Description
The time value of money is a very important concept in agricultural finance. It is widely used in investment evaluation, particularly for discounted cash flow analysis of investment alternatives. It is also very useful in determining values for financial transactions.
Decem bar 1993
A.E. Ext. 93-18
:
Time Value of Money
Financial Examples and Calculations
Using Tables and Calculators
by
Eddy L. LaDue
Department of Agricultural, Resource and Managerial Economics
College of Agriculture and Life Sciences
Cornell University
Ithaca, NY
It is the policy of Cornell University actively to support equality
of educational and employment opportunity. No person shall I
be denied admission to any educational program or activity or
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I
l
discrimination involving, but not limited to, such factors as race,
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affirmative action programs which will assure the continuation
of such equality of opportunity.
1
Table of Contents
Future Value
Future value of a sum (compound interest)
1. Savings accounts
2. Inflating farm values
Future value of a nonuniform series
3. Saving different amounts each year
4. IRA
Future value of a uniform series
5. Sinking fund
6. IRA
7. Saving to start farming
Present Value
Present value of a future sum
8. Stripped treasurer
9. Zero coupon bonds
10. Unamortized loan
Present value of a nonun iform series
11. Bond
12. Unequal payments from sale
Present value of a uniform series
13. Land contract
14. Loan principal
15. Payments on a loan
16. Farm machinery early purchase payments
17. Financial lease (annual payment)
18. Financial lease (monthly payments)
19. Loan principal
20. Monthly payments on a loan
Present value of a uniform infinite series
21. Value of farmland
Amortization
22. Monthly payment loan
23. Annual payment loan
24. Partial amortization
25. Lease paym ents
2
TIME VALUE OF MONEY
Financial Examples and Calculations
Using Tables and Calculators
by
Eddy L. LaDue
1
The time value of money is a very important concept in agricultural finance. It is
widely used in investment evaluation, particularly for discounted cash flow analysis of
investment alternatives. It is also very useful in determining values for financial
transactions. For these transactions, the interest rate reflects the time value of money
and the monetary values used are determined by time value concepts.
This publication provides a discussion of the time value of money concepts and
their application to financial transactions.
2
A series of examples are presented to illustrate
the application and calculations for each concept. Example calculations are given in
detail to allow readers to learn the calculation procedure. Agricultural finance examples
are used throughout.
Two approaches to the calculation are provided. One approach uses tabled values
of time value functions. Specifically, the tables provided in "Present Value, Future Value
and Amortization: Formulas and Tables" Cornell University Agricultural Economics
Extension 90-17 are used. Whenever a table number (Le. Table 6) is used in calculation,
reference is to a table in that publication.
In the second approach, calculations are made using a calculator with financial
functions. The calculator referred to is a Texas Instruments BAli Plus. The calculations
can, of course, be made with other calculators, usually with slightly different key
identification and key stroke configuration. In some cases, more complex variations of
a problem that cannot be handled with the tables because table values are presented for
only selected situations, are illustrated using the calculator. The financial function keys
referred to in this publication are:
Definition
N Number of payments (or number of compoundings if no
payments are involved)
IIV Interest rate per year
Professor of Agricultural Finance, Cornell University
2
For discussion of time value as applied to investment decision see Casler, Anderson and
Aplin, ·Capitallnvestment Analysis Using Disoounted Cash Flows."
3
PV Present value (value now)
PMT Payment (per month, year, quarter, etc.)
FV Future value (value at the end of the period under
consideration)
PlY Payments per year
BEG Payments occur at the beginning of each period (month, year,
etc.)
END Payments occur at the end of each period (month, year, etc.)
CY Compoundings per year (automatically equals payments per
year unless specifically set).
It is assumed that the number of compoundings per year equals the number of payments
per year and that payments all occur at the end of the period (year, month, etc.) unless
specifically indicated otherwise (BEG is assumed to equal END unless specifically
indicated otherwise).
A reader who plans to use the tables can use this publication by skipping the
sections labeled "Using Calculators". Similarly, a reader who plans to use a calculator
can use this publication by skipping the sections labeled "Using Tables'·.
Time value relationships can be divided into two basic categories: (1) the future
value of a monetary amount or series of amounts, and (2) the present value of a
monetary amount or series of amounts. Although present value and future value
represent a continuum of the same relationships, we are normally interested in
determining a value for one point in time, usually the present, or some point in the future.
Amortization is really an application of future value concepts. The discussion of these
concepts is presented in three sections: (1) future value, (2) present value and (3)
amortization.
FUTURE VALUE
Future value techniques are designed to determ ine the value of an amount, or
series of amounts of money as of some fixed point in time in the future. Examples
include the value of a savings account in 'five years or the value of a farm in 19 years
when land values are increasing at five percent per year.
4
1. Future value of a present sum (Compound Interest)
•
When we are dealing with the future value of a single amount of money, we use
compound interest, or the future value of a present sum. The future value of a present
sum is calculated using equation (1):
FV =PV(1+i)n (1 )
Where: FV = Value in period n (n periods in the future)
PV = Value in period 0 (now)
=Interest rate per conversion period
n =Number of conversion periods
If each period is a year, n is the number of years into the future on which the value
is to be calculated, and i is the annual interest rate. In this case, the formula results in
normal compound interest with annual compounding. If payments are made more
frequently than annually, i is the annual rate divided by the number of payments per year
and n is the number of payments per year multiplied by the number of years.
Example 1
If $1 is invested in a savings account with a fixed interest rate of eight percent and
annual compounding, how much will be in the account in 10 years? Direct application of
the equation results in the following:
$1(1+.08)10 =1(2.16) =$2.16
Using Tables
The calculations can be made using the compound interest table (Table 1). The
factor from Table 1 for 10 years at eight percent interest is 2.158924. Multiplying the
factor times the investment gives the correct value.
2.158924($1) = $2.16
5
Using Calculators
PlY =1
N = 10
FlY = 8 FV = $2.16
PN =1
PMT =0
FV =?
If interest were compounded quarterly instead of annually:
PlY
=
4
N
=
40 (4 payments per year x 10 years)
IIY
=
8
PV
=
1
PMT
=
0 FV =$2.21
FV
=
?
Example .2
Pete now has a farm that is worth $200,000. If farm values inflate at an average
rate of four percent over the next 15 years, what will the farm be worth at the end of the
15 year period?
Using (1): $200,000 (1 +.04) 15 =$360,189.
Using Tables
$200,000 (1.800943) =$360,189
?
Table 1, 15 years, four percent
Using Calculators
PlY =1
N =15
IIY =4
PV =200,000
PMT =0 FV =$360,189
FV =?
6
2. Future Value of a Nonuniform Series
The future value of a series of different amounts can be determined using the
futurevalueofaseries. Thefuturevalueofaseriesofsums iscalculated using equation
(2).
(2)
Where: FV .=Value in period (year) n
PV0 =Amount invested in period 0 (now)
PV
1
=Amount invested in period 1
PV
n
= Amount invested in period n
i = Interest rate
n =Number of years to future time for which the value is
calculated
Ifyou lookatthisequation closely, you willseethatthisisjustaseries ofequation
(1) calculations addedtogether. Thus, Table 1can also be usedtodeterminethefuture
value.
Example
Jason, a beginning high school freshman, is planning to save money from his
weekend and summer work on the farm to buy a car to drive back and forth to Cornell
while he is attending college. He invested $200 at the end of this summer (now) and
expects he can invest $400 one year from now, $500 in three years, and $600 in four
years(heplanstoworkon amidwestfarm between hisSophomoreandJunioryearsand
willspend allhemakes). Jason'sfatherhasagreedtoborrowthesefunds andpayJason
10 percent interest. How much money will Jason have at the end of four years?
FV
4
=200(1+.10)4+ 400(1+.10)3 + 500(1+.10)1 + 600
=200(1.4641) + 400(1.3310) + 500(1.10) + 600
=292.82 +532.40 + 550 + 600
=$1,975.22
7
Using Tables
FV = 200 (1.4641) + 400 (1.3310) + 500 (1.1) + 600
,
Table 1, 10%, 4 years Table 1, 10%, 3 years
= $1,975.22
Using Calculator
PlY = 1 PlY
=
1 PIX
= 1
N = 4
N
=
3 N
= 1
FlY = 10 FV = $292,82 IIY
=
10 FV = $532.40 IIY = 10 FV = $550,00
PV =200 PV = 400 PV
=500
PMT = 0 PMT = 0 PMT = 0
FV =? FV
=
? FV
= ?
