Transcript Compound Interest - KCEE - economic education for students

Compound Interest
• Suppose you invest $100 in an account that will
pay 10% interest per year. How much will be in
the account after three years?
– Year 1: Interest = $100*.10 = $10, total value in
account = $100 + $10 = $110 = $100 +
100*r=($100)*(1+r)
– Year 2: Interest = $110 *.10 = $11, total value in
account = $110 + $11 = $121 = $110 + 110*r =
$110*(1+r) = $100*(1+r)*(1+r) = $100*(1+r)^2
– Year 3: Interest = $121*.10 = $12.10, total value in
account = $121+$12.10 = $133.10 = $121 + $121*r=
$121*(1+r) = [$100*(1+r)^2]*[1+r] = $100*(1+r)^3
Compound Interest
• Generalizing we get:
– FV= PV(1 + r)t
• Finding PVs is discounting, and it’s the reverse
of compounding.
t
FV
 1 
-t
PV =
= FV
  FV 1  r 
t
1 + r 
1+ r 
Compound Interest
• Let’s suppose that you decide to save $50 per
month starting at age 18 and ending at age 65.
• How much money would you have in your
savings account?
• Total amount saved
– 47 years * 12 months * $50 = $28,200
– Is that how much money you will have in the
future?
Compound Interest
• How much money would you have in your
savings account if you earned
– 4%?
– 8%?
– 12%?
• http://www.lei.ncee.net/interactives/compou
nd/
Compound Interest
• The principle of compounding means that you
earn interest on interest
• Three things to consider
– Invest early
– Invest often
– Have patience
Compound Interest
• Finding PVs is discounting, and it’s the reverse
of compounding.
t
FV
 1 
-t
PV =
= FV
  FV 1  r 
t
1 + r 
1+ r 
Compound Interest
• PV = value today of a future cash flow or
series of cash flows= Equilibrium value of an
investment
– price at which investors are indifferent between
buying and selling a security
• Opportunity cost rate = the rate of return on
the best available alternative investment of
equal risk
What’s the PV of $100 due in 3 years
if r = 10%?
Finding PVs is discounting, and it’s the
reverse of compounding.
0
PV = ?
10%
1
2
3
100
Present value
3
1 

PV = $100 
 =
 1.10 
= $100 0.7513  = $75.13.
Amortization
• Construct an amortization schedule for a
$1,000, 10% annual rate loan with 3 equal
payments of $402.11.
Step 1: Find interest charge for Year 1.
INTt = Beg balt (r)
INT1 = $1,000(0.10) = $100.
Step 2: Find repayment of principal in
Year 1.
Repmt = PMT - INT
= $402.11 - $100
= $302.11.
Step 3: Find ending balance after Year 1.
End bal
= Beg bal - Repmt
= $1,000 - $302.11 = $697.89.
Repeat these steps for Years 2 and 3
to complete the amortization table.
BEG
BAL
YR
1
2
3
TOT
$1,000
698
366
PMT
INT
PRIN
PMT
$402
402
402
1,206.34
$100
70
37
206.34
$302
332
366
1,000
Interest declines. Tax implications.
END
BAL
$698
366
0
Project Example Information
• You are looking at a new project and you have
estimated the following cash flows:
–
–
–
–
Year 0:
Year 1:
Year 2:
Year 3:
CF = -165,000
CF = 63,120;
CF = 70,800
CF = 91,080;
• Your required return for assets of this risk is 12%.
Net Present Value
• The difference between the market value of a project
and its cost
• How much value is created from undertaking an
investment?
– The first step is to estimate the expected future cash flows.
– The second step is to estimate the required return for
projects of this risk level.
– The third step is to find the present value of the cash flows
and subtract the initial investment.
NPV – Decision Rule
• If the NPV is positive, accept the project
• A positive NPV means that the project is
expected to add value to the firm and will
therefore increase the wealth of the owners.
• Since our goal is to increase owner wealth,
NPV is a direct measure of how well this
project will meet our goal.
Computing NPV for the Project
• Using the formulas:
– NPV = 63,120/(1.12) + 70,800/(1.12)2 +
91,080/(1.12)3 – 165,000 = 12,627.42
• Do we accept or reject the project?
Internal Rate of Return
• This is the most important alternative to NPV
• It is often used in practice and is intuitively
appealing
• It is based entirely on the estimated cash
flows and is independent of interest rates
found elsewhere
IRR – Definition and Decision Rule
• Definition: IRR is the return that makes the
NPV = 0
• Decision Rule: Accept the project if the IRR is
greater than the required return
Computing IRR For The Project
• If you do not have a financial calculator, then
this becomes a trial and error process
• Calculator
– Enter the cash flows as you did with NPV
– Press IRR and then CPT
– IRR = 16.13% > 12% required return
• Do we accept or reject the project?
NPV Profile For The Project
70,000
60,000
50,000
NPV
40,000
30,000
20,000
IRR = 16.13%
10,000
0
-10,000 0
0.02 0.04 0.06 0.08
0.1
0.12 0.14 0.16 0.18
-20,000
Discount Rate
0.2
0.22