have accumulated in the savings account at the end of the five-year period? Round to the nearest dollar.

use future value of an ordinary annuity for 20 periods at 2% to get = 26.1833 $200 x 26.1833 = $5,237

present value of an ordinary annuity; Rita Grant wants to accumulate a sum of money to pay for graduate school. She wants to invest a single amount today in a savings account earning 10% interest compounded annually that is equivalent to investing $10,000 at the END of each of the next three years.

use present value of an ordinary annuity of $1 n=3 i=10% PVA = $10,000 (annuity amount) x 2.48685 = $24,868

valuation of long-term bonds; On June 30, 2021, Fumatsu Electric issued 10% stated rate bonds with a face amount of $200 million. The bonds mature on June 30, 2041 (20 years). The market rate of interest for similar issues was 12%. Interest is paid semiannually (5%) on June 30 and December 31, beginning December 31, 2021. The interest payment is $10 million (5% × $200 million). What was the price of the bond issue? What amount of interest expense will Fumatsu record for the bonds in 2021?

use present value of an ordinary annuity of $1 n=40, i=6% = 15.04630 PVA=$10 million (annuity amount) x 15.04630 = $150,463,000 PV=$200 million (lump-sum) x .09722 (PV of $1, n=40, i=6%) =19,444,000 now we find the price of the bond issue by subtracting: 150,463,000-19,444,000 =$169,907,000 to find interest expense: $169,907,000 x 6% =$10,194,420

present value of an annuity due; In the previous illustration, suppose that the three equal payments of $10,000 are to be made at the BEGINNING of each of the three years. Recall from the previous slide that the future value of this annuity is $36,410. What is the PVAD

using the PVAD table to calculate the present value; n=3, i=10% $10,000 (annuity amount) x 2.73554 = $27,355

(determining an unknown interest rate) suppose a friend asks to borrow $500 today and promises to repay you $605 two years from now. what is the annual interest rate you would be agreeing to?

we know n=2; present value=500; future value=605 i=? *we use the present value table to get .82645 also, we can divide 500/605 to get the same answer.