Value =292.82 + 532.40 + 550 + 600 =$1,975.22
Example
Three years ago, Will invested $2,000 in an individual retirement account (IRA).
Two years ago, he invested $1,800 and last year he invested $1,500. If this IRA earns
interest at eight percent, how much is currently in the account?
Using Tables
Value =2,000(1.259712) + 1,800(1)664) + 1,500(1.08)
Table 1, 8%, 2 years
=2,519.42 + 2,099.52 + 1,620
=$6,238.94
Using Calculators
PlY = 1 PlY =1 PIX =1
N =3 FV= N =2 FV= N =1 FV=
FlY = 8 $2,519.42 IIY = 8 $2,099,52 IIY =8 $1,620
PV = 2,000 PV = 1,800 PV = 1,500
PMT = 0 PMT= 0 PMT = 0
FV =? FV =? FV =?
Value =$2,519.42 + $2,099.52 + 1,600 =$6,238.94
8
3. Future Value of a Uniform Series
If the sum involved (invested) is the same each period (year), the future value can
be determined more easily, with a single calculation, using equation (3).
FV = ~ ( 1+ i Jn -1 ]
(3)
Where: FV = Value in period (year) n (future)
PMT = Amount invested each period (year, month, etc.)
=Interest rate per period (year, month, etc.)
n =Number of periods (years or months or ... )
Using Tables
Values of the equation 3 expression in the brackets, for situations where the period
is one year, are presented in Table 4. Thus, the future value can be calculated by simply
multiplying the amount received each year by the appropriate value from Table 4.
Table 4 assumes that investments (payments) are made at the end of each year,
including the final year (at the time for which the future value is being calculated). If this
is not the case, adjustments must be made. If an investment is made at the beginning
of the period (as well as the end), use the number of years plus one in selecting the
coefficient from Table 4. If investments are made at the beginn ing of each year but none
is made at the end of the period (the time for which the future values are being
calculated), use the number of years in the period plus one in selecting the coefficient
and subtract one (1.0) from the value of the coefficient found. When the first payment
is made in one year and no payment is made at the end of the last year, subtract one
(1.0) from the coefficient for the number of years involved. The following table indicates
the appropriate values for different situations.
Payment at Table
Payment End of Value Coefficient
Payment Plan Now? Period to Use to Use
End of each year No Yes n Table
Beg. of each year Yes No n+1 Table-1
Beg. and end Yes Yes n+1 Table
End of each year
except last No No n Table-1
9
Table4can alsobe used todeterminetheinvestmentrequired each yearto meet
agoal orprovideagiven sum ofmoneyatthefuturepoint in time.. This isaccomplished
by dividing the coefficientfrom Table 4 into the amount required (the goal).
Example 5
Loren'sbankerhas required thatLoren developasinkingfund toprovidethefunds
for a neworchard sprayerwhich he will need in five years.
a. If he invests $5,000 per yearat the end of each yearfor the next five years and
the funds earn interest at 12 percent, how much money will be available for
purchase of the sprayer?
Using Tables
$5,000(6.3528) =$31,764
r
Table 4, five years, 12 percent
Using Calculators
PlY = 1
N =5
IIY =12 FV =$31,764
PV = 0
PMT =-5,000
FV =?
If interest were compounded monthly instead of annually:
PlY = 1
CIY = 12
N =5
IIY = 12 FV =$32,198
PV =0
PMT =-5,000
FV =?
b. If Lorenexpectsthesprayertocost$50,000 in fiveyears, andfunds areputinthe
same 12 percent account with annual compounding, how much must he invest
each year?
10
Using Tables
$50,000 =$7,871
6.3528
Using Calculators
PlY
=1
N =5
FlY = 12 PMT =$7,870.49
PV =0
FV =50,000
PMT =?
If interest were compounded monthly instead of annually:
PlY
=1
CIY =12
N =5
IIY
=12
PMT =$7,764.51
PV =0
FV =50,000
PMT =?
c. Ifinvestmentsweremadeatthebeginningofeachyearinsteadoftheend (annual
compounding), how much must be invested each year?
Using Tables
$50,000 =$7,027
7.1152
f
(Table 4, six years, 12 percent) - 1
Using Calculators
BEG = BEG
PlY = 1
N =5 PMT =$7,027
IIY = 12
PV = 0
FV =50,000
PMT =?
d. If payments were made now and at the end of each year, how much must be
invested?
11
Using Tables
$50,000 =$6,161
8.1152
Using Calculators
Making payment now is the same as making a payment at the end of last year
(year 0). Thus, this payment is like adding another year (year 0) to the front end
of the payment stream. The number of years (periods) becomes 6.
BEG =END
PlY
=1
N =6
IIY =12
PV =0 PMT =$6,161
FV =50,000
PMT =?
Example 6
If Joe invests $2,000 in an IRA now and at the end of each year for the next 15
years, how much will be in the account after he makes his 15 year payment if the account
earns nine percent per year?
Using Tables
$2,000(33.0034) =$66,007
/"
Table 4, 16 years, nine percent
Using Calculators
PlY =1
N =16
IIY =9 FV =$66,007
PV =0
PMT = -2,000
FV =?
12
If interest were compounded quarterly:
PlY = 1
CIY =4
N = 16 FV =$67,764
IIY =9
PV =0
PMT =-2,000
FV =?
Example 7
Sandy is planning to start farming in 10 years. To get the initial funds needed, she
is working for Farm Credit and plans to set aside $4,000 per year starting now from her
salary in a mutual fund account that she expects to earn 11 percent per year. How much
money will she have for farm investment in 10 years if moving and other costs keep her
from investing at the end of the 10th year?
Using Tables:
$4,000(18.5614) =$74,246
./'
(Table 4, 11 years, 11 percent) - 1
Using Calculators
BEG =BEG
PlY =1
N =10
IIY =11 FV =$74,246
PV =0
PMT =-4,000
FV =?
If she knows she will need $100,000 to start, how much will she need to invest
each year?
Using Tables:
$100,000 =$5,387.52
18.5614
13
Using Calculators:
BEG = BEG
PlY =1
N =10 PMT =$5,387.52
IIY = 11
PV =0
FV =100,000
PMT =?
PRESENT VALUE
Present value or "discounting" is essentially the reverse of future value. Present
value determines the current value of an amount or series of amounts to be received in
the future. In general terms, present value is the amount that you could be paid now and
be equally well off compared to receiving a specified payment or series of payments at
some specific time in the future.
4. Present Value of a Future Sum
Present value of a future sum is used to calculate the amount that one could
receive now that would be equal in value to the receipt of a specified single amount at
some point in the future. Remember, the compound interest formula told us the future
value of a current sum. Well, the present value of a sum tells us the current value of
a future sum. We can modify the compound interest formula,
FV = PV(1+i)n
by dividing both sides by (1+i)n, to get the present value formula.
PV= Vn
( I +r) n
(4)
This can be reformulated as:
PV =FV (1+i)"n (5)
14
Using Tables
Thebracketedvalue (1+ithas been placed in tabularform in Table2. Returning
toourExample 1situation, ifwewill receive $2.16 in 10yearsandfunds received
now could earn eight percent over the next 10 years, the present value of the
$2.16
= $2.16(.463193) = $1
I
Table 2, 10year, eight percent
Note that .463193 = 1
f 2.158924,
Table 2 Table 1
Thus, if you have Table 2you can calculate Table 1values and vice versa.
Table 1 = 1
Table 2
Table 2= 1
Table 1
Thus, the answer to Example 2 could be calculated using Table 2 values as shown
below. The modest difference between the values results from rounding in the tables.
$200,000 ( 1.0 ) =$360,188
.555265
?
Table 2, 15 years, four percent
Example 8
Tobroaden the rangeof investmentalternativesavailable,securitiesdealershave
developedstrippedtreasurieswhereeachoftheinterestpaymentsandthefinal principal
payment on a treasury note or bond are separated into separate instruments and sold
individually. Thus, you can buy the next year's interest payment on a million dollar
treasurynoteortheyear2000principal payment'on a$100,000treasurybond. Similarly,
zerocoupon bonds represent an agreement of the issuer to pay the face amountof the
bond atthe maturitydate. What is the presentvalue of the $100,000 principal payment
(or zero coupon bond) to be received in 12 years if you could earn ninepercent interest
on funds invested overthe period?
