3. Basic Principle: A dollar received today is worth
more than a dollar received in the future.
This is due to opportunity cost. The opportunity cost
of receiving $1 in the future is the interest we could
have earned if we had received the $1 sooner.
This concept is so important in understanding
financial management
4.
Translate $1 today into its equivalent in the future
(compounding) – Future Value (FV)
today
future
$1
$?
Translate $1 in the future into its equivalent today
(discounting) – Present Value (PV)
today
$?
future
$1
5.
Compound interest occurs when interest paid on the
investment during the first period is added to the principal;
then, during the second period, interest is earned on this
new sum.
Compounding is the process of determining the Future Value
(FV) of cash flow.
The compounded amount (FV) is equal to the beginning
amount plus interest earned.
6.
Example: If we place RM1,000 in savings account
paying 5% interest compounded annually. How
much will it be worth at the end of each year?
RM1000
0
i = 5%
1
Year 1: RM1000 (1.05)
Year 2: RM1050 (1.05)
Year 3: RM1102.50 (1.05)
Year 4: RM1157.63 (1.05)
2
3
= RM1050.00
= RM1102.50
= RM1157.63
= RM1215.51
4
n..
7.
FVn = PV (1 +
i)n
or
FVn = PV (FVIF i,n )
where;
FVn = the FV of the investment at the end of n year
n = the number of years
i = the annual interest rate
PV = original amount invested at beginning of the first year
(1 + i) is also known as compounding factor
8. If we place RM1,000 in a savings account paying 5% interest
compounded annually. How much will our account accrue in 4
years?
PV = RM1,000 i = 5% n = 4 years
FVn = PV (1 + i)n
= 1,000(1 + 0.05)4
= 1,000 (1.2155)
= RM1,215.50
FVn = PV (FVIF i,n )
= 1,000(FVIF 5%,4 )
= 1,000 (1.2155)
= RM1,215.50
9. If Anuar invests RM10,000 in a bank where it will earn 6%
interest compounded annually. How much will it be worth at the
end of a) 1 year and b) 5 years
a)
Compounded for 1 year
FV1 = RM10,000 (1 + 0.06)1
= RM10,000 (1.06)1
= RM10,600
FV1 = RM10,000 (FVIF 6%,1 )
= RM10,000 (1.0600)
= RM10,600
10. b)
Compounded for 5 years
FV5 = RM10,000 (1 + 0.06)5
= RM10,000 (1.06)5
= RM13,380
FV1 = RM10,000 (FVIF 6%,5 )
= RM10,000 (1.3382)
= RM13,382
11. 1.
If Danial invests RM20,000 in a bank where it will earn 8%
interest compounded annually. How much will it be worth
at the end of a) 5 years and b) 15 years
12. 2.
If the interest rate increases to 10%, how much will the
Danial’s savings grow?
13. At what annual rate would the following have to be invested;
$500 to grow to RM1183.70 in 10 years.
FVn
1183.70
1183.70/500
2.3674
i
= PV (FVIF i,n )
= 500 (FVIF i,10 )
= (FVIF i,10 )
= (FVIF i,10 ) refer to FVIF table
= 9%
14. How many years will the following investment takes? $100 to
grow to $672.75 if invested at 10% compounded annually
FVn
672.75
672.75/100
6.7272
n
= PV (FVIF i,n )
= 100 (FVIF 10%,n )
= (FVIF 10%,n )
= (FVIF 10%,n ) refer to FVIF table
= 20 years
15. 1.
How many years will the following investments take:
2.
$100 to grow to $298.60 if invested at 20% compounded annually
$550 to grow to $1044.05 if invested at 6% compounded annually
At what annual rate would the following investments have to
be invested:
$200 to grow to $497.65 in 5 years
$180 to grow to $485.93 in 6 years
16. Non-annual periods occurs semiannually, quarterly or
monthly
if semiannually compounding:
FV = PV (1 + i/2)nx2 or
FV = PV (FVIF i/2,nx2 )
if quarterly compounding:
FV = PV (1 + i/4)nx4 or
FV = PV (FVIF i/4,nx4 )
if monthly compounding:
FV = PV (1 + i/12)nx12 or
FV = PV (FVIF i/12,nx12 )
17. If you deposit $100 in an account earning 6% with semiannually
compounding, how much would you have in the account after 5
years?
FV = PV (1 + i/2)nx2
= 100 (1 + 6%/2)5x2
= 100 (1 + 0.03)10
= 100 (1.3439)
= $134.39
FV = PV (FVIF i/2,nx2 )
= 100 (FVIF 6%/2,5x2 )
= 100 (FVIF 3%,10 )
= 100 (1.3439)
= $134.39
18. If you deposit $1000 in an account earning 6% with quarterly
compounding, how much would you have in the account after 5
years?
FV = PV (1 + i/4)nx4
= 1000 (1 + 6%/4)5x4
= 1000 (1 + 0.015)20
= $1,346.86
*6% /4 = 1.5% (can’t use FVIF table)
19. 1.
If you deposit $1,000 in an account earning 8% with
quarterly compounding, how much would you have in the
account after 3 years?
