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Chapter 3 – Important Stuff

• Mechanics of compounding / discounting• PV, FV, PMT – lump sums and annuities• Relationships – time, interest rates, etc• Calculations: PV’s, FV’s, loan payments,

interest rates

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Time Value of Money (TVM)

• Time Value of Money – relationship between value at two points in time– Today versus tomorrow; today versus

yesterday– Because an invested dollar can earn interest,

its future value is greater than today’s value• Problem types: monthly loan payments,

growth of savings account; time to goal

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Financial Calculator Keys

• PV - Present value• FV - Future value• PMT- Amount of the payment• N - Number of periods

(years?)• I/Y - Interest rate per period

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TI Calculator ManualStrongly Suggested Readings

• Getting Started – page 6 and 7• Overview – page 1-4, 1-10 and 1-20• Worksheets – pages 2-14 and 2-15• TVM – 3-1 to 3-9

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Calculator Tips

Decimals and Compounding Periods• 2nd (gray), Format (bottom row), 4,

enter, CE/C (lower left) - hit twice• Compounding: 2nd , I/Y, 1, enter,

CE/C – extremely important !!• Right arrow key fixes “misteaks”• One cash flow must be negative or error

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Concept of Compounding

• Compound Interest – basically interest paid on interest

• Takes interest earned on an investment and reinvests it– Earn interest on the principal and reinvested

interest

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Compound Interest @ 6%

Year Begin Interest Future Val

1 $100.00 $6.00 $106.00

2 106.00 6.36 112.36

3 112.36 6.74 119.10

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Future Value Interest Factor

Year @2% @6% @10%

1 1.020 1.060 1.1002 1.040 1.124 1.2103 1.104 1.191 1.611 10 1.219 1.791 2.594

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Future Value (FV)

Algebraically FVn = PV (1 + i)n

Underlies all TVM calculations

Keystrokes: 100 +/- PV; 3 N; 0 PMT;6 I/Y; CPT FV = 119.10

One cash flow must be negative

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FV – Other Keystrokes• How long for an investment to grow from

$15,444 to $20,000 if earn 9% when compounded annually? Must solve for N.

• 15444 +/- PV; 20000 FV; 0 PMT; I/Y 9; CPT N = 3 years

• What rate earned if start at $15,444 to reach $20,000 in 3 years? Solve for I/Y.

• 15444 +/- PV; 20000 FV; 0 PMT; 3 N; CPT I/Y = 9%

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FV Can Be Increased By

1. Increasing the length of time it is compounded

2. Compounding at a higher rateAnd/or

3. Compounding more frequently

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FV at Different Rates and Periods

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Present Value (PV)

If I earn 10%, how much must I deposittoday to have $100 in three years? $75.10This is “inverse compounding”Discount rate – interest rate used to bring

(discount) future money back to presentFor lump sums (only) PV and FV are

reciprocals

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Present Value Interest Factor

@2% @5% @10% Year 1 .980 .952 .909 Year 2 .961 .907 .826Year 3 .942 .864 .751Year 10 .820 .614 .386

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Present Value Formula

[ 1 ]PV = FVn [ (1 + i) n ]

PVIF and FVIF for lump sums only are reciprocals. For 5% over ten years

FVIF = 1.629 = 1 / .614PVIF = .614 = 1 / 1.629

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Keystrokes$100 @5% for ten years

• For PV 100 FV; 0 PMT; 5 I/Y; 10 N; CPT PV = 61.39• For I/Y 100 FV; 0 PMT; +/-61.39 PV; 10 N; CPT I/Y = 5• For N 100 FV; +/-61.39 PV; 0 PMT; 5 I/Y; CPT N = 10 years

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PV Decreases If

1. Number of compounding periods (time) increases,

2. The discount rate increases,

And/or3. Compounding frequency increases

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Present Value – Different Rates

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Annuities

• Series of equal dollar payments– Usually at the end of the year/period

• If I deposit $100 in the bank each year starting a year from now, how much will I have at the end of three years if I earn 6%? $318.36

• We are solving for the FV of the series by summing FV of each payment.

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FV of $100 Annuity @ 6%

End ofPMTFVIF $

Year 3 $100 1.0000 * $100.00Year 2 100 1.0600 106.00Year 1 100 1.1236 112.36

$318.36* The payment at end Year 3 earns nothing

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Annuity Keystrokes

What will I have if deposit $100 per year starting at the end of the year for three years and earn 6%?

0 PV; 100+/- PMT; 3 N; 6 I/Y;CPT FV = 318.36

PV is zero - nothing in the bank today

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Present Value of an Annuity

Amount we must put in bank today towithdraw $500 at end of next three yearswith nothing left at the end if we earn 6%?Present valuing each of three paymentsKeystrokes: 500+/- PMT; 0 FV; 3 N; 6 I/Y; CPT PV = 1,336.51

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Nonannual Compounding

• Invest for ten years at 12% compounded quarterly. What are we really doing?– Investing for 40 periods (10 * 4) at 3%

(12%/4)• Make sure 2nd I/Y is set to 1.• Need to adjust rate per period downward

which is offset by increase in N

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Nonannual Compounding

• FVn = PV ( 1 = i/m) m * n

• m = number of compounding periods per year so per period rate is i/m

• And m * n is the number of years times the compounding frequency which adjusts to the rate per period

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Compounding $100 @10%

Compounding One Year 10 YearsAnnually $110.00 $259.37Semiannually 110.25 265.33

Quarterly 110.38 268.51Monthly 110.47 270.70

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Amortizing Loans

• Paid off in equal installments– Makes it an annuity

• Payment pays interest first, remainder goes to principal (which declines)

• $600 loan at 15% over four years with equal annual payments of $210.16

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$600 Loan Amortization

Total To Int To Prin End Bal

Year 1 210.16 90.00 120.16 479.84Year 2 210.16 71.98 138.18 341.66Year 3 210.16 51.25 158.91 182.75Year 4 210.16 27.41 182.75 0

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Loan Interest & Principal Payment

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Calculate a Loan Payment

• $8,000 car loan payable monthly over three years at 12%. What is your payment?

How many monthly periods in 3 yrs? 36 NMonthly rate? 12%/12 = 1%/mo = I/YWhat is FV? Zero because loan paid out8000+/- PV; 0 FV; 1.0 I/Y; 36 N; CPT PMT=265.71

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Perpetuities

• Equal payments that continue forever– Like Energizer Bunny and preferred stock

• Present Value = Payment Amount Interest RatePreferred stock pays $8/yr, disc. rate- 10%Payment fixed at $8/ .10 = $80 market price