15
Using Tables
$100,000 (.355535) = $35,553
/
Table 2, 12 years, nine percent
Using Calculators
PlY
=1
N =12
IIY =9 PV = 35,553
PMT =0
FV = 100,000
PV =?
Example 9
TwoyearsagoJohn loanedhisson, JohnJr.,alocalgrapegrower, $20,000tobuy
additionalvineyard. The loan wasforfiveyearswithinterestcompoundedannuallyat 10
percent. but the loan was unamortized and no interest or principal was to be paid until
the end of the five yearperiod.
a. How much will John receive at the end of the five year period?
Using Tables
$20,000(1.61051) = $32,210
/"
Table 1, five years, 10 percent
Using Calculators
PlY =1
N =5
IIY =10 FV=$32,210
PV = 20,000
PMT =0
FV =
b. If John were tosell the loan to an investortoobtainfundsforotheractivities, how
much could the investor afford to pay if she expected to earn 16 percent on all
investments?
16
Using Tables
$32,210(.640658) =$20,636
./'
Table 2, three years, 16 percent
Using Calculators
PN =1
N =3
IN = 16 PV = $20,636
PMT = °
FV =32,210
PV =?
5. Present Value of a Nonuniform Series
When a series of amounts are to be received in the future and the amounts are
not all of the same magnitude, equation (6) can be used.
PM?;
+ (6)
(1 + i)n
For ease in use, equation (6) can be reformulated as:
PV = PMI:: (1 + i) -1 + ~ 1 + i) -2 + • •. + PM?;( 1 + i) n (7)
In equation (7) form, it is easy to see that this is just the sum of a series equation 5
calculations.
Thus, the present value of the series is just the sum of the present value of the individual
amounts in 'that series. That is, it is the sum of a series of calculations using
equation 5.
Example 10
What is the value of a Federal Farm Credit Bond with a face value of $1,000, a 10
percent interest rate with annual payments and a three year maturity, if increasing interest
rates have increased the normal yield on such bonds to 15 percent (or, you could earn
15 percent on available funds if invested elsewhere)?
17
You will receive the following amounts:
Year 1 ($1,000 x .10) = $ 100
Year 2 ($1,000 x .10) = 100
Year 3
plus
($1,000 x .10)
$1,000 face value = $1,100
Using Tables
PV = 100(.869565) + 100(.756144) + 1,100(.657516)
t
Table 2 Table 2
one year three years
15 percent 15 percent
= 86.96 + 75.61 + 723.27
= $885.84
If you paid $885.84 for this bond, how much would you earn? Obviously 15 percent.
Using Calculators
PlY = 1
N =3
IIY = 15 PV = 885.84
PMT = 100
FV = 1000
PV =?
If bond interest payments were semiannual:
PlY = 2
N =6
FlY = 15 PV = 882.65
PMT = 50
FV = 1000
PV =?
Why is the present value lower with more frequent payments? The semiannual
discounting of the $1000 reduces the present value more than the earlier receipt of part
of the interest payments increases the present value.
18
Example 11
In negotiating the sale of some machinery to his son, Harry agreed to accept
interest and principal payments of $10,000 in one year; $15,000 in two years, and
$22,000 in three years (in line with the son's ability to pay). If Harry can earn nine
percent on invested funds, how much is he receiving for the machinery?
Using Tables
PV = $10,000(.917431) + 15,000(.841680) + 22,000(.772183)
?
= 9,174.31 + 12,625.20 + 16,988.03 Table 2,9%,3 years
= $38,787.54
Using Calculators
PlY =1 PlY
=1
PlY
=1
N =1
N =2 N =3
IIY =9 PV = 9,174.31 IIY =9 PV = $12,625.20 IIY =9 PV = 16,988.04
PMT =0 PMT =0 PMT =0
FV = 10,000 FV =15,000
FV = 22,000
PV =? PV =? PV =?
Value = 9,174.31 + 12,625.20 + 16,988.04 = $38,787.55
6. Present Value of a Uniform Series
If the amounts to be received in the future are the same for all consecutive years,
the present value can be obtained more simply using equation (8).
PV= PMT [1-( 1/i) -nj (8)
Where: PV =The value in period zero (now)
PMT =The amount received each period (year, month, etc.)
= The interest rate
(ann ual interest rate divided by num ber of payments per year)
n = The number of periods the amount PMT is to be received
19
Example 12
Ann is an only child who became a banker rather than returning to the home farm.
At the time her father retired from farming, he sold the farm to Dave on a land contract.
The contract called for annual payments of $3,000 per year with interest at 10 percent.
At the time of Ann's father's death, there were 10 years remaining on the contract and
she could earn 16 percent on funds available for investment at the time. Dave offers
$15,000 to payoff the contract. Should she accept Dave's offer?
Using Tables
The bracketed part of equation 8 has been calculated for a number of values of
i and n, and presented in Table 3 for annual payments and table 5 for monthly payments.
Present values are calculated by multiplying the amount received each year by the correct
value from Table 3 or 5.
$3,000(4.83323) =$14,500
1
Table 3, 10 years, 16 percent
Using Calculators
PlY
=1
N
=10
IIY
=16 PV =$14,500
PMT =-3,000
FV =0
PV =?
Since she can earn 16 percent on money invested, the $3,000 per year for 10
years is equivalent to $14,500. Dave has offered $15,000. She should accept the offer.
Present value can be used to determine the principal outstanding on a loan.
Discounting removes the earnings that could be obtained on funds if they are received
now rather than in the future. For an amortized loan, the earnings generated for the
holder of the loan is the interest to be received. Thus, discounting removes the interest
from the loan payments and the present value is the current principal balance. Only the
number of payments remaining, the amount of each payment and the interest rate must
be known. Thus, the principal outstanding on Ann's contract can be calculated by
discounting the payments at the interest rate specified by the contract (10 percent).
20
Using Tables
$3,000(6.14457) = $18,434 = principal owed
'\
Table 3, 10 years, 10 percent
Using Calculators
PlY = 1
N =10
IIY =10 PV = $18,434 = principal
PMT = -3000
FV = 0
PV =?
Example 13
Ten years ago, the Vegies National Bank made a 25 year loan at six percent
interest to a local vegetable farmer to purchase a farm. The annual payments are
$8,719.
1. The principal outstanding is:
Using Tables
$8,719(9.7122) = $84,681
1
Table 3, 15 years, six percent
Using Calculators
PlY
=1
N
=15
IIY =6 PV = $84,681
PMT =-$8,719
FV
=0
PV =?
2. If an insurance company (or someone else) that is currently earning 16 percent on
newly invested funds were to buy this loan, the amount they can afford to pay for the loan
is:
21
Using Tables
$8,719(5.5755) =$48,613
t
Table 3, 15 years, 16 percent
Using Calculators
PlY
=1
N
=15
IIY
=16
PV = #48,612
PMT =- 8,719
FV =0 (results differ due to rounding on tabled values)
PV =?
Example 14
Kelly borrowed the funds to buy new greenhouse space. The terms of the loan
are:
$50,000 borrowed
15 percent interest
25 years repayment period
Annual payments
Using Tables
The annual payments calculated using Table 3 are:
$50,000 =$7,735.03
6.4641
,
Table 3, 25 years, 15 percent
This result can be checked using amortization tables (Table 7) where the payment
per $1 for loans for 15 percent for 25 years is $0.1547.
$.1547 x 50,000 =$7,735
The difference between the two results is due to rounding.
Example 15
In an effort to stimulate lagging farm machinery sales, John Deere recently offered
a delayed payment plan which required no interest until March 15 and no payment until
22
December 15. The interest rate charged was 12 percent and interest was charged from
the date of the purchase or March 15, whichever occurs last. The loan is for five years
with six equal payments starting December 15. Mark purchased $100,000 worth of
machinery on January 15, and took advantage of this special repayment plan.
Calculating the amount of each payment in this case requires first determining the
amount outstanding on December 15. The interest can be calculated as,
=$9,000
12
leaving a total of $109,000 outstanding on December 15.
Using Tables
Since the first payment is due on December 15, the Table 3 value must be modified to
reflect the immediate payment. As of December 15, the present value coefficient for the
December 15, payment must be 1.0, thus, adjusting for the immediate payment involves
adding 1.0 to the coefficient for the remaining five payments. The annual payments are:
$109.000 =$23,671.05
4.60478,
(Table 3, five years, 12 percent) + 1.0
Clearly, if the finance plan required "no interest until December 15", the payment would
be:
$100.000 =$21,716.56
4.60478
Using Calculators
No Interest Until March 15 No Interest Until December 15
BGN = BGN BGN =BGN
PlY
=1
PlY
=1
N =6 N =6
IIY =12 PMT =23,671.07 IIY =12 PMT = 21,716.58
PV = 109,000 PV = 100,000
FV =0 FV =0
PMT =? PMT =?
