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TVM Preferences
• More vs. Less
• Sooner vs. Later
• More Now vs. Less Later
• Less Now vs. More Later ????
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?
Compare and Contrast
1970 2011
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%
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
Compound Interest Calculation
n
1-n
1 Periodsi
n
1Periods
i) (1 *PV FV
Principal
Rate Interest * Interest
Rate Interest * Principal ValueFuture
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
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
Lump Sum vs. Periodic Pmts• Lump Sum
–Single Payment–At time zero–Present ValueOR–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
Annuities
• Must be–Equal Amounts–Occurring in every compounding
period–Ordinary Annuity – End of Period–Annuity Due – Beginning of Period
Lump Sum & Periodic Payment
• Combination–Single Payment–With periodic payments for life of
investment–PV & PMT
Recommended StructureFuture ValuePresent ValuePaymentAnnual RateYearsBeg / EndCompounding PeriodsRate / PeriodYears * Periods
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?
Future Value of Lump Sum
Future Value ?Present Value (1,000.00) PaymentAnnual Rate 10.00%Years 5Beg / EndCompounding PeriodsRate / PeriodYears * Periods
Future Value of Lump Sum
Future Value 1,610.51 Present Value (1,000.00) PaymentAnnual Rate 10.00%Years 5Beg / EndCompounding PeriodsRate / PeriodYears * Periods
Future Value of Lump Sum
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?
Future ValuePresent ValuePaymentAnnual RateYearsBeg / EndCompounding PeriodsRate / PeriodYears * Periods
Future Value of Lump Sum
Future Value ?Present Value (10,000.00) PaymentAnnual Rate 12.00%Years 50Beg / EndCompounding PeriodsRate / PeriodYears * Periods
Future Value
Future Value 2,890,021.90 Present Value (10,000.00) PaymentAnnual Rate 12.00%Years 50Beg / EndCompounding PeriodsRate / PeriodYears * Periods
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?
Future ValuePresent ValuePaymentAnnual RateYearsBeg / EndCompounding PeriodsRate / PeriodYears * Periods
Future Value of an Annuity
Future Value ?Present ValuePayment (10,000.00) Annual Rate 12.00%Years 5Beg / End EndCompounding PeriodsRate / PeriodYears * Periods
Future Value of an Annuity
Future Value 63,528.47 Present ValuePayment (10,000.00) Annual Rate 12.00%Years 5Beg / End EndCompounding PeriodsRate / PeriodYears * Periods
Future Value of an Annuity
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?
Future ValuePresent ValuePaymentAnnual RateYearsBeg / EndCompounding PeriodsRate / PeriodYears * Periods
Future Value of an Annuity
Future Value ?Present ValuePayment (10,000.00) Annual Rate 12.00%Years 5Beg / End BegCompounding PeriodsRate / PeriodYears * Periods
Future Value of an Annuity
Future Value 71,151.89 Present ValuePayment (10,000.00) Annual Rate 12.00%Years 5Beg / End BegCompounding PeriodsRate / PeriodYears * Periods
Ordinary Annuity
Time Payment Return FV
0
1 10,000 12% / 4 yrs 15,735.19
2 10,000 12% / 3 yrs 14,049.28
3 10,000 12% / 2 yrs 12,544.00
4 10,000 12% / 1 yr 11,200.00
5 10,000 12% / 0 yrs 10,000.00
Total 63,528.47
Annuity Due
Time Payment Return FV
0 10,000 12% / 5 yrs 17623.42
1 10,000 12% / 4 yrs 15,735.19
2 10,000 12% / 3 yrs 14,049.28
3 10,000 12% / 2 yrs 12,544.00
4 10,000 12% / 1 yr 11,200.00
5
Total 71,151.89
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?
Future Value of a Combination
Future ValuePresent ValuePaymentAnnual RateYearsBeg / EndCompounding PeriodsRate / PeriodYears * Periods
Future Value
Future Value ?Present Value (10,000.00) Payment (1,000.00) Annual Rate 12.00%Years 5Beg / End EndCompounding PeriodsRate / PeriodYears * Periods
Future Value
Future Value 23,976.26 Present Value (10,000.00) Payment (1,000.00) Annual Rate 12.00%Years 5Beg / End EndCompounding PeriodsRate / PeriodYears * Periods
Combination Investment
Time Payment Return FV
0 10,000 12% / 5 yrs 17,623.42
1 1,000 12% / 4 yrs 1,573.52
2 1,000 12% / 3 yrs 1,404.93
3 1,000 12% / 2 yrs 1,254.40
4 1,000 12% / 1 yr 1,120.00
5 1,000 12% / 0 yrs 1,000.00
Total 23,976.26
Annual Rate of Return
TVM can also solve for the rate of return required for a PV to reach a FV in n years.
