Usury
Chapter 6
Accounting and the Time Value of Money
Declining the purchase power of money (currency) due to
inflation (increases in prices)
Basic Time Value Concepts
In accounting (and finance), the phrase time value of money indicates a relationship between time
and money—that a dollar received today is worth more than a dollar promised at some time in the
future.
The Nature of Interest
Interest is payment for the use of money. It is the excess cash received or repaid over and above
the amount lent or borrowed (principal/ Present Value).
The Types of Interest
1
Simple Interest
‫فائدة البسيطة‬
2
Compound Interest
‫الفائدة المركبة‬
Simple Interest
Companies compute simple interest on the amount of the principal only. It is the return on (or
growth of) the principal for one time period. The following equation expresses simple interest.
Interest = p × r × n
where
p = principal
r = rate of interest for a single period
n = number of periods
Example:
X borrows $50,000 for 4 years with a simple interest rate of 11% per year.
Required: computes the total interest.
Interest = p × r × n
= $50,000 ×0.11 × 4
= $22000 for all periods (4 year) or [5500 per year] [4]= 22000
Annual interest = 50000 x 0.11 = 5500 per year.
1
Example:
If X borrows $20,000 for 5 months at 12% annual interest.
Interest = $20,000 × 0.12 × 5/12
= $1000
Compound Interest
We compute compound interest on principal and on any interest earned that has not been paid or
withdrawn. It is the return on (or growth of) the principal for two or more time periods.
Compounding computes interest not only on the principal but also on the interest earned to date
on that principal,
Example: simple interest
Assume that Kamal deposits $10,000 in the Arab Bank, where he will earn simple interest of 3%
per year. He invested the amount for three years.
Interest = [10000][0.03][3]= 900
Year 1 : I = 10000x 0.03 = 300
Year 2: I = 10000x 0.03 = 300
Year 3: I = 10000x 0.03 = 300
2
Example: Compound interest
Assume that Kamal deposits $10,000 in the Arab Bank, where he will earn compound interest of
3% per year compounded annually. In both cases, Kamal will not withdraw any interest until 3
years from the date of deposit.
Total Interest = Future Value – Principal
Total Interest = ([principal][1+r]n ) – [Principal ]
Total Interest = ([10000][1+0.03]3 ) – [10000 ]
Total Interest = 927.27
Year 1: I = [10000][0.03] = 300
Year 2: I = [10300][0.03]= 309
Year 3: I= [10609][0.03]=318.27
3
Year
Interest for first year
Interest for second year
Interest for third year
Equation
= ([principal][1+r] ) - Principle = [10000][1.03]1 -10000 = 300
= ([principal][1+r]2 ) - ([principal][1+r]1 ) = [10000][1.03]2 –
[10000][1.03]1 = 10609 -10300= 309
= ([principal][1+r]3 ) - ([principal][1+r]2 ) = [10000][1.03]3 –
[10000][1.03]2 = 10927.27 -10609= 318.27
1
The Future Value
Single-Sum Problems ‫دفعة واحدة‬
Future value = (Present Value) (1+r)n
Where
FV = future value
PV = present value (principal or single sum)
r = interest rate
n = time periods.
Example 1: Ali wants to determine the future value of $50,000 invested for 5 years compounded
annually at an interest rate of 6%.
Future value = (Present Value) (1+r)n
Future value = (50000) (1+0.06)5 = 66911.2788
At the end of fifth year Ali will receive 66911.2788
Example 2: Assume that Ali deposited $10000 in an escrow account with Arab Bank at the
beginning of 2020 as a commitment toward a power plant to be completed December 31, 2025.
How much will Ali have on deposit at the end of 6 years if interest is 12% annually, compounded
semiannually?
Future value = (10000) (1+0.06)12
Future value = 20122
Example 3: Assume that Ali deposited $10000 in an escrow account with Arab Bank at the
beginning of 2020 as a commitment toward a power plant to be completed December 31, 2025.
How much will Ali have on deposit at the end of 6 years if interest is 12% annually, compounded
quarterly?
Future value = (10000) (1+0.03)24
Future value = 20328
Example 4: Assume that Ali deposited $10000 in an escrow account with Arab Bank at the
beginning of 2020 as a commitment toward a power plant to be completed December 31, 2025.
