- Present value is the current worth of a future sum of money or stream of cash flows given a specified rate of return. - Discounting is the process of determining the present value of future cash flows. - The document provides examples of using formulas to calculate future and present values under different interest rates and time periods, demonstrating the impact of compounding.

1. Time value of Money

2. Learning Objectives
. Understand what is Future Value
. Understand what is compounding
. Understand the impact of compounding on interest rate and time
. Understand present values.
. Understand discounting
. Understand the impact of discounting on interest rate and time

3. Introduction
Item Definition
1 Future value The future value of an asset in the future
2 Compounding The process where the value of an investment
increases because the revenue on both capital and
interest, earn interest as time passes.
It is a direct realization of the time value of money.
If we invest some money today, what will be the
amount we get at a future date.
3 Present value The current value of an asset in the future
4 Discounting The process of determining the Present Value of a
payment or a stream of payments that is to be
received in the future.
What should be the amount we need to invest
today, to get a specific amount in future.
5 Nominal interest rate The stated interest rate on a financial product
6 Effective annual
interest rate
The interest rate that is actually earned or
paid due to the result of compounding

5. Future value
Compounding

6. Money and Time
Consider 2 persons :- (example 1)
Mr. A have $1,000 he put in his house for 10 years
Mr. B have $1,000 he put into a saving accounts for 10 years
at 10 % interest.
What are their future values?
Future values after 10 years :- Mr. A = $1,000 (no change)
Mr. B = $2,593.74
Future values depends on what you do with the current money
Year Beginning interest ending
1 1,000.00 100.00 1,100.00
2 1,100.00 110.00 1,210.00
3 1,210.00 121.00 1,331.00
4 1,331.00 133.10 1,464.10
5 1,464.10 146.41 1,610.51
6 1,610.51 161.05 1,771.56
7 1,771.56 177.16 1,948.72
8 1,948.72 194.87 2,143.59
9 2,143.59 214.36 2,357.95
10 2,357.95 235.79 2,593.74

7. Future value
Future value :- The value of an asset at a specific future date
Present value X ( 1 + interest rate )
where n is the number of years
n
Excel Formula =FV(Interest rate,time period,0, initial amount)
= FV(10%,10,0,1000)
= 2,594
Using Excel – formulas, financial
Ignore – sign, Excel treat FV as an investment function,
put $1,000 as an investment, to get $2,594
Using Financial Calculator
Enter 10 10 -1000 0
N I/Y PV PMT FV
2,594

8. Future Value table

9. Compounding
Consider example 1 Mr. B $1,000 for 10 years at 10 % interest :=–
Why after 10 years the FV is $ 2,594 and not $2,000
((1,000 + (10% *10 years * 1,000))
The answer = Characteristic of Compounding
= From 2nd year to 10th year the annual interest
calculation is based on previous years
accumulated balance of both principal and
interests compounded.

10. Compounding – Exercise 1
Your company has $ 1,000 extra cash and wish to put into
a fixed deposit account at 5% interest per annum.
What is the FV amount in 1 year, 3 years, 5 years and 10 years time ?
Use Excel or a financial calculator

11. Compounding – Answer 1
Excel =FV(interest,period,0,PV)
= -1,050 (1 year)
= -1,158 (3 years)
= -1,276 (5 years)
= -1,629 (10 years)
Financial calculator
Enter 1 5 -1,000 0
3
5
10
N I/Y PV PMT FV
1,050
1,158
1,276
1,629

12. Compounding and interest rate
Consider example 1 Mr. B under various interest rates : -
Future value increase :-
a) not directly proportional to interest rates
b) the higher the interest rate per se the more the growth
c) the lower the interest rate per se the lesser the growth

13. Compounding and time
Consider example 1 Mr. B under various time zones : -
Future value increase :-
a) not directly proportional to time
b) exponentially (more and more) in timeLife Insurance
EPF