The difference between values using calculator and tables is due to rounding of table
values.
23
Example 16
The appropriate value to enter on the balance sheet for a financial lease is the
present value of the future lease payments discounted at the firms incremental borrowing
rate or the implicit rate on the lease. Thus, present value can be used to calculate the
value of leases.
Jim is in the process of completing his 12/31 (end of year) balance sheet. On
March 31, two years and nine months ago, Jim leased a farrowing unit from Telmark.
The payments were $15,000 per year for seven years with all payments at the beginning
of the year. The incremental borrowing rate for Jim's business was 12 percent. Three
payments have been made (two years nine months ago, one year nine months ago, and
nine months ago). The remaining four payments are due at the end of each of the next
four years (ending March 31), so normal present values can be used for them. The
present value of the remaining payments as of the last payment date (March 31, of the
coming year) are:
Using Tables
$15,000(3.03735) =$45,560
'/
Table 3, four years, 12 percent
Using Calculators
PIV =1
N =4
IIV =12
PMT = -15,000 PV =$45,560
FV =0
PV =
Example 17
Lynda is currently leasing a tractor for $500 per month. The implicit interest rate
on the lease is 11 percent. She has 29 payments left to make. What is the present
value of the lease?
Using Tables
$500(25.3641) =$12,682
'\
Table 5, 29 months, 11 percent
24
Using Calculator
PlY =12
N = 29
IIY = 11 PV = $12,682
PMT = -500
FV = 0
PV =?
If the implicit rate on the lease were 11.23 percent, all entries are the same except IIY
= 11.23 and the present value of the lease is $12,648.
Example 18
A number of years ago, Dan borrowed funds at 13 percent interest to purchase an
orchard sprayer. His payments are $157.60 per month. He still has 47 payments to
make. What is the outstanding principal on the loan?
Using Tables
$157.60(36.6790) = $5,780.61
?
Table 5, 47 months, 13 percent
Using Calculators
PlY =12
N = 47
IIY = 13 PV = $5,780.61
PMT = 157.60
FV = 0
PV =?
Example 19
Mark is borrowing $10,000 to buy a pickup. If it is borrowed at nine percent for 30
months, how much will his monthly payments be?
Using Tables
$10.000 =$373.48
26.7751
r
Table 5, 30 months, nine percent
25
Using Calculators
PlY =12
•
N =30
IIY = 9 PV = $373.48
•
PV = 10,000
FV = 0
PV =?
If Mark uses dealer financing he can get interest at 2.99 percent by giving up a $500
rebate. In this case, all entries would bethe same except IIYwouldequal2.99 and PV
would equal 10,500. The payment would be $363.68.
7. Present Value of a Uniform Infinite Series
If the same amountwill be received each yearforever, the present value can be
calculated even moresimply. In this case, the annual amount is divided bythe interest
or discount rate, equation (9).
PV = Ali (9)
Example 20
One of the methods used to determine the value of farmland is to capitalize the
annual rent the land is expected to earn (known as the income approach to appraisal).
Thisis referredtoasthecapitalized valueof land. Ifthenetreturn tolandforaparticular
appleorchardisestimatedat$315 peracreandtheappropriatecapitalization rate isnine
percent, the capitalized value of the land equals:
$315=$3,500/acre
.09
However, if theexpectedapple price drops $0.20/bushel, and the orchard has an
average yield of 500 bushels, the expected annuaI income declines to $215 and the
capitalized value equals:
$215 =$2,389/acre
.09
AMORTIZATION
As impliedearlier, theequationforthepresentvalue ofa uniform series (equation
8) can be rearrangedtoprovide an equation forcalculatingthe payments required on a
loan, oramortization values. The amortization equation, thus, is equation 10.
26
(10)
•
•
Where PV = the amount to be amortized or the value of the loan in period zero
(now).
PMT = the amount of each payment
=
the interest rate per period (annual rate divided by the number of
payments per year)
n
=
number of payments (periods) the PMT amount is to be paid.
Using Tables
From equation 10 it is clear that the amount of the payments on a loan can be calculated
by dividing the loan amount by the present value factor from tables 3 or 5 (as illustrated
in Examples 14 and 15). It is also clear that the amount of payment per dollar of loan
can be calculated as the reciprocal of the bracketed amount in equation 10.
To make amortization calculations more straightforward values of that reciprocal
1
[1-( \+i) -n]
are calculated and presented in tables 7 and 8. These tables indicate the amount of
payment per dollar of loan.
Example 21
Aaron is selling the farm real estate to his son, Alan. They have agreed that the price
of the real estate is $450,000 and that Alan will pay 7.5 percent interest. How much are
Alan's monthly payments if the money is paid back over 20 years?
Using Tables
450,000 (.008056)
,
=$3,625.20
Table 8, 20 years, 7.5%
27
Using Calculators
PlY =12
•
N = 240
IIY =7.5 PMT = 3,625.17
PV = 450,000
FV = 0
PMT =?
The difference between results determ ined using tables and those using calculators is
due to roundin g of table values.
Example 22
Steve is borrowing the money from his mother to get started in farming. She has agreed
to lend him $75,000 and take a second priority lien on his line of equipment. They have
agreed that he will repay the loan over 7 years at 6.5 percent interest with annual
payments. How much are his payments?
Using Tables
75,000 (.182332) = $13,675
I
Table 7, 7 years, 6.5 percent
Using Calculators
PlY = 1
N =7
IIY = 6.5 PMT = $13,675
PV = 75,000
FV =0
PMT =?
Example 23
Mary is planning to buy the home farm from her parents. The farm is valued at
$500,000. They expect to ultimately will her half of the farm and plan to put the principal
Mary repays in a separate account which, along with the remaining monthly payments,
will be willed to Mary's nonfarming sister, Susan. Thus they want a loan that will amortize
half of the principal (partial amortization) and provide Mary's parents with income on the
•
complete farm value during their lifetimes. With this type of loan, how much will Mary's
monthly payments be if the loan is for 25 years at 8.25 percent interest.
•
28
Using Calculators
PN =12
N =300 (25 x 12)
IN = 8.25 PMT = $3,690
PV =500,000
FV =-250,000 (500,000 x .5)
PMT =?
Example 24
George plans to lease his line of equipm ent to his daughter, Amanda. They have agreed
that the equipment is now worth $150,000 and at the end of the 5 year lease period it will
be worth about $25,000. If, through the lease, George wants to earn 7.8 percent interest
on the money he has invested, and be paid for the machinery that is "used up", at what
level should they set Amanda's monthly lease payments?
Using Calculators
PN =12
N
=60
IN =7.8 PMT = $2,685.10
PV =150,000
FV =-25,000
PMT =?
•
OTHER AGRICULTURAL ECONOMICS EXTENSION PUBLICATIONS
No. 93-09
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No. 93-10
No. 93-11
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No. 93-13
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Dairy Farm Business Summary
Northern Hudson Region 1992
Dairy Farm Business Summary
Southeastern New York Region 1992
Dairy Farm Business Summary Oneida-
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Dairy Farm Business Summary
Western Plateau Region 1992
Winding Down Your Farm operation
Dairy Farm Business Summary
Eastern New York Renter Summary
1992
Supercenters: The Emerging Force
in Food Retailing
Farm Income Tax Management and
Reporting Reference Manual
New York Economic Handbook 1994
Agricultural Situation and Outlook
Stuart F. Smith
Linda D. Putnam
Cathy S. Wickswat
John M. Thurgood
Stuart F. Smith
Linda D. Putnam
Alan S. White
Gerald J. Skoda
Stephen E. Hadcock
Larry R. Hul1e
Eddy L. LaDue
Jacqueline M. Mierek
Charles Z. Radick
George L. Casler
Andrew N. Dufresne
Joan S. Petzen
Michael L. Stratton
Linda D. Putnam
John R. Brake
Stuart F. Smith
Linda D. Putnam
Gene A. German
Gerard Hawkes
Debra Perosio
George L.Cas1er
Stuart F. Smith
•
Ag Ec Staff
,.
