This chapter discusses financial mathematics and time value of money concepts. It covers calculating future and present values of single amounts as well as annuities. Key points covered include: - Understanding why time value of money is important for accounting, management, marketing, and other business functions. - Using formulas to calculate future and present values, as well as future and present values of ordinary, due, deferred, and forfeited annuities. - Examples are provided to demonstrate calculating interest earned over time, compounding periods, and valuations of cash flows at different points in time.

1. Financial management chapter 5
CHAPTER 5: FINANCIAL MATHEMATICS
Learning
Objectives
1- The use of
computational
tools, and the
basic of time
value of money.
2- Understand
the concepts of
future and
present value,
their calculation
for single
amounts, and
the relationship
of present value
to future value.
3- Find the
future value
and the present
value of both
an ordinary
annuity and an
annuity due
and find the
present value of
a perpetuity.
1

2. Financial management chapter 5
Why do you need to learn about time value of money?
Accounting:
You need to
understand
time-value-
of-money
calculations
in order to
account for
certain
transactions
such as loan
amortization,
lease
payments,
and bond
interest rates.
Information
systems: You
need to
understand
time-value-
of- money
calculations
in order to
design
systems that
optimize the
firm’s cash
flows.
Management:
You need to
understand
time-value-of-
money
calculations so
that you can
plan cash
collections
Marketing:
You need to
understand
time value of
money
because
funding for
new
programs and
products
must be
justified
financially
using time-
value-of-
money
techniques.
Operations:
You need to
understand
time value of
money
because
investments
in new
equipment,
in inventory,
and in
production
quantities
will be
affected by
time-value-
of-money
techniques.
2

3. Financial management chapter 5
Hint
The time value of money is one of the
most important concepts in finance.
Money that the firm has in its
possession today is more valuable
than future payments because the
money it now has can be invested
and earn positive returns.
3

4. Financial management chapter 5
Future Value versus Present Value
Financial values and decisions can be assessed by using either future value or
present value techniques. Although these
techniques will result in the same decisions, they view the decision differently.
Future value techniques typically measure cash flows at the end of a project’s life.
Present value techniques measure cash flows at the start of a project’s life (time zero).
Future value is cash you will receive at a given future date, and present value is just
like cash in hand today.
4

5. Financial management chapter 5
• Example
• IF Sara deposits 100.00 EGP into her saving account at a compound interest rate of 5% per year
(Sara receives interest on both principal plus earned interest). Calculate the interest and the
total amount of money she has at the end of each year for five years.
• Year Beginning Balance Interest Earned Ending
Balance
• 1 100 5.00 105.00
• 2 105 5.25 110.25
• 3 110.25 5.51 115.76
• 4 115.76 5.79 121.55
• 5 121.55 6.08 127.63
5

6. Financial management chapter 5
• Example
• If Sara deposits 100.00 EGP into her saving account at a compound interest rate of 1.25% per 3-months
period. Sara receives interest quarterly on both principal and on earned interest. Calculate the interest
and the total amount of money she has at the end of each 3-months period.
• Period Beginning Balance Interest Earned Ending Balance
• 1 100 1.25 101.25
• 2 101.25 1.27 102.52
• 3 102.52 1.28 103.80
• 4 103.80 1.29 105.09
• 5 105.09 1.32 106.41
• 6 106.41 1.33 107.74
• 7 107.74 1.35 109.09
• 8 109.09 1.36 110.45
• 9 110.45 1.38 111.83
• 10 111.83 1.40 113.23
• 11 113.23 1.41 114.64
• 12 114.64 1.44 116.08
6

7. Financial management chapter 5
• FINANCIAL TABLE
7

8. Financial management chapter 5
Future Value of a Single Amount
• Imagine that at age 25 you
began making annual purchases of $2,000 of an investment
that earns a guaranteed 5 percent annually.
At the end of 40 years, at age 65, you
would have invested a total of $80,000 (40 years X $2,000 per year).
Assuming that all funds remain invested, how much would you have
accumulated at the end of the fortieth year?
$100,000? $150,000? $200,000? No,
your $80,000 would have grown to $242,000! Why?
Because the time value of money allowed your investments to
generate returns that built on each other over the 40 years
8

9. Financial management chapter 5
The Equation for Future Value
• FVn = future value at the end of period n
• PV = initial principal, or present value
• i = annual rate of interest paid.
• n = number of p
• FVn = PV X (1 + i)³ ²
• Future value interest factor = FVIFi,n = (1 + i)² ³
• FVn = PV X (FVIFi,n)
9

10. Financial management chapter 5
Present Value of a Single Amount
• J Conversion m: number of i = J /m n for 3 years
n for 18 month
• period conversions per year
• 12% annually 12 months 1 12% 3
1 full conversion and
• 6
months as part of period 12% semiannually 6 months 2
6% 6 3
• 12% quarterly 3 months 4 3% 12
6
• 12% monthly 1 month 12 1% 36
18
• Compound Amount Formula: S = P(1+i)² ³
• 10