14. FV - Exercise 2
You have $20,000.
a) If you put it into an account at 7 % interest annually,
how many years needed to get $30,000
b) if you put in for 10.25 years, how much you will get
c) How many years to get $30,000 if the interest is 3%
d) How many years to get $30,000 if the interest is 11%
e) What does this tell you about relationship among
interest rates, time and FV

15. FV - Answer 2 a
a) Excel = 6 years (RM30,014.61) 7% 6 years, get $30,000
b) Excel = S40,014 (RM40,014.16) 7% 10.25 years get $40,000
c) Excel = 13.8 years (RM30,073.48) 3% 13.8 years get $30,000
d) Excel = 3.9 years (RM30,046.20) 11% 3.9 years get $30,000
= FV(7%,6,0,20000)
= FV(7%,10.25,0,20000)
= FV(3%,13.8,0,20000)
= FV(11%,3.9,0,20000)
Recall =FV(interest,period,0,PV)
Solving for period is tough, so we use trial and error :-

16. FV - Answer 2 b
AVG Interest
per year
7% 6 years, get $30,000 $1,667
7% 10.25 years get $40,000 $1,951 Longer time
3% 13.8 years get $30,000 $724 Low interest rate
11% 3.9 years get $30,000 $2,564 High interest rate

17. Compounding – occurrence frequencies
We have seen compounding on annual basis.
Consider compounding on a daily, weekly, monthly, quarterly, semi annually.
The formula :-
FV = Present value X ( 1 + Annual interest rate (i) )
Compounding periods
per year (m)
where n is the number of years
m is the number of periods in the year
mn
The corresponding period is thus 365, 52, 12, 4, 2

18. Compounding – occurrence frequencies
Present value 1,000 1,000 1,000 1,000 1,000 1,000
Interest rate 10% 10% 10% 10% 10% 10%
Years 10 10 10 10 10 10
Interest compounded Daily weekly monthly quarterly semi annually annually
Compounding period 365 52 12 4 2 1
Future Value -2,718 -2,716 -2,707 -2,685 -2,653 -2,594
=FV(10%/12,10*12,0,1000)
Compare Monthly vs Annually = FV 4.4 % more for monthly
Present value 1,000 1,000 1,000 1,000 1,000 1,000
Interest rate 2% 3% 5% 7% 8% 10%
Years 10 10 10 10 10 10
Interest compounded Daily weekly monthly quarterly semi annually annually
Compounding period 365 52 12 4 2 1
Future Value -1,221 -1,350 -1,647 -2,002 -2,191 -2,594
=FV(2%/365,10*365,0,1000)
Compare 10 % annually vs 2 % daily = Interest rate drop 5 times but FV only drop 2 times

19. Compounding occurrence frquencies – Exercise 3
Your company has $ 100,000 extra cash and wish to put into
a fixed deposit account at 10% interest per annum.
What is the FV amount in 10 years time if the interest is compounded
daily, weekly, monthly, quarterly, semi annually and annually ?
Use Excel or a financial calculator

20. Compounding occurrence frquencies – Answer 3 (a)
Excel =FV(interest/compounding period per year,
Compounding period per year *number of years, 0, PV)

21. Compounding occurrence frequencies – Answer 3 (b)
Financial calculator
1) divide the yearly interest rate into each compounding period rate
2) input this rate into the calculator
Yearly Compounding Compounding
Rate Occurance Rate
10% 365 0.027397%
10% 52 0.192308%
10% 12 0.833333%
10% 4 2.5%
10% 2 5%
10% 1 10%
Enter 3650 0.027397 -100,000 0
520 0.192308
120 0.833333
40 2.5
20 5
10 10
N I/Y PV PMT FV

