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Time Value of Money Introduction

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TVM Preferences More vs. Less Sooner vs. Later More Now vs. Less Later Less Now vs. More Later ????

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TVM Questions What will my investment grow to? How much do I need today? How fast must my investment grow? How long will it take?

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Compare and Contrast 19702011 Cost of a first-class stamp: $ 0.06 $ 0.44 Cost of a gallon of gas: $ 0.36 $ 2.98 Cost of a dozen eggs: $ 0.62 $ 2.20 Cost of a gallon of Milk: $ 1.15 $ 3.69 TVM 4.98% 5.29% 3.14% 2.88%

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TVM Basic Concepts Simple vs. Compound Interest Simple Interest = interest earned only on principal (amount loaned) Compound Interest = interest earned on principal and any unpaid interest earned in an earlier time period

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Simple Interest Calculation

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Interest Example Principal$1,000 Interest Rate 10% Term5 years

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Interest Example FV = (1,000 x.10 x 5) + 1,000 FV = 500 +1,000 FV = 1,500

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Simple Interest Example Principal$1,000 Total Interest 500 Ending Balance$1,500

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Compound Interest Calculation

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Compound Interest Example

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Principal$1,000 Total Interest Rate 611 Ending Balance$1,611

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Time Value of Money

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Calculator Tips Set Calculator to 4 decimal points Set P/Yr to 1 and do not change Clear calculator before calculation Use recommended format Learn to use special features Read carefully Know the concepts of TVM

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TVM Concepts Use a time line Use + or - to indicate cash flow Periodic Cash flows can be at Beginning or End of Period Calculators use Percentages Excel uses decimals

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Lump Sum vs. Periodic Pmts Lump Sum –Single Payment –At time zero –Present Value OR –Single Payment –At end of time –Future Value Periodic Payment –Ordinary Annuity Pmt at end of periods For life of investment –Annuity Due Pmt at beg. of periods For life of investment –PMT

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Annuities Must be –Equal Amounts –Occurring in every compounding period –Ordinary Annuity – End of Period –Annuity Due – Beginning of Period

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Annuity? 0 1100 2 3 4 5

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Annuity? 0 1 2 3 4 5

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Annuity? 0 1 2 3 4 5

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Annuity? 0 1 2200 3300 4400 5500

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Lump Sum & Periodic Payment Combination –Single Payment –With periodic payments for life of investment –PV & PMT

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Recommended Structure

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Future Value of Lump Sum If you invest $1,000 in a savings account earning 10% compounded annually, how much will you have after 5 years?

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Future Value of Lump Sum

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If you invest $10,000 in a mutual fund that is expected to earn a 12% compound after-tax return, how much will you have at the end of 50 years?

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Future Value of Lump Sum

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Future Value

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Future Value of an Annuity If you invest $10,000 at the end of each year in a mutual fund that is expected to earn a 12% compound after-tax return, how much will you have at the end of 5 years?

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Future Value of an Annuity

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If you invest $10,000 at the beginning of each year in a mutual fund that is expected to earn a 12% compound after-tax return, how much will you have at the end of 5 years?

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Future Value of an Annuity

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Ordinary Annuity TimePaymentReturnFV 0 110,00012% / 4 yrs15,735.19 210,00012% / 3 yrs14,049.28 310,00012% / 2 yrs12,544.00 410,00012% / 1 yr11,200.00 510,00012% / 0 yrs10,000.00 Total63,528.47

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Annuity Due TimePaymentReturnFV 010,00012% / 5 yrs17623.42 110,00012% / 4 yrs15,735.19 210,00012% / 3 yrs14,049.28 310,00012% / 2 yrs12,544.00 410,00012% / 1 yr11,200.00 5 Total71,151.89

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Future Value of a Combination If you invest $10,000 today and $1,000 at the end of each year in a mutual fund that is expected to earn a 12% compound after-tax return, how much will you have at the end of 5 years?

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Future Value of a Combination

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Future Value

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Combination Investment TimePaymentReturnFV 010,00012% / 5 yrs17,623.42 11,00012% / 4 yrs1,573.52 21,00012% / 3 yrs1,404.93 31,00012% / 2 yrs1,254.40 41,00012% / 1 yr1,120.00 51,00012% / 0 yrs1,000.00 Total23,976.26

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Annual Rate of Return TVM can also solve for the rate of return required for a PV to reach a FV in n years.

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Annual Rate of Return What rate of return is required for $10,000 to grow to $16,000 in 5 years?

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Annual Rate of Return

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If you invest $2,000 at the end of each year for 5 years, what rate of return must your investment earn for you to have $16,000 at the end of that period?

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Annual Rate of Return

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If you invest $10,000 today and $500 at the end of each year for the next 5 years, what rate of return must you earn to have $16,000 at the end of that period?

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Annual Rate of Return

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Number of Periods TVM can also solve for the holding period required for a PV, a series of Payments or a combination of PV and Payments to reach a FV given a specific rate of return

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Number of Periods How long will it take for a $10,000 investment to grow to $24,000 if it earns 11.25% compounded annually?

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Number of Periods

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If you deposit $3,000 at the beginning of each year in a savings account earning 9.75%, how long will it take for you to save for a $20,000 down payment for a house?

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Number of Periods

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Present Value TVM can also solve for the price you would pay for a FV, a series of Payments, or a combination of a series of Payments and a FV given a specific rate of return and holding period.

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Present Value of a Future Amount What would you pay for the right to collect $8,000 in 7 years, if your required return is 8.75%?

