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-

t

t r1FV r+1

1FV =

r+1

FV = PV

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/compound/

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-

t

t r1FV r+1

1FV =

r+1

FV = PV

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

10%

What’s the PV of $100 due in 3 years if r = 10%?

Finding PVs is discounting, and it’s the reverse of compounding.

100

0 1 2 3

PV = ?

Present value

( )PV = $100

11.10

=

= $100 0.7513 = $75.13.

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3

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 3to complete the amortization table.

Interest declines. Tax implications.

BEG PRIN ENDYR BAL PMT INT PMT BAL

1 $1,000 $402 $100 $302 $6982 698 402 70 332 3663 366 402 37 366 0

TOT 1,206.34 206.34 1,000

Project Example Information

• You are looking at a new project and you have estimated the following cash flows:– Year 0: CF = -165,000– Year 1: CF = 63,120; – Year 2: CF = 70,800– Year 3: 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

-20,000

-10,000

0

10,000

20,000

30,000

40,000

50,000

60,000

70,000

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2 0.22

Discount Rate

NP

V

IRR = 16.13%