,)
i
doc_540197109.pdf
The time value of money is a very important concept in agricultural finance. It is widely used in investment evaluation, particularly for discounted cash flow analysis of investment alternatives. It is also very useful in determining values for financial transactions.
Decem bar 1993
A.E. Ext. 93-18
:
Time Value of Money
Financial Examples and Calculations
Using Tables and Calculators
by
Eddy L. LaDue
Department of Agricultural, Resource and Managerial Economics
College of Agriculture and Life Sciences
Cornell University
Ithaca, NY
It is the policy of Cornell University actively to support equality
of educational and employment opportunity. No person shall I
be denied admission to any educational program or activity or
be denied employment on the basis of any legally prohibited
I
l
discrimination involving, but not limited to, such factors as race,
color, creed, religion, national or ethnic origin, sex, age or
handicap. The University is committed to the maintenance of
affirmative action programs which will assure the continuation
of such equality of opportunity.
1
Table of Contents
Future Value
Future value of a sum (compound interest)
1. Savings accounts
2. Inflating farm values
Future value of a nonuniform series
3. Saving different amounts each year
4. IRA
Future value of a uniform series
5. Sinking fund
6. IRA
7. Saving to start farming
Present Value
Present value of a future sum
8. Stripped treasurer
9. Zero coupon bonds
10. Unamortized loan
Present value of a nonun iform series
11. Bond
12. Unequal payments from sale
Present value of a uniform series
13. Land contract
14. Loan principal
15. Payments on a loan
16. Farm machinery early purchase payments
17. Financial lease (annual payment)
18. Financial lease (monthly payments)
19. Loan principal
20. Monthly payments on a loan
Present value of a uniform infinite series
21. Value of farmland
Amortization
22. Monthly payment loan
23. Annual payment loan
24. Partial amortization
25. Lease paym ents
2
TIME VALUE OF MONEY
Financial Examples and Calculations
Using Tables and Calculators
by
Eddy L. LaDue
1
The time value of money is a very important concept in agricultural finance. It is
widely used in investment evaluation, particularly for discounted cash flow analysis of
investment alternatives. It is also very useful in determining values for financial
transactions. For these transactions, the interest rate reflects the time value of money
and the monetary values used are determined by time value concepts.
This publication provides a discussion of the time value of money concepts and
their application to financial transactions.
2
A series of examples are presented to illustrate
the application and calculations for each concept. Example calculations are given in
detail to allow readers to learn the calculation procedure. Agricultural finance examples
are used throughout.
Two approaches to the calculation are provided. One approach uses tabled values
of time value functions. Specifically, the tables provided in "Present Value, Future Value
and Amortization: Formulas and Tables" Cornell University Agricultural Economics
Extension 90-17 are used. Whenever a table number (Le. Table 6) is used in calculation,
reference is to a table in that publication.
In the second approach, calculations are made using a calculator with financial
functions. The calculator referred to is a Texas Instruments BAli Plus. The calculations
can, of course, be made with other calculators, usually with slightly different key
identification and key stroke configuration. In some cases, more complex variations of
a problem that cannot be handled with the tables because table values are presented for
only selected situations, are illustrated using the calculator. The financial function keys
referred to in this publication are:
Definition
N Number of payments (or number of compoundings if no
payments are involved)
IIV Interest rate per year
Professor of Agricultural Finance, Cornell University
2
For discussion of time value as applied to investment decision see Casler, Anderson and
Aplin, ·Capitallnvestment Analysis Using Disoounted Cash Flows."
3
PV Present value (value now)
PMT Payment (per month, year, quarter, etc.)
FV Future value (value at the end of the period under
consideration)
PlY Payments per year
BEG Payments occur at the beginning of each period (month, year,
etc.)
END Payments occur at the end of each period (month, year, etc.)
CY Compoundings per year (automatically equals payments per
year unless specifically set).
It is assumed that the number of compoundings per year equals the number of payments
per year and that payments all occur at the end of the period (year, month, etc.) unless
specifically indicated otherwise (BEG is assumed to equal END unless specifically
indicated otherwise).
A reader who plans to use the tables can use this publication by skipping the
sections labeled "Using Calculators". Similarly, a reader who plans to use a calculator
can use this publication by skipping the sections labeled "Using Tables'·.
Time value relationships can be divided into two basic categories: (1) the future
value of a monetary amount or series of amounts, and (2) the present value of a
monetary amount or series of amounts. Although present value and future value
represent a continuum of the same relationships, we are normally interested in
determining a value for one point in time, usually the present, or some point in the future.
Amortization is really an application of future value concepts. The discussion of these
concepts is presented in three sections: (1) future value, (2) present value and (3)
amortization.
FUTURE VALUE
Future value techniques are designed to determ ine the value of an amount, or
series of amounts of money as of some fixed point in time in the future. Examples
include the value of a savings account in 'five years or the value of a farm in 19 years
when land values are increasing at five percent per year.
4
1. Future value of a present sum (Compound Interest)
•
When we are dealing with the future value of a single amount of money, we use
compound interest, or the future value of a present sum. The future value of a present
sum is calculated using equation (1):
FV =PV(1+i)n (1 )
Where: FV = Value in period n (n periods in the future)
PV = Value in period 0 (now)
=Interest rate per conversion period
n =Number of conversion periods
If each period is a year, n is the number of years into the future on which the value
is to be calculated, and i is the annual interest rate. In this case, the formula results in
normal compound interest with annual compounding. If payments are made more
frequently than annually, i is the annual rate divided by the number of payments per year
and n is the number of payments per year multiplied by the number of years.
Example 1
If $1 is invested in a savings account with a fixed interest rate of eight percent and
annual compounding, how much will be in the account in 10 years? Direct application of
the equation results in the following:
$1(1+.08)10 =1(2.16) =$2.16
Using Tables
The calculations can be made using the compound interest table (Table 1). The
factor from Table 1 for 10 years at eight percent interest is 2.158924. Multiplying the
factor times the investment gives the correct value.
2.158924($1) = $2.16
5
Using Calculators
PlY =1
N = 10
FlY = 8 FV = $2.16
PN =1
PMT =0
FV =?
If interest were compounded quarterly instead of annually:
PlY
=
4
N
=
40 (4 payments per year x 10 years)
IIY
=
8
PV
=
1
PMT
=
0 FV =$2.21
FV
=
?
Example .2
Pete now has a farm that is worth $200,000. If farm values inflate at an average
rate of four percent over the next 15 years, what will the farm be worth at the end of the
15 year period?
Using (1): $200,000 (1 +.04) 15 =$360,189.
Using Tables
$200,000 (1.800943) =$360,189
?
Table 1, 15 years, four percent
Using Calculators
PlY =1
N =15
IIY =4
PV =200,000
PMT =0 FV =$360,189
FV =?
6
2. Future Value of a Nonuniform Series
The future value of a series of different amounts can be determined using the
futurevalueofaseries. Thefuturevalueofaseriesofsums iscalculated using equation
(2).
(2)
Where: FV .=Value in period (year) n
PV0 =Amount invested in period 0 (now)
PV
1
=Amount invested in period 1
PV
n
= Amount invested in period n
i = Interest rate
n =Number of years to future time for which the value is
calculated
Ifyou lookatthisequation closely, you willseethatthisisjustaseries ofequation
(1) calculations addedtogether. Thus, Table 1can also be usedtodeterminethefuture
value.
Example
Jason, a beginning high school freshman, is planning to save money from his
weekend and summer work on the farm to buy a car to drive back and forth to Cornell
while he is attending college. He invested $200 at the end of this summer (now) and
expects he can invest $400 one year from now, $500 in three years, and $600 in four
years(heplanstoworkon amidwestfarm between hisSophomoreandJunioryearsand
willspend allhemakes). Jason'sfatherhasagreedtoborrowthesefunds andpayJason
10 percent interest. How much money will Jason have at the end of four years?
FV
4
=200(1+.10)4+ 400(1+.10)3 + 500(1+.10)1 + 600
=200(1.4641) + 400(1.3310) + 500(1.10) + 600
=292.82 +532.40 + 550 + 600
=$1,975.22
7
Using Tables
FV = 200 (1.4641) + 400 (1.3310) + 500 (1.1) + 600
,
Table 1, 10%, 4 years Table 1, 10%, 3 years
= $1,975.22
Using Calculator
PlY = 1 PlY
=
1 PIX
= 1
N = 4
N
=
3 N
= 1
FlY = 10 FV = $292,82 IIY
=
10 FV = $532.40 IIY = 10 FV = $550,00
PV =200 PV = 400 PV
=500
PMT = 0 PMT = 0 PMT = 0
FV =? FV
=
? FV
= ?