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Present Value of a Future Amount

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Stop

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Present Value of Periodic Payments What would you pay for the right to collect $8,000 at the beginning of each year for 7 years, if your required return is 8.75%?

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Present Value of Periodic Payment

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Present Value of a Combination What would you pay for the right to collect $800 at the end of each year for 7 years and an additional $10,000 at the end of the period, if your required return is 7.25%?

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Present Value of a Combination

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Time Value of Money Compounding Periods Shorter than One Year

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Compounding Periods Cash Flows are often more frequent than annually –Monthly, quarterly, semi-annually If Compound periods < annual –Effective Interest Rate is higher –FV is higher and PV is lower

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Compound Interest Formula with Compounding Periods less than 1 Year Where m = the number of compounding periods within the year.

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Adjustments for Compounding Periods < Annual Compounding Periods = m Divide Annual rate by m i/m Multiply Years by m n x m Input i/m for I/Y Input (n x m) for N

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Future Value of Lump Sum If you invest $6,000 in a savings account earning 10% compounded quarterly, how much will you have after 5 years?

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Future Value of Lump Sum

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If you invest $1,000 in a savings account earning 10% compounded daily, how much will you have after 5 years?

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Future Value of Lump Sum

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Future Value of an Annuity If you invest $1,000 at the end of each month in a mutual fund that is expected to earn a 12% after- tax return, how much will you have at the end of 5 years?

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Future Value of an Annuity

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If you invest $1,000 at the beginning of each month in a mutual fund that is expected to earn a 12% after-tax return, how much will you have at the end of 5 years?

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Future Value of an Annuity

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Annual Rate of Return If you invest $2,000 at the end of each quarter for 5 years, what rate of return must your investment earn for you to have $60,000 at the end of that period?

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Annual Rate of Return

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If you invest $10,000 today and $500 at the end of each month for the next 5 years, what rate of return must you earn to have $60,000 at the end of that period?

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Annual Rate of Return

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Number of Periods If you deposit $300 at the beginning of each month in a savings account earning 9.75%, how long will it take for you to save for a $20,000 down payment for a house?

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Number of Periods

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Uneven Cash Flows How do you calculate Present Value when your required return is 9.0% and you expect to receive the following cash flows: Year 12,000 Year 23,000 Year 51,000

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Uneven Cash Flows Alternative One – The Hard Way 1.Draw a Time Line 2.Calculate the PV of each cash flow 3.Total the Present Values

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Uneven Cash Flows

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Alternative Two – Use the CF Register 1.Draw Time Line 2.Input Cash Flows into CF Register 3.Go to NPV Register 1.Input Rate of Return 2.Compute NPV

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Uneven Cash Flows Example 1 – Alternative Two 1.Draw Time Line 2.Push CF button 3.Clear CF register 2 nd CLR Work 4.Input Cash Flows

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Cash Flow Register Inputs –CF 0 = Investment, Price, Cost at Time 0 We are solving for PV so CF 0 should be 0 Since CF0 already = 0, –C01 = Cash Flow at the end of Period 1 –F01 = Frequency of C01 The number of times that C01 occurred consecutively

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Uneven Cash Flows Example 1 1.Draw Time Line 2.Clear the CF Register 3.Input Cash Flows a.CF 0 = 0, b.C01 = 2,000; F01 = 1, c.C02 = 3,000; F02 =1, d.C03 = 0; F03 = 2, e.C04 = 1,000; F04 = 1,

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Uneven Cash Flows Example 1 1.Check Inputs 2.Go To NPV Register 3.Input I 9 ENTER, 4.CPT NPV = 5,009.83

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Uneven Cash Flows Example 2 What would you be willing to pay for a real estate investment that has the following expected cash flows: Yr. 1 $500, Yrs. 2-6 $1,000, Yr. 7-10 $1,500, and Yr. 11 $30,000? Assume your required return for this type of investment is 17.0%.

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Uneven Cash Flows Example 2 1.Draw Time Line 2.Input Cash Flows a.CF 0 = 0 b.C01 = 500; F01 = 1 c.C02 = 1,000; F02 = 5 d.C03 = 1,500; F03 = 4 e.C04 = 30,000; F04 = 1

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Uneven Cash Flows Example 2 3.Check your Inputs 4.Go to “NPV” Register 1.Enter I = 17.0; 2.Press “CPT” NPV = ?

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Uneven Cash Flows Example 2 NPV = 10,100.25

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Uneven Cash Flows The CF Register can also be used to find the rate of return associated with uneven cash flows. This cannot be done easily any other way.

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Uneven Cash Flows Inputs –CF Register Steps are the same –Go to IRR Register CPT IRR IRR = the Internal Rate of Return IRR = the rate of return on the investment

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Effective Interest Rate Calculation

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The annual rate of return actually earned when compounding or payment periods are less than 1 year.

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Effective Interest Rate Nominal rate = i –The nominal rate is the rate “named” in the information. –“The credit card rate is for 18.0% compounded monthly.” 18.0% is the nominal rate

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EIR Calculations What is the Effective Interest Rate for a credit card with an 18% nominal interest rate if the card is not paid off each month?

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Effective Interest Rate with Compounding Periods < 1 Year Where m = the number of compounding periods within the year.

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EIR Calculations

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EIR CALCULATIONS Use “I Conv” Register for easy Effective Interest Rate calculations.

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I Conv Register Steps 1.2 nd I Conv 2.Input Nominal Rate, ENTER 3.Arrow Down Twice 4.Input C/Y (Compounding Periods per Year) 5.Arrow Up 6.CPT EFF (Effective Interest Rate)

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Chapter 4 The Time Value of Money

Chapter 4 The Time Value of Money

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