11. Financial management chapter 5
• Example
• Find the compound amount of $4500 invested for 8 years and 6 months at a.12%
compounded semiannually
• a.8% compounded monthly
• b.6% compounded quarterly
• c.7% compounded monthly.
• Solution
• s= p(1+i) n
• a. n = 8.5 × 2 = 17, s = 4500(1.06)17 =$12,117.48
• b. n = 8.5 × 4 = 34, s = 4500(1.015)34 =$7465.48
• c. n = 8.5 × 12 = 102, s =4500(1.006666666)102=$8862.43
11

12. Financial management chapter 5
Annuities
The formula’ for the amount of an ordinary annuity is:
• 𝑆𝑛 = 𝑅. (1+𝑖)³ ² −1 / 𝑖
• R = 100 (per quarter),
• i = 1% (per quarter),
• n = 4 (quarterly payments)
• Substituting the values in Formula:
• 𝑆𝑛= 𝑆4=100.(1+1%)4−1%=100.1.04060401−10.01
• =100.0.040604010.01=100(4.060401)=$406.0401, 𝑜𝑟 $406.04
12

13. Financial management chapter 5
Example
• Find the amount of an annuity of $2,000 payable at the end of each year for 45
years, if the interest rate is 10% compounded annually.
• Solution
• R = $2,000 (per year), i = 10% (per year), n = 45 (annual payments) OR
• Thus, if a person, aged 20, made an initial deposit of $2000 in his account that
pays 10% compounded annually, after 45 equal annual deposits he or she will
have $1,437,809.67 at age 65.
• Total deposits = $2000 X 45 = $90,000
• Total interest = 1,437.809.67 — 90,000.00
• = $1,347,809.67
13

14. Financial management chapter 5
PRESENT VALUE OF AN ANNUITY
• The present value of an annuity is the value at the beginning of the term of the
annuity.
• Let R = size of each regular payment (or periodic rent)
• i = interest rate per conversion period
• n = number of payments during the term of an annuity
• An = the present value of an ordinary annuity
• 𝐴𝑛=𝑅 .1− (1+𝑖)−𝑛 / 𝑖
14

15. Financial management chapter 5
EXAMPLE
• If a man wishes to receive $200 at the end of each month for three years from a bank that
pays interest at 6% compounded monthly, how much must he deposit in the hank now?
• Solution R = $200; i = 6%/12 = 12% (per month),
• n = 3 X 12 = 36 (monthly payments). 𝐴𝑛=𝑅 .1− (1+𝑖)−𝑛 / 𝑖 an
• = 200(32.87102) = $6,574.20
• The present value of the annuity is $6,574.20. Thus, the man must deposit $6,574.20 in the
bank now. The sum of the compound interest paid by the bank on the 36 payments is
computed as follows: Payments = 200 × 36 = $7,200
• Compound interest = 7,200 – 6,574.20 = $625.80
15

16. Financial management chapter 5
OTHER TYPES OF ANNUITIES CERTAIN: DUE,
DEFERRED, and FORBORNE
• Amount Ordinary: 𝑆𝑛=𝑅.(1+𝑖)𝑛−1 / 𝑖
• Present Value Ordinary: 𝐴𝑛=𝑅 .1− (1+𝑖)−𝑛 / 𝑖
• an Amount Due:
• Sn(Due) = Sn (1+i) 𝑆𝑛=𝑅.(1+𝑖)𝑛−1 / 𝑖
• Present Value Ordinary:
• An (Due) = An (1+i)
• 𝐴𝑛=𝑅 .1− (1+𝑖)−𝑛 / 𝑖
• Amount Forborne:
• Sn(For) = Sn (1+i)p 𝑆𝑛=𝑅.(1+𝑖)𝑛−1 / 𝑖
• Present Value deferred:
• An (Def) = An (1+i)-m 𝐴𝑛=𝑅 .1− (1+𝑖)−𝑛 / 𝑖
16

17. Financial management chapter 5
EXAMPLE
• If $20 is deposited at the beginning of each month in a fund that earns interest at 18%
compounded monthly, what is the final value at the end of three years?
• Solution
• R = $20 (per month),
• i = 18%/12 = 112% (per month),
• n = 3 X 12 = 36 (months or payments)
• Sn(Due) = Sn (1+i) 𝑆𝑛=𝑅.(1+𝑖)𝑛−1𝑖
• S36 (due) = $949,70
17

18. Financial management chapter 5
Present Value of an Annuity Due, An (due)
• EXAMPLE
• What is the present value of an annuity due if the size of each payment is $100 payable at the
beginning of each quarter for one year and the interest rate is 4% compounded quarterly?
• Solution
• R = $100 (per quarter),
• i = 4%/4 = 1% (per quarter),
• n = 4 (quarterly payments)
• An (Due) = An (1+i)
• 𝐴𝑛=𝑅 .1− (1+𝑖)−𝑛𝑖
• An(due) = 100(3.90197)(1.01) = $394.10
18