22. Compounding and other things – using Excel
Consider current market for wireless printer is 1,000
What will be the market if 10 % increase after 1, 2, 3, 4, 5 years ?
What will be the market if 10 % decrease after 1, 2, 3, 4, 5 years ?
Presentvalue 1,000 1,000 1,000 1,000 1,000
Rateincrease 10% 10% 10% 10% 10%
Years 1 2 3 4 5
FutureValue -1,100 -1,210 -1,331 -1,464 -1,611
Presentvalue 1,000 1,000 1,000 1,000 1,000
Ratedecrease -10% -10% -10% -10% -10%
Years 1 2 3 4 5
FutureValue -900 -810 -729 -656 -590
Increasing rate
Decreasing rate (put – sign at the rate)

23. Thoughts on Future Values and Compounding
1. Theoretically – sounds reasonable
2. Realistically - NO
= FV is on monetary unit and doesn’t consider inflation
= very few investment give constant rate of returns
over a long period except Preference Shares
and Debentures, Bonds and Treasury Bills.
They are low risk investments thus carrying
low rates of interest. So calculating FV
base on high interest rates are not realistic
= If the investment is in a foreign currency then there
is an additional risk of foreign exchange losses
or vice versa

24. Present value
Discounting

25. Present Value
Recall :-
Future Value = Present value X ( 1 + interest rate )
where n is the number of years
Then :-
n
Present value = Future Value
( 1 + interest rate )
where n is the number of years
n

26. Present Value
Consider :-
Mr. C wants to have $1,000 in 10 years time.
What is the amount he has to put into a saving account now
at 10 % compounding interest?
Present value = Future Value
( 1 + interest rate ) n
where n is the number of years
= 1,000
(1 + 10 % )10
= 1000
2.59374
= 385.54
To the
Year power
1 1.10000
2 1.21000
3 1.33100
4 1.46410
5 1.61051
6 1.77156
7 1.94872
8 2.14359
9 2.35795
10 2.59374

27. Present value
Excel Formula =PV(Interest rate,time period,0, future value)
= PV(10%,10,0,1000)
= -385.54
Using Excel – formulas, financial
Ignore – sign, Excel treat PV as a investment function,
to get 1,000 need to outgo -385.54 now
Using Financial Calculator
Enter 10 10 -1000 0
N I/Y FV PMT PV
385.54

28. Present Value Table

29. Discounting when periods are not known
Solving the period is difficult , there are 2 ways :-
a) Use trial and error of FV formula in Excel – input various periods
= FV(interest rate, period, payment, initial investment)
b) Use Excel formula in Excel
=NPER(interest, 0, - initial investment, FV needed)
(Initial investment must be – sign)
Rule of 72 – Roughly if you want to know how long it takes
for an investment to double :-
take 72 divide by interest rate
example 72/8 % = 9 years

30. Discounting when interest rate is not known
Solving the period is difficult , there are 2 ways :-
a) Use trial and error of FV formula in Excel – input various interest
= FV(interest rate, period, payment, initial investment)
b) Use Excel formula in Excel
=RATE(period, payment, - initial investment, FV needed)
(Initial investment must be – sign)

31. Present value - Exercise 4
You want to retire in 35 years with $2,000,000 and the bank is
offering 4 % interest rate compounded annually.
a) How much money you need to put in now
b) How many years you need to invest if the interest rate is 14%
and the initial investment is same as (a) above
c) What is the interest rate if you have only $200,000 now and
need $2,000,000 in 35 years time.

32. Present value - Answer 4
Recall =PV(Interest rate,time period,0, future value)
a) = PV(4%,35,0,2000000)
(RM506,830.94)=
= FV(14%,10.48,0,506830)
= 10.48 years
b)
Use trial and error to get period
= FV(6.8%,35,0,200000)
c)
Use trial and error to get interest
= 6.8 %
Recall =NPER(interest, 0, - initial investment, FV needed)
=NPER(14%,0,-506830,2000000)
= 10.48 years
=RATE(period, payment, - initial investment, FV needed)Recall
=RATE(35,0,-200000,2000000)
= 6.8 %