Value =292.82 + 532.40 + 550 + 600 =$1,975.22
Example
Three years ago, Will invested $2,000 in an individual retirement account (IRA).
Two years ago, he invested $1,800 and last year he invested $1,500. If this IRA earns
interest at eight percent, how much is currently in the account?
Using Tables
Value =2,000(1.259712) + 1,800(1)664) + 1,500(1.08)
Table 1, 8%, 2 years
=2,519.42 + 2,099.52 + 1,620
=$6,238.94
Using Calculators
PlY = 1 PlY =1 PIX =1
N =3 FV= N =2 FV= N =1 FV=
FlY = 8 $2,519.42 IIY = 8 $2,099,52 IIY =8 $1,620
PV = 2,000 PV = 1,800 PV = 1,500
PMT = 0 PMT= 0 PMT = 0
FV =? FV =? FV =?
Value =$2,519.42 + $2,099.52 + 1,600 =$6,238.94
8
3. Future Value of a Uniform Series
If the sum involved (invested) is the same each period (year), the future value can
be determined more easily, with a single calculation, using equation (3).
FV = ~ ( 1+ i Jn -1 ]
(3)
Where: FV = Value in period (year) n (future)
PMT = Amount invested each period (year, month, etc.)
=Interest rate per period (year, month, etc.)
n =Number of periods (years or months or ... )
Using Tables
Values of the equation 3 expression in the brackets, for situations where the period
is one year, are presented in Table 4. Thus, the future value can be calculated by simply
multiplying the amount received each year by the appropriate value from Table 4.
Table 4 assumes that investments (payments) are made at the end of each year,
including the final year (at the time for which the future value is being calculated). If this
is not the case, adjustments must be made. If an investment is made at the beginning
of the period (as well as the end), use the number of years plus one in selecting the
coefficient from Table 4. If investments are made at the beginn ing of each year but none
is made at the end of the period (the time for which the future values are being
calculated), use the number of years in the period plus one in selecting the coefficient
and subtract one (1.0) from the value of the coefficient found. When the first payment
is made in one year and no payment is made at the end of the last year, subtract one
(1.0) from the coefficient for the number of years involved. The following table indicates
the appropriate values for different situations.
Payment at Table
Payment End of Value Coefficient
Payment Plan Now? Period to Use to Use
End of each year No Yes n Table
Beg. of each year Yes No n+1 Table-1
Beg. and end Yes Yes n+1 Table
End of each year
except last No No n Table-1
9
Table4can alsobe used todeterminetheinvestmentrequired each yearto meet
agoal orprovideagiven sum ofmoneyatthefuturepoint in time.. This isaccomplished
by dividing the coefficientfrom Table 4 into the amount required (the goal).
Example 5
Loren'sbankerhas required thatLoren developasinkingfund toprovidethefunds
for a neworchard sprayerwhich he will need in five years.
a. If he invests $5,000 per yearat the end of each yearfor the next five years and
the funds earn interest at 12 percent, how much money will be available for
purchase of the sprayer?
Using Tables
$5,000(6.3528) =$31,764
r
Table 4, five years, 12 percent
Using Calculators
PlY = 1
N =5
IIY =12 FV =$31,764
PV = 0
PMT =-5,000
FV =?
If interest were compounded monthly instead of annually:
PlY = 1
CIY = 12
N =5
IIY = 12 FV =$32,198
PV =0
PMT =-5,000
FV =?
b. If Lorenexpectsthesprayertocost$50,000 in fiveyears, andfunds areputinthe
same 12 percent account with annual compounding, how much must he invest
each year?
10
Using Tables
$50,000 =$7,871
6.3528
Using Calculators
PlY
=1
N =5
FlY = 12 PMT =$7,870.49
PV =0
FV =50,000
PMT =?
If interest were compounded monthly instead of annually:
PlY
=1
CIY =12
N =5
IIY
=12
PMT =$7,764.51
PV =0
FV =50,000
PMT =?
c. Ifinvestmentsweremadeatthebeginningofeachyearinsteadoftheend (annual
compounding), how much must be invested each year?
Using Tables
$50,000 =$7,027
7.1152
f
(Table 4, six years, 12 percent) - 1
Using Calculators
BEG = BEG
PlY = 1
N =5 PMT =$7,027
IIY = 12
PV = 0
FV =50,000
PMT =?
d. If payments were made now and at the end of each year, how much must be
invested?
11
Using Tables
$50,000 =$6,161
8.1152
Using Calculators
Making payment now is the same as making a payment at the end of last year
(year 0). Thus, this payment is like adding another year (year 0) to the front end
of the payment stream. The number of years (periods) becomes 6.
BEG =END
PlY
=1
N =6
IIY =12
PV =0 PMT =$6,161
FV =50,000
PMT =?
Example 6
If Joe invests $2,000 in an IRA now and at the end of each year for the next 15
years, how much will be in the account after he makes his 15 year payment if the account
earns nine percent per year?
Using Tables
$2,000(33.0034) =$66,007
/"
Table 4, 16 years, nine percent
Using Calculators
PlY =1
N =16
IIY =9 FV =$66,007
PV =0
PMT = -2,000
FV =?
12
If interest were compounded quarterly:
PlY = 1
CIY =4
N = 16 FV =$67,764
IIY =9
PV =0
PMT =-2,000
FV =?
Example 7
Sandy is planning to start farming in 10 years. To get the initial funds needed, she
is working for Farm Credit and plans to set aside $4,000 per year starting now from her
salary in a mutual fund account that she expects to earn 11 percent per year. How much
money will she have for farm investment in 10 years if moving and other costs keep her
from investing at the end of the 10th year?
Using Tables:
$4,000(18.5614) =$74,246
./'
(Table 4, 11 years, 11 percent) - 1
Using Calculators
BEG =BEG
PlY =1
N =10
IIY =11 FV =$74,246
PV =0
PMT =-4,000
FV =?
If she knows she will need $100,000 to start, how much will she need to invest
each year?
Using Tables:
$100,000 =$5,387.52
18.5614
13
Using Calculators:
BEG = BEG
PlY =1
N =10 PMT =$5,387.52
IIY = 11
PV =0
FV =100,000
PMT =?
PRESENT VALUE
Present value or "discounting" is essentially the reverse of future value. Present
value determines the current value of an amount or series of amounts to be received in
the future. In general terms, present value is the amount that you could be paid now and
be equally well off compared to receiving a specified payment or series of payments at
some specific time in the future.
4. Present Value of a Future Sum
Present value of a future sum is used to calculate the amount that one could
receive now that would be equal in value to the receipt of a specified single amount at
some point in the future. Remember, the compound interest formula told us the future
value of a current sum. Well, the present value of a sum tells us the current value of
a future sum. We can modify the compound interest formula,
FV = PV(1+i)n
by dividing both sides by (1+i)n, to get the present value formula.
PV= Vn
( I +r) n
(4)
This can be reformulated as:
PV =FV (1+i)"n (5)
14
Using Tables
Thebracketedvalue (1+ithas been placed in tabularform in Table2. Returning
toourExample 1situation, ifwewill receive $2.16 in 10yearsandfunds received
now could earn eight percent over the next 10 years, the present value of the
$2.16
= $2.16(.463193) = $1
I
Table 2, 10year, eight percent
Note that .463193 = 1
f 2.158924,
Table 2 Table 1
Thus, if you have Table 2you can calculate Table 1values and vice versa.
Table 1 = 1
Table 2
Table 2= 1
Table 1
Thus, the answer to Example 2 could be calculated using Table 2 values as shown
below. The modest difference between the values results from rounding in the tables.
$200,000 ( 1.0 ) =$360,188
.555265
?
Table 2, 15 years, four percent
Example 8
Tobroaden the rangeof investmentalternativesavailable,securitiesdealershave
developedstrippedtreasurieswhereeachoftheinterestpaymentsandthefinal principal
payment on a treasury note or bond are separated into separate instruments and sold
individually. Thus, you can buy the next year's interest payment on a million dollar
treasurynoteortheyear2000principal payment'on a$100,000treasurybond. Similarly,
zerocoupon bonds represent an agreement of the issuer to pay the face amountof the
bond atthe maturitydate. What is the presentvalue of the $100,000 principal payment
(or zero coupon bond) to be received in 12 years if you could earn ninepercent interest
on funds invested overthe period?