Annual Rate of Return
Future ValuePresent ValuePaymentAnnual RateYearsBeg / EndCompounding PeriodsRate / PeriodYears * Periods
Annual Rate of Return
Future Value 16,000.00 Present Value (10,000.00) PaymentAnnual Rate ?Years 5Beg / EndCompounding PeriodsRate / Period
Annual Rate of ReturnFuture Value 16,000.00 Present Value (10,000.00) PaymentAnnual Rate 9.86%Years 5Beg / EndCompounding PeriodsRate / Period
Annual Rate of Return
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?
Future ValuePresent ValuePaymentAnnual RateYearsBeg / EndCompounding PeriodsRate / PeriodYears * Periods
Annual Rate of Return
Future Value 16,000.00 Present ValuePayment (2,000.00) Annual Rate ?Years 5Beg / End EndCompounding PeriodsRate / PeriodYears * Periods
Annual Rate of Return
Future Value 16,000.00 Present ValuePayment (2,000.00) Annual Rate 23.69%Years 5Beg / End EndCompounding PeriodsRate / PeriodYears * Periods
Annual Rate of Return
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?
Future ValuePresent ValuePaymentAnnual RateYearsBeg / EndCompounding PeriodsRate / PeriodYears * Periods
Annual Rate of ReturnFuture Value 16,000.00 Present Value (10,000.00) Payment (500.00) Annual Rate ?Years 5Beg / End EndCompounding PeriodsRate / PeriodYears * Periods
Annual Rate of ReturnFuture Value 16,000.00 Present Value (10,000.00) Payment (500.00) Annual Rate 5.71%Years 5Beg / End EndCompounding PeriodsRate / PeriodYears * Periods
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
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?
Future ValuePresent ValuePaymentAnnual RateYearsBeg / EndCompounding PeriodsRate / PeriodYears * Periods
Number of Periods
Future Value 24,000.00 Present Value (10,000.00) PaymentAnnual Rate 11.25%Years ?Beg / EndCompounding PeriodsRate / PeriodYears * Periods
Number of Periods
Future Value 24,000.00 Present Value (10,000.00) PaymentAnnual RateYears 8.21Beg / EndCompounding PeriodsRate / PeriodYears * Periods
Number of Periods
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?
Future ValuePresent ValuePaymentAnnual RateYearsBeg / EndCompounding PeriodsRate / PeriodYears * Periods
Number of PeriodsFuture Value 20,000.00 Present ValuePayment (3,000.00) Annual Rate 9.75%Years ?Beg / End BegCompounding PeriodsRate / PeriodYears * Periods
Number of Periods
Future Value 20,000.00 Present ValuePayment (3,000.00) Annual Rate 9.75%Years 5Beg / End BegCompounding PeriodsRate / PeriodYears * Periods
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.
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%?
Future ValuePresent ValuePaymentAnnual RateYearsBeg / EndCompounding PeriodsRate / PeriodYears * Periods
Present Value of a Future Amount
Future Value 8,000.00 Present Value ?PaymentAnnual Rate 8.75%Years 7Beg / EndCompounding PeriodsRate / PeriodYears * Periods
Present Value of a Future Amount
Future Value 8,000.00 Present Value (4,447.18) PaymentAnnual Rate 8.75%Years 7Beg / EndCompounding PeriodsRate / PeriodYears * Periods
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%?
Future ValuePresent ValuePaymentAnnual RateYearsBeg / EndCompounding PeriodsRate / PeriodYears * Periods
Present Value of Periodic Payment
Future ValuePresent Value ?Payment 8,000.00 Annual Rate 8.75%Years 7Beg / End BegCompounding PeriodsRate / PeriodYears * Periods
Present Value of Periodic Payment
Future ValuePresent Value (44,156.42) Payment 8,000.00 Annual Rate 8.75%Years 7Beg / End BegCompounding PeriodsRate / PeriodYears * Periods
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%?