How much will Ali have on deposit at the end of 6 years if interest is 12% annually, compounded
monthly?
Future value = (10000) (1+0.01)72
Future value = 20328
4
Interest Due: (annual interest 0.12, and periods = 5 years)
Interest Due: (annual interest 0.08, and periods = 4)
Annually
R= 0.08
Periods = 4 x 1 =4
Semiannually
R=0.08/2=0.04
Periods = 4 x 2 =8
Quarterly
R=0.08/4=0.02
Periods = 4 x 4 =16
Monthly
R=0.08/12=0.00667
Periods = 4 x 12 =48
Present Value of a Single Sum
Present Value=
Future Value
(1+r)n
Example: How many dollars should be invested today at an annually compounded interest rate
of 6% to receive 66912.5 after 5 years?
Future Value
Present Value=
(1+r)n
Present Value=
66912.5
(1+0.06)5
Present Value= 50000
Natural Logarithm
5
Example—Computation of the Number of Periods [n=???]
Ali wants to accumulate $70,000 for the construction. At the beginning of the current year, Ali
deposited $47,811 in Arab Bank that earns 10% interest compounded annually. How many years
will it take to accumulate $70,000?
Future value = (Present Value) (1+r)n
70000 = (47811) (1+0.10)n
70000=[47811][1.1]n
70000/47811= [1.1]n
1.46409822 = [1.1]n
Compute the natural logarithm for the two sides
Log (1.46409822) = Log (1.1)n
Log (1.46409822) = [n] [Log (1.1)]
N=
Log (1.46409822)
[Log (1.1)]
N=
0.165570212
0.041392685
N= 4
prove
Future value = (47811) (1+0.10)4 = 70000
Example—Computation of the Interest Rate
X needs $1,070,584 for basic research 5 years from now. X currently has $800,000 to invest for
that purpose. At what rate of interest must it invest the $800,000 to fund basic research projects of
$1,070,584, 5 years from now?
Future value = (Present Value) (1+r)n
1070584=(800000) (1+r)5
1.33823 = (1+r)5
Log (1.33823) = log (1+r)5
0.12653 =5 Log X
0.025306=log x
100.025306 = X
X= 1.06
1+r=1.06
R=0.06
0.05888888
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Annuities (Future Value) ‫دفعات دورية‬
The preceding discussion involved only the accumulation or discounting of a single principal sum.
However, many situations arise in which a series of dollar amounts are paid or received
periodically, such as installment loans or sales; regular, partially recovered invested funds; or a
series of realized cost savings.
Note that the rents may occur at either the beginning or the end of the periods.
1. If the rents occur at the end of each period, an annuity is classified as an ordinary annuity.
2. If the rents occur at the beginning of each period, an annuity is classified as an annuity
due.
Case 1: Future Value of an Ordinary Annuity
Example: What is the future value of five $5,000 deposits made at the end of each of the next 5
years, earning interest of 6%?
Ahmad decided to invest $5000 in Arab Bank at the end of each year with annual interest of 0.06.
If you know that the time period of investments is 5 years. What is the amount of money that will
be received at the end of period?
FV=
[Rent]
(1+r)n - 1
r
FV=
[5000]
(1+0.06)5 - 1
0.06
FV=
[5000]
1.338225578 - 1
0.06
FV=
[5000]
0.338225578
0.06
FV=
[5000]
[5.63709296]
FV=28185.465
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Future Value of an Annuity Due
FV=
(1+r)n - 1
r
[Rent]
[1+r]
What is the future value of five $5,000 deposits made at the beginning of each of the next 5 years,
earning interest of 6%?
FV=
[5000]
(1+0.06)5 - 1
0.06
[1+0.06]
FV=
[5000]
(1+0.06)5 - 1
0.06
[1+0.06]
=29876.6
Example: Ali invests 5000 at the end of each accounting period to receive 28185, the annual
interest is 0.06. compute the number of years.
