Module 1 – Lecture 4
MONEY TIME
RELATIONSHIP
Prof. Dr. M.F. El-Refaie
Compound interest factors:
Based on the concept of time value of money and bound to the mechanism of
compound interest, it is often required to interrelate some of the following quantities:
P is the value or sum of money at a time denoted as the present, m.u.
F is the value or sum of money at a future time, m.u.
A is the series of consecutive, equal, end-of-period amounts of money, m.u per unit
time.
n is the number of interest periods.
i is the interest rate.
The transformation multipliers (or time-value conversions) are called the “Compound
Interest Factors" or the “Engineering Economy Factors".
The names of these factors together with their standard abbreviations and notations are
given in Table 4.1.
Table 4.1: Standard factor notations
Factor name
Standard notation
Single-payment present worth (SPPWF)
 P / F , i%, n 
Single-payment compound amount (SPCAF)
 F / P , i%, n 
Uniform series present worth (USPWF)
 P / A, i%, n 
Capital recovery (CRF)
 A / P, i%, n 
Sinking fund (SFF)
 A / F , i%, n 
Uniform series compound amount (USCAF)
 F / A, i%, n 
Table 4.2: Computations using standard notation
To find
given
factor
formula
p
F
 P / F , i%, n 
F
p
 F / P , i%, n 
F  P  F / P, i%, n 
p
A
 P / A, i%, n 
P  A  P / A, i%, n 
A
P
 A / P, i%, n 
A  P  A / P, i%, n 
A
F
 A / F , i%, n 
A  F  A / F , i%, n 
F
A
 F / A, i%, n 
F  A  F / A, i%, n 
P  F  P / F , i%, n 
SINGLE-PAYMENT COMPOUND
AMOUNT FACTOR (SPCAF)
(SPCAF )
P= given
0
1
2
3
4
n-2
n-1
n
F= ??
This factor was implicitly derived previously.
F  P 1  i 
n
F / P   F / P , i , n   1  i 
n
Example 4.1
If you have L.E.2000 now and invest it at 10%, how much will it be
worth in 8 years?
F= ??
i = 10 %
0
1
2
3
4
years
2000
5
6
7
8
Solution
Given: P = L.E. 2000, i = 10% per year, and n = 8 years
Find: F
Using a calculator: Use a calculator to evaluate the (1+i)n term
F = L.E.2000 (1 + 0.10)8
= L.E.4287.18
Using a spreadsheet package as Excel, the future worth calculation could be
determined from FV Function:
FV returns the future value of an investment based on periodic, constant
payments and a constant interest rate.
Syntax
FV (rate, nper, pmt, pv, type)
Where
Rate
is the interest rate per period.
nper
is the total number of payment periods in an annuity.
Pmt
is the payment made each period; it cannot change over the life of the annuity.
Typically, pmt contains principal and interest but no other fees or taxes.
Pv
is the present value, or the lump-sum amount that a series of future payments is
worth right now.
Type is the number 0 or 1 and indicates when payments are due. 0 or omitted at the end
of the period, 1 at the beginning of the period
F = FV (10%, 8,, 2000, 0)
Example 4.2
The following sums are to be invested at 5 % interest rate:
6x105 m.u. now, 3x105 m.u. two years from now and 4x105
m.u. five years from now. How much will be accumulated ten
years from now?
Solution
F=
??
i= 5 %
0
years
1
2
3x105
m.u.
6x105 m.u.
3
4
5
6
7
8
4 x105
m.u.
F = P1 (F/P, 5, 10) + P2 (F/P, 5, 8) + P3 (F/P, 5, 5)
= 6 × 105 × (1.05)10 + 3×105 × (1.05)8 + 4×105 × (1.05)5
= 1.9311×106 m.u.
9
10
Alternative solution
F=
??
i= 5 %
0
years
1
2
3
3x105
m.u.
4
5
4 x105
m.u.
6
7
6x105 m.u.
P’
P' = 6 × 105 (F/P, 5, 5) + 3 × 105 (F/P, 5, 3) + 4 × 105
= 6 × 105 × (1.05)5 + 3 × 105 × (1.05)3 + 4 × 105