15
Using Tables
$100,000 (.355535) = $35,553
/
Table 2, 12 years, nine percent
Using Calculators
PlY
=1
N =12
IIY =9 PV = 35,553
PMT =0
FV = 100,000
PV =?
Example 9
TwoyearsagoJohn loanedhisson, JohnJr.,alocalgrapegrower, $20,000tobuy
additionalvineyard. The loan wasforfiveyearswithinterestcompoundedannuallyat 10
percent. but the loan was unamortized and no interest or principal was to be paid until
the end of the five yearperiod.
a. How much will John receive at the end of the five year period?
Using Tables
$20,000(1.61051) = $32,210
/"
Table 1, five years, 10 percent
Using Calculators
PlY =1
N =5
IIY =10 FV=$32,210
PV = 20,000
PMT =0
FV =
b. If John were tosell the loan to an investortoobtainfundsforotheractivities, how
much could the investor afford to pay if she expected to earn 16 percent on all
investments?
16
Using Tables
$32,210(.640658) =$20,636
./'
Table 2, three years, 16 percent
Using Calculators
PN =1
N =3
IN = 16 PV = $20,636
PMT = °
FV =32,210
PV =?
5. Present Value of a Nonuniform Series
When a series of amounts are to be received in the future and the amounts are
not all of the same magnitude, equation (6) can be used.
PM?;
+ (6)
(1 + i)n
For ease in use, equation (6) can be reformulated as:
PV = PMI:: (1 + i) -1 + ~ 1 + i) -2 + • •. + PM?;( 1 + i) n (7)
In equation (7) form, it is easy to see that this is just the sum of a series equation 5
calculations.
Thus, the present value of the series is just the sum of the present value of the individual
amounts in 'that series. That is, it is the sum of a series of calculations using
equation 5.
Example 10
What is the value of a Federal Farm Credit Bond with a face value of $1,000, a 10
percent interest rate with annual payments and a three year maturity, if increasing interest
rates have increased the normal yield on such bonds to 15 percent (or, you could earn
15 percent on available funds if invested elsewhere)?
17
You will receive the following amounts:
Year 1 ($1,000 x .10) = $ 100
Year 2 ($1,000 x .10) = 100
Year 3
plus
($1,000 x .10)
$1,000 face value = $1,100
Using Tables
PV = 100(.869565) + 100(.756144) + 1,100(.657516)
t
Table 2 Table 2
one year three years
15 percent 15 percent
= 86.96 + 75.61 + 723.27
= $885.84
If you paid $885.84 for this bond, how much would you earn? Obviously 15 percent.
Using Calculators
PlY = 1
N =3
IIY = 15 PV = 885.84
PMT = 100
FV = 1000
PV =?
If bond interest payments were semiannual:
PlY = 2
N =6
FlY = 15 PV = 882.65
PMT = 50
FV = 1000
PV =?
Why is the present value lower with more frequent payments? The semiannual
discounting of the $1000 reduces the present value more than the earlier receipt of part
of the interest payments increases the present value.
18
Example 11
In negotiating the sale of some machinery to his son, Harry agreed to accept
interest and principal payments of $10,000 in one year; $15,000 in two years, and
$22,000 in three years (in line with the son's ability to pay). If Harry can earn nine
percent on invested funds, how much is he receiving for the machinery?
Using Tables
PV = $10,000(.917431) + 15,000(.841680) + 22,000(.772183)
?
= 9,174.31 + 12,625.20 + 16,988.03 Table 2,9%,3 years
= $38,787.54
Using Calculators
PlY =1 PlY
=1
PlY
=1
N =1
N =2 N =3
IIY =9 PV = 9,174.31 IIY =9 PV = $12,625.20 IIY =9 PV = 16,988.04
PMT =0 PMT =0 PMT =0
FV = 10,000 FV =15,000
FV = 22,000
PV =? PV =? PV =?
Value = 9,174.31 + 12,625.20 + 16,988.04 = $38,787.55
6. Present Value of a Uniform Series
If the amounts to be received in the future are the same for all consecutive years,
the present value can be obtained more simply using equation (8).
PV= PMT [1-( 1/i) -nj (8)
Where: PV =The value in period zero (now)
PMT =The amount received each period (year, month, etc.)
= The interest rate
(ann ual interest rate divided by num ber of payments per year)
n = The number of periods the amount PMT is to be received
19
Example 12
Ann is an only child who became a banker rather than returning to the home farm.
At the time her father retired from farming, he sold the farm to Dave on a land contract.
The contract called for annual payments of $3,000 per year with interest at 10 percent.
At the time of Ann's father's death, there were 10 years remaining on the contract and
she could earn 16 percent on funds available for investment at the time. Dave offers
$15,000 to payoff the contract. Should she accept Dave's offer?
Using Tables
The bracketed part of equation 8 has been calculated for a number of values of
i and n, and presented in Table 3 for annual payments and table 5 for monthly payments.
Present values are calculated by multiplying the amount received each year by the correct
value from Table 3 or 5.
$3,000(4.83323) =$14,500
1
Table 3, 10 years, 16 percent
Using Calculators
PlY
=1
N
=10
IIY
=16 PV =$14,500
PMT =-3,000
FV =0
PV =?
Since she can earn 16 percent on money invested, the $3,000 per year for 10
years is equivalent to $14,500. Dave has offered $15,000. She should accept the offer.
Present value can be used to determine the principal outstanding on a loan.
Discounting removes the earnings that could be obtained on funds if they are received
now rather than in the future. For an amortized loan, the earnings generated for the
holder of the loan is the interest to be received. Thus, discounting removes the interest
from the loan payments and the present value is the current principal balance. Only the
number of payments remaining, the amount of each payment and the interest rate must
be known. Thus, the principal outstanding on Ann's contract can be calculated by
discounting the payments at the interest rate specified by the contract (10 percent).
20
Using Tables
$3,000(6.14457) = $18,434 = principal owed
'\
Table 3, 10 years, 10 percent
Using Calculators
PlY = 1
N =10
IIY =10 PV = $18,434 = principal
PMT = -3000
FV = 0
PV =?
Example 13
Ten years ago, the Vegies National Bank made a 25 year loan at six percent
interest to a local vegetable farmer to purchase a farm. The annual payments are
$8,719.
1. The principal outstanding is:
Using Tables
$8,719(9.7122) = $84,681
1
Table 3, 15 years, six percent
Using Calculators
PlY
=1
N
=15
IIY =6 PV = $84,681
PMT =-$8,719
FV
=0
PV =?
2. If an insurance company (or someone else) that is currently earning 16 percent on
newly invested funds were to buy this loan, the amount they can afford to pay for the loan
is:
21
Using Tables
$8,719(5.5755) =$48,613
t
Table 3, 15 years, 16 percent
Using Calculators
PlY
=1
N
=15
IIY
=16
PV = #48,612
PMT =- 8,719
FV =0 (results differ due to rounding on tabled values)
PV =?
Example 14
Kelly borrowed the funds to buy new greenhouse space. The terms of the loan
are:
$50,000 borrowed
15 percent interest
25 years repayment period
Annual payments
Using Tables
The annual payments calculated using Table 3 are:
$50,000 =$7,735.03
6.4641
,
Table 3, 25 years, 15 percent
This result can be checked using amortization tables (Table 7) where the payment
per $1 for loans for 15 percent for 25 years is $0.1547.
$.1547 x 50,000 =$7,735
The difference between the two results is due to rounding.
Example 15
In an effort to stimulate lagging farm machinery sales, John Deere recently offered
a delayed payment plan which required no interest until March 15 and no payment until
22
December 15. The interest rate charged was 12 percent and interest was charged from
the date of the purchase or March 15, whichever occurs last. The loan is for five years
with six equal payments starting December 15. Mark purchased $100,000 worth of
machinery on January 15, and took advantage of this special repayment plan.
Calculating the amount of each payment in this case requires first determining the
amount outstanding on December 15. The interest can be calculated as,
=$9,000
12
leaving a total of $109,000 outstanding on December 15.