FV=
[Rent]
(1+r)n - 1
r
28185=
[5000]
(1+0.06)n - 1
0.06
(1+0.06)n - 1
0.06
5.6371=
[5.6371][0.06]= (1.06)n
-1
0.338225577 = (1.06)n – 1
1.338225577 = (1.06)n
Log[1.338225577] = Log(1.06)n
0.126529326 = n log [1.06]
0.126529326 = n [0.025305]
N= 5 year
Annuities (Present Value)
Present Value of an Ordinary Annuity
PV=
(
Rent
r
)
(
1-
1
(1+r)n
)
Example: What is the present value of rental receipts of $6,000 each, to be received at the end of
each of the next 5 years when discounted at 6%?
PV=
6000
0.06
1-
1
(1+0.06)5
PV=25274
8
Present Value of Annuity due
Rent
1
1(1+r)
r
(1+r)n
To illustrate, Space Odyssey, Inc., rents a communications satellite for 4 years with annual rental payments of $4.8
million to be made at the beginning of each year. If the relevant annual interest rate is 5%, what is the present value
of the rental obligations?
4.8
1
PV=
1(1+0.05)
0.05
(1+0.05)4
PV= 17,871,590
PV=
Application: Selling price of bond (intermediate accounting 2:
Chapter 14 (long-term liabilities: bonds and notes)
There are 3 situations
1- Issuing bonds at face value (sated interest = Market interest).
2- Issuing bond at discount (market interest > stated interest).
3- Issuing bond at premium (market interest < stated).
Bond certificate
# of bonds
Face value = coupon value or nominal value $1000 per bond
Stated interest rate 5% (annual)
Market interest rate = Effective interest rate
Valuation of Long-Term Bonds
Selling price of bond = present value (pv)
PV = Present value of principal + present value of rent (interest)
PV=
Principal
(1+r)n
+
[(
Rent
r
)
(
1-
1
(1+r)n
R= market interest rate.
N= periods.
Rent = interest expense = [principal][stated interest]
9
)]
1- Issuing bonds at face value (sated interest = Market interest).
X Corporation on January 1, 2020, issues $100,000 (100 bond @ $1000 face value) of 5% bonds
due in 5 years with interest payable annually at year-end. The current market rate of interest for
bonds of similar risk is 5%. What will the buyers pay for this bond issue?
PV=
100000
(1+0.05)5
+
[(
5000
0.05
)
(
1-
1
(1+0.05)5
)]
PV= 78352.617 + 21647.38335 = 100000
No discount or premium
Note: rent = interest expense = [principal][stated interest] = [100000][0.05] = 5000
X Corporation will record this journal entry
Dr. Cash 100000
Cr. Bonds Payable 100000
2- Issuing bond at discount (market interest > stated interest).
X Corporation on January 1, 2020, issues $100,000 (100 bond @ $1000 face value) of 5%
bonds due in 5 years with interest payable annually at year-end. The current market rate of
interest for bonds of similar risk is 6%. What will the buyers pay for this bond issue?
PV=
100000
(1+0.06)5
+
[(
5000
0.06
)
(
1-
1
(1+0.06)5
)]
PV= 74725.817 + 21061.819 = 95787 selling price
X Corporation will record this journal entry
Dr. Cash
95787
Dr. Discount on bonds 4213
Cr. Bonds Payable 100000
Balance sheet
Long –term liabilities
Bonds payable 100000
Less: discount on bonds 4213
Adjunct account (add) versus contra account (deduct)
A/R
Less: ADA
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3- Issuing bond at premium (market interest < stated).
X Corporation on January 1, 2020, issues $100,000 (100 bond @ $1000 face value) of 5%
bonds due in 5 years with interest payable annually at year-end. The current market rate of
interest for bonds of similar risk is 4%. What will the buyers pay for this bond issue?
PV=
100000
(1+0.04)5
+
[(
5000
0.04
)
(
1-
1
(1+0.04)5
)]
PV= 82192.711 + 22259.911 = 104452 selling price
X Corporation will record this journal entry
Dr. Cash
104452
Cr. Bonds Payable 100000
Cr. Premium on bonds 4452
Balance sheet
Long –term liabilities
Bonds payable
100000
Plus: premium on bonds 4452
Notes:
1 – The discount on bonds is deferred cost.
2 – The premium on bonds is deferred revenue.
3 – The discount or premium should be allocated (amortized) among useful life by using a – The straight-line
methods. b. Effective interest method. (In chapter 14 of intermediate II we will discuss this issue).
4 – In Chapter 7, we will apply the time value of money (valuation of notes receivable).
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