= 15.1306 ×105 m.u.
F = P' (F/P, 5, 5)
= 15.1306 × 105 × (1.05)5
= 19.311 × 105
8
9
10
Solution
Solution of the Excel sheet is shown in figure where we can see in the equation bar that:
F = FV (5%, 10,, 600000, 0) + FV (5%, 8,, 300000, 0) + FV (5%, 5,, 400000, 0)
SINGLE-PAYMENT PRESENT
WORTH FACTOR (SPPWF)
(SPPWF)
P= ??
0
1
2
3
4
n-2
n-1
n
F= given
P  F / 1  i 
n
P / F  P / F ,i , n  
1
1  i 
n
Example 4.3
Suppose that L.E.1000 is to be received after 5 years. At an
annual interest rate of 12%,
What is the present worth of this amount?
Solution
1000 L.E.
i = 12 %
0
1
2
3
4
5
years
P = ??
This example could be solved by one of three ways:
1-Using a calculator (Mathematical Formulas)
P = 1000 × (1 + 0.12)-5 = 1000 × (0.5674) = L.E. 567.40
In order To have L.E.1000 in your savings account at the end of 5 years,
you must deposit L.E.567.40 now.
2. Using interest tables
Interest Table For Discrete Compounding; i = 12 %
Periods
Compound
Amount
Factor
Present
Worth
Factor
Compound
Amount
Factor
Present
Worth
Factor
n
F/P
P/F
F/A
P/A
4
1.5735
0.6355
4.7793
3.0373
5
1.7623
0.5674
6.3528
3.6048
6
1.9738
0.5066
8.1152
4.1114
(0.5674)
P = 1,000 (P/F, 12%, 5) = L.E.567.40
Note: this method might be obsolete as it requires lots of data tables. These tables are
formed using the simple formulae developed here in this lecture
3. Using Spreadsheet as Excel, the present value calculation is determined from the
function PV:
PV functions returns the present value of an investment.
Syntax
PV (rate, nper ,pmt, fv, type)
Rate
is the interest rate per period
Nper is the total number of payment periods in an annuity.
Pmt
is the payment made each period and cannot change over the life of the annuity.
Fv is the future value, or a cash balance you want to attain after the last payment is
made. If fv is omitted, it is assumed to be 0
Type is the number 0 or 1 and indicates when payments are due. 0 or omitted at the
end of the period, 1 at the beginning of the period.
Then the Solution of the above example is.
P = PV (12%, 5, 0, 1000, 0).
Example 4.4
Suppose you buy a share for L.E.10 and sell it for L.E.20, your profit is L.E.10. If
that happens within a year, your rate of return is an impressive 100% (L.E.10 /
L.E.10 = 1). If that takes 5 years, what would be the average annual rate of return
on your investment?
L.E.20
i=?%
years
0
L.E.10
1
2
3
4
5
Solution
This example could be solved by either of the following methods:
1. F = P (1 + i)n
20 = 10(1 + i) 5
Then
i = 2(1/5) – 1 = 14.87 %
2. Using spreadsheet application such as Excel. A financial
function such as RATE (n, 0, P, F) allows us to calculate an
unknown interest rate. The precise command statement would
be as follows:
i = RATE (5, 0, -10, 20) = 14.87%
UNIFORM-SERIES COMPOUNDAMOUNT FACTOR (USCAF)
(USCAF)
0
1
2
3
4
F= ??
n-2
n-1
n
A =given
A series of equal installments paid at the ends of consecutive interest periods
will accumulate to a certain future value after n periods.
Dealing with each payment separately we can deduce the formula for determining
the future value F for any series of equal installments as shown in table.
A series of equal payments
Payment at end of period
No.
1
2
3
Accumulate to
1  i n1
1  i n2
1  i n3
.
.
.
.
.
n-1
A 1  .i 
n
A
The future compound amount is the summation of the right-hand column. Thus,
F  A 1  i 