Using Tables
Since the first payment is due on December 15, the Table 3 value must be modified to
reflect the immediate payment. As of December 15, the present value coefficient for the
December 15, payment must be 1.0, thus, adjusting for the immediate payment involves
adding 1.0 to the coefficient for the remaining five payments. The annual payments are:
$109.000 =$23,671.05
4.60478,
(Table 3, five years, 12 percent) + 1.0
Clearly, if the finance plan required "no interest until December 15", the payment would
be:
$100.000 =$21,716.56
4.60478
Using Calculators
No Interest Until March 15 No Interest Until December 15
BGN = BGN BGN =BGN
PlY
=1
PlY
=1
N =6 N =6
IIY =12 PMT =23,671.07 IIY =12 PMT = 21,716.58
PV = 109,000 PV = 100,000
FV =0 FV =0
PMT =? PMT =?
The difference between values using calculator and tables is due to rounding of table
values.
23
Example 16
The appropriate value to enter on the balance sheet for a financial lease is the
present value of the future lease payments discounted at the firms incremental borrowing
rate or the implicit rate on the lease. Thus, present value can be used to calculate the
value of leases.
Jim is in the process of completing his 12/31 (end of year) balance sheet. On
March 31, two years and nine months ago, Jim leased a farrowing unit from Telmark.
The payments were $15,000 per year for seven years with all payments at the beginning
of the year. The incremental borrowing rate for Jim's business was 12 percent. Three
payments have been made (two years nine months ago, one year nine months ago, and
nine months ago). The remaining four payments are due at the end of each of the next
four years (ending March 31), so normal present values can be used for them. The
present value of the remaining payments as of the last payment date (March 31, of the
coming year) are:
Using Tables
$15,000(3.03735) =$45,560
'/
Table 3, four years, 12 percent
Using Calculators
PIV =1
N =4
IIV =12
PMT = -15,000 PV =$45,560
FV =0
PV =
Example 17
Lynda is currently leasing a tractor for $500 per month. The implicit interest rate
on the lease is 11 percent. She has 29 payments left to make. What is the present
value of the lease?
Using Tables
$500(25.3641) =$12,682
'\
Table 5, 29 months, 11 percent
24
Using Calculator
PlY =12
N = 29
IIY = 11 PV = $12,682
PMT = -500
FV = 0
PV =?
If the implicit rate on the lease were 11.23 percent, all entries are the same except IIY
= 11.23 and the present value of the lease is $12,648.
Example 18
A number of years ago, Dan borrowed funds at 13 percent interest to purchase an
orchard sprayer. His payments are $157.60 per month. He still has 47 payments to
make. What is the outstanding principal on the loan?
Using Tables
$157.60(36.6790) = $5,780.61
?
Table 5, 47 months, 13 percent
Using Calculators
PlY =12
N = 47
IIY = 13 PV = $5,780.61
PMT = 157.60
FV = 0
PV =?
Example 19
Mark is borrowing $10,000 to buy a pickup. If it is borrowed at nine percent for 30
months, how much will his monthly payments be?
Using Tables
$10.000 =$373.48
26.7751
r
Table 5, 30 months, nine percent
25
Using Calculators
PlY =12
•
N =30
IIY = 9 PV = $373.48
•
PV = 10,000
FV = 0
PV =?
If Mark uses dealer financing he can get interest at 2.99 percent by giving up a $500
rebate. In this case, all entries would bethe same except IIYwouldequal2.99 and PV
would equal 10,500. The payment would be $363.68.
7. Present Value of a Uniform Infinite Series
If the same amountwill be received each yearforever, the present value can be
calculated even moresimply. In this case, the annual amount is divided bythe interest
or discount rate, equation (9).
PV = Ali (9)
Example 20
One of the methods used to determine the value of farmland is to capitalize the
annual rent the land is expected to earn (known as the income approach to appraisal).
Thisis referredtoasthecapitalized valueof land. Ifthenetreturn tolandforaparticular
appleorchardisestimatedat$315 peracreandtheappropriatecapitalization rate isnine
percent, the capitalized value of the land equals:
$315=$3,500/acre
.09
However, if theexpectedapple price drops $0.20/bushel, and the orchard has an
average yield of 500 bushels, the expected annuaI income declines to $215 and the
capitalized value equals:
$215 =$2,389/acre
.09
AMORTIZATION
As impliedearlier, theequationforthepresentvalue ofa uniform series (equation
8) can be rearrangedtoprovide an equation forcalculatingthe payments required on a
loan, oramortization values. The amortization equation, thus, is equation 10.
26
(10)
•
•
Where PV = the amount to be amortized or the value of the loan in period zero
(now).
PMT = the amount of each payment
=
the interest rate per period (annual rate divided by the number of
payments per year)
n
=
number of payments (periods) the PMT amount is to be paid.
Using Tables
From equation 10 it is clear that the amount of the payments on a loan can be calculated
by dividing the loan amount by the present value factor from tables 3 or 5 (as illustrated
in Examples 14 and 15). It is also clear that the amount of payment per dollar of loan
can be calculated as the reciprocal of the bracketed amount in equation 10.
To make amortization calculations more straightforward values of that reciprocal
1
[1-( \+i) -n]
are calculated and presented in tables 7 and 8. These tables indicate the amount of
payment per dollar of loan.
Example 21
Aaron is selling the farm real estate to his son, Alan. They have agreed that the price
of the real estate is $450,000 and that Alan will pay 7.5 percent interest. How much are
Alan's monthly payments if the money is paid back over 20 years?
Using Tables
450,000 (.008056)
,
=$3,625.20
Table 8, 20 years, 7.5%
27
Using Calculators
PlY =12
•
N = 240
IIY =7.5 PMT = 3,625.17
PV = 450,000
FV = 0
PMT =?
The difference between results determ ined using tables and those using calculators is
due to roundin g of table values.
Example 22
Steve is borrowing the money from his mother to get started in farming. She has agreed
to lend him $75,000 and take a second priority lien on his line of equipment. They have
agreed that he will repay the loan over 7 years at 6.5 percent interest with annual
payments. How much are his payments?
Using Tables
75,000 (.182332) = $13,675
I
Table 7, 7 years, 6.5 percent
Using Calculators
PlY = 1
N =7
IIY = 6.5 PMT = $13,675
PV = 75,000
FV =0
PMT =?
Example 23
Mary is planning to buy the home farm from her parents. The farm is valued at
$500,000. They expect to ultimately will her half of the farm and plan to put the principal
Mary repays in a separate account which, along with the remaining monthly payments,
will be willed to Mary's nonfarming sister, Susan. Thus they want a loan that will amortize
half of the principal (partial amortization) and provide Mary's parents with income on the
•
complete farm value during their lifetimes. With this type of loan, how much will Mary's
monthly payments be if the loan is for 25 years at 8.25 percent interest.
•
28
Using Calculators
PN =12
N =300 (25 x 12)
IN = 8.25 PMT = $3,690
PV =500,000
FV =-250,000 (500,000 x .5)
PMT =?
Example 24
George plans to lease his line of equipm ent to his daughter, Amanda. They have agreed
that the equipment is now worth $150,000 and at the end of the 5 year lease period it will
be worth about $25,000. If, through the lease, George wants to earn 7.8 percent interest
on the money he has invested, and be paid for the machinery that is "used up", at what
level should they set Amanda's monthly lease payments?
Using Calculators
PN =12
N
=60
IN =7.8 PMT = $2,685.10
PV =150,000
FV =-25,000
PMT =?
•
OTHER AGRICULTURAL ECONOMICS EXTENSION PUBLICATIONS
No. 93-09
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Dairy Farm Business Summary
Northern Hudson Region 1992
Dairy Farm Business Summary
Southeastern New York Region 1992
Dairy Farm Business Summary Oneida-
Mohawk Region 1992
Dairy Farm Business Summary
Western Plateau Region 1992
Winding Down Your Farm operation
Dairy Farm Business Summary
Eastern New York Renter Summary
1992
Supercenters: The Emerging Force
in Food Retailing
Farm Income Tax Management and
Reporting Reference Manual
New York Economic Handbook 1994
Agricultural Situation and Outlook
Stuart F. Smith
Linda D. Putnam
Cathy S. Wickswat
John M. Thurgood
Stuart F. Smith
Linda D. Putnam
Alan S. White
Gerald J. Skoda
Stephen E. Hadcock
Larry R. Hul1e
Eddy L. LaDue
Jacqueline M. Mierek
Charles Z. Radick
George L. Casler
Andrew N. Dufresne
Joan S. Petzen
Michael L. Stratton
Linda D. Putnam
John R. Brake
Stuart F. Smith
Linda D. Putnam
Gene A. German
Gerard Hawkes
Debra Perosio
George L.Cas1er
Stuart F. Smith
•
Ag Ec Staff
,.
,)
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