# 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.

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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.

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/ 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.

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 0123 PV = ?

Present value  PV= \$100 1 1.10 = = \$1000.7513 = \$75.13.       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. INT t = Beg bal t (r) INT 1 = \$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.

Interest declines. Tax implications. BEGPRINEND YRBALPMTINTPMTBAL 1\$1,000\$402\$100\$302\$698 269840270332366 3366402373660 TOT1,206.34206.341,000

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 IRR = 16.13%

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