Future ValuePresent ValuePaymentAnnual RateYearsBeg / EndCompounding PeriodsRate / PeriodYears * Periods
Present Value of a Combination
Future Value 10,000.00 Present Value ?Payment 800.00 Annual Rate 7.25%Years 7Beg / End EndCompounding PeriodsRate / PeriodYears * Periods
Present Value of a Combination
Future Value 10,000.00 Present Value (10,400.70) Payment 800.00 Annual Rate 7.25%Years 7Beg / End EndCompounding PeriodsRate / PeriodYears * Periods
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
Compound Interest Formula with Compounding Periods less
than 1 Year
mn*
m
i1*PVFV
Where m = the number of compounding periods within the year.
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
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?
Future ValuePresent ValuePaymentAnnual RateYearsBeg / EndCompounding PeriodsRate / PeriodYears * Periods
Future Value of Lump Sum
Future Value ?Present Value (6,000.00) PaymentAnnual Rate 10.00%Years 5Beg / EndCompounding Periods 4Rate / Period 2.50%Years * Periods 20
Future Value of Lump Sum
Future Value $9,831.70Present Value (6,000.00) PaymentAnnual Rate 10.00%Years 5Beg / EndCompounding Periods 4Rate / Period 2.50%Years * Periods 20
Future Value of Lump Sum
If you invest $1,000 in a savings account earning 10% compounded daily, how much will you have after 5 years?
Future ValuePresent ValuePaymentAnnual RateYearsBeg / EndCompounding PeriodsRate / PeriodYears * Periods
Future Value of Lump Sum
Future Value ?Present Value (1,000.00) PaymentAnnual Rate 10.00%Years 5Beg / EndCompounding Periods 365Rate / Period 0.0274%Years * Periods 1,825
Future Value of Lump Sum
Future Value $1,648.61Present Value (1,000.00) PaymentAnnual Rate 10.00%Years 5Beg / EndCompounding Periods 365Rate / Period 0.0274%Years * Periods 1,825
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?
Future ValuePresent ValuePaymentAnnual RateYearsBeg / EndCompounding PeriodsRate / PeriodYears * Periods
Future Value of an Annuity
Future Value ?Present ValuePayment (1,000.00) Annual Rate 12.00%Years 5Beg / End EndCompounding Periods 12Rate / Period 1.00%Years * Periods 60
Future Value of an Annuity
Future Value $81,669.67Present ValuePayment (1,000.00) Annual Rate 12.00%Years 5Beg / End EndCompounding Periods 12Rate / Period 1.00%Years * Periods 60
Future Value of an Annuity
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?
Future ValuePresent ValuePaymentAnnual RateYearsBeg / EndCompounding PeriodsRate / PeriodYears * Periods
Future Value of an Annuity
Future Value ?Present ValuePayment (1,000.00) Annual Rate 12.00%Years 5Beg / End BegCompounding Periods 12Rate / Period 1.00%Years * Periods 60
Future Value of an Annuity
Future Value $82,486.37Present ValuePayment (1,000.00) Annual Rate 12.00%Years 5Beg / End BegCompounding Periods 12Rate / Period 1.00%Years * Periods 60
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?
Future ValuePresent ValuePaymentAnnual RateYearsBeg / EndCompounding PeriodsRate / PeriodYears * Periods
Annual Rate of Return
Future Value 60,000.00 Present ValuePayment (2,000.00) Annual Rate ?Years 5Beg / End EndCompounding Periods 4Rate / Period ?Years * Periods 20
Annual Rate of Return
Future Value 60,000.00 Present ValuePayment (2,000.00) Annual Rate ?Years 5Beg / End EndCompounding Periods 4Rate / Period 4.07%Years * Periods 20
Annual Rate of Return
Future Value 60,000.00 Present ValuePayment (2,000.00) Annual Rate 16.29%Years 5Beg / End EndCompounding Periods 4Rate / Period 4.07%Years * Periods 20
Annual Rate of Return
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?
Future ValuePresent ValuePaymentAnnual RateYearsBeg / EndCompounding PeriodsRate / PeriodYears * Periods
Annual Rate of ReturnFuture Value 60,000.00 Present Value (10,000.00) Payment (500.00) Annual Rate ?Years 5Beg / End EndCompounding Periods 12Rate / PeriodYears * Periods 60
Annual Rate of ReturnFuture Value 60,000.00 Present Value (10,000.00) Payment (500.00) Annual