n 1
 1  i 
n2
 1  i 
n 3
 1  i   1


Multiplying by 1  i 
1  i  F  A 1  i   1  i 
n
Subtracting

n 1
 1  i 
n2


iF  A 1  i   1
n
1  i  1

F
F
 
, i, n 
A
A
i


n
 1  i 

Example 4.5
Suppose you make 15 equal annual deposits of 1000 L.E. each
into a bank account paying 5% interest per year. The first deposit
will be made one year from today. How much money can be
withdrawn from this bank account immediately after the 15th
deposit?
Solution:
F= ??
i = 5%
years
0
1
2
3
4
5
6
7
8
9
10
11
12 13
14
15
A = L.E.1000
The value of A is L.E.1000, n = 15 years, and i = 5% per year.
Immediately after the 15th payment, the future equivalent amount is.
F = l000 (F/A, 5%, 15) = 1000 (1.0515 – 1) / 0.05
= 1000 × (21.5786)
= L.E. 21578.60
Example 4.6
Suppose you make an annual contribution of L.E.3000 to your saving
account at the end of each for 10 years. If your saving account earns 7%
interest annually how much can be withdrawn at the end of ten years
F = ??
i=7%
years
0
1
2
3
4
5
A = L.E.3000
6
7
8
9
10
Solution
Given: A = L.E.3000, n = 10 years, and i = 7% per year.
Find: F
F = 3000(F/A, 7%, 10) = L.E.3000 (13.8164)
= L.E.41449.20.
To obtain the future value of the annuity on Excel, we may use the following
financial command:
= FV (7%, 10, 3000, 0, 0)
Example 4.7
You have an opportunity to purchase a piece of vacant land for
m.u.30000 cash. If you bought it, you would plan to hold the
property for 15 years and then sell it at a profit. During this period,
you would have to pay annual property taxes of m.u.600. You
would have no income from the property. Assuming that you want a
10 % rate of return from the investment, at what net price would
you have to sell it 15 years hence?
Solution
F = ??
i= 15 %
0
years
1
2
3
30 000 m.u.
4
5
6
7
8
9 10
11 12 13 14 15
A = 600 m.u.
F = 30000 (F/P,10%,15) + 600 (F/A,10%,15)
= 30000 (1.08)15 + 600 ((1.0815 – 1) / 0.08)
= m.u. 144380.93
Using Spreadsheets (Excel) the above example could be solved as,
FV= (10%,15, 600, 30000, 0)
SINKING FUND FACTOR (SFF)
AF
i
1  i n  1


A
i
A

, i, n 
n
F
F
1  i   1
(SSF)
0
1
2
3
4
A = ??
F = given
n-2
n-1
n
Example 4.8
An enterprising student is planning to have personal savings
totaling L.E.1000000 when she retires at age 65. She is now 20
years old. If the annual interest rate will average 7% over the next
45 years on her savings account, what equal end-of-year amount
must she save to accomplish her goal?
Solution
The future amount, F, is L.E.1000000. The equal annual amount this student
must place in a sinking fund that grows to L.E.1000000 in 45 years at 7%
annual interest
A = 1000000 (A/ F, 7%, 45)
A= 1000000 (0.07/((1+0.07)45 –1))
A= 1000000(0.0035) = L.E.3500
Also the above example could be solved by spreadsheet as Excel using function called
PMT,
PMT function Calculates the payment for a loan based on constant payments and a
constant interest rate.
Syntax
PMT (Rate, Nper, Pv, Fv, Type)
Description of the arguments in PMT function,
Rate is the interest rate for the loan.
Nper is the total number of payments for the loan.
Pv is the present value, or the total amount that a series of future payments is worth
now; also known as the principal.
Fv is the future value, or a cash balance you want to attain after the last payment is
made. If Fv is omitted, it is assumed to be 0 (zero), that is, the future value of a loan is 0.
Type is the number 0 (zero) or 1 and indicates when payments are due.
The Excel solution then is,
PMT (7%, 45, 0, 1000000, 0)
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Educational Material:
Engineering Economy
Module 1: Lectures 1-8
Mohamed F. El-Refaie
 (Sunday
 Dr.
and Tuesday, L3,L4,L5 and L6)
Sayed Kaseb
 (Saturday
and Wednesday, L1,L2,L7 and L8)
Module 1 Teaching Team
 Dr.