2.
To what amount will the following investments accumulate:
$5,000 invested for 5 years at 10% with quarterly compounding
$4,000 invested for 6 years at 6% with semiannually compounding
20. PV is the current value of futures sum
Finding PV is called discounting and can be
calculated by using this equation:
or
n
PV = [ FVn / (1 + i) ]
PVn = FV (PVIF i,n )
[1/ (1 + i)n ] is also known as discounting factor
21. What is the PV of $800 to be received 10 years from today if our
discount rate is 10%?
PV = 800 / (1.10)10
= $308.43
PV = 800 (PVIF10%,10 )
= 800 (0.3855)
= $308.40
22. Find the PV of $10,000 to be received 10 years from today if our
discount rates:
a) 5%
b)
10%
c)
20%
23. What is the PV of an investment that yields $300 to be received
in 2 years and $450 to be received in 8 years if the discount rate
is 5%?
PV = 300 (PVIF5%,2 ) + 450 (PVIF5%,8 )
= 300 (0.907) + 450 (0.677)
= 272.10 + 304.65
= $576.75
24. 1.
What is the PV of an investment that yields $500 to be
received in 3 years and $750 to be received in 5 years if the
discount rate is 5%?
2.
What is the PV of an investment that yields $1,000 to be
received in 2 years and $2,500 to be received in 4 years if
the discount rate is 6%?
25. An annuity is a series of equal payments for a specified
number of years.
100
0
100
1
100
100
2
3
100
4
There are 2 type of annuities:
Ordinary annuity
Annuity due
*in finance, ordinary annuities are used much more frequent
compared to annuities due
26. Ordinary annuity is an annuity which the payments occur at the
end of each period
a.
Present Value Annuity (PVA)
PVAn = PMT / (1+i)n
b.
or
PVAn = PMT (PVIFA i, n )
Future Value Annuity (FVA)
FVAn = PMT (1 +i)n
FVAn = PMT (FVIFA i,n )
27. Find the PV of $500 received at the end of each year of the next
3 years discounted back to the present at 10%?
PVA3 = 500/1.101 + 500/1.102 + 500/1.103
= 454.55 + 413.22 +375.66
= $1,243.43
OR
PVA3 = 500 (PVIFA 10%, 3 )
= 500 (2.487)
= $1,243.50
28. We are going to deposit $15,000 at the end of each year for the
next 5 years in a bank where it will earn 9% interest. How much
will we get at the end of 5 years?
FVA5 = 15000 (1.09)4 + 15000 (1.09)3 + 15000 (1.09)2 +
15000 (1.09)1 + 15000
= $89,770.66
OR
FVA5 = 15000(FVIFA 9%,5 )
= 15000 (5.9847)
= $89,770.50
29. 1.
What is the accumulated sum of each of the following
streams of payments
2.
$500 a year for 15 years compounded annually at 5%
$850 a year for 10 years compounded annually at 7%
What is the PV of the following annuities
$2500 ay ear for 15 years discounted back to the present at 8%
$280 a year for 5 years discounted back to the present at 9%
30. Annuity due is an annuity in which the payments occur at the
beginning of each period
a.
Present Value of Annuity Due(PVAD)
PVADn = PMT (PVIFA i, n )(1 + i)
b.
Future Value of Annuity Due(FVAD)
FVADn = PMT (FVIFA i,n )(1 + i)
31. Find the PV of $500 at the beginning of each year of the next 5
years discounted back to the present at 6%?
PVAD = 500 (PVIFA 6%,5 ) (1 + 0.06)
= 500 (4.212) (1.06)
= $2,232.36
We are going to deposit $1,000 at the beginning of each year for
the next 5 years in a bank where it will earn 5% interest. How
much will we get at the end of 5 years?
FVAD = 1000 (FVIFA 5%,5 ) (1 + 0.05)
= 1000 (5.526) (1.05)
= $5,802.30
32.
Perpetuity is an annuity that continues forever
The equation representing the PV of annuity
PV = PP / i
Example: What is PV of $1,000 perpetuity discounted back
to the present at 8%?
PV = PP / i
= 1000 / 0.08
= $12,500
33. What is the present value of the following:
A $100 perpetuity discounted back to the present at 12%
A $95 perpetuity discounted back to the present at 5%
34. Making Interest Rates Comparable
We cannot compare rates with different
compounding periods. For example, 5%
compounded annually is not the same as 5%
percent compounded quarterly.
To make the rates comparable, we compute the
annual percentage yield (APY) or effective annual
rate (EAR).
35.
Quoted rate could be very different from the
effective rate if compounding is not done annually.
Example: $1 invested at 1% per month will grow to
$1.126825 (= $1.00(1.01)12) in one year. Thus even
though the interest rate may be quoted as 12%
compounded monthly, the effective annual rate
(EAR) or APY is 12.68%.
36.
APY = (1 + quoted rate/m)m – 1
Where m = number of compounding periods
= (1 + .12/12)12 – 1
= (1.01)12 – 1
= .126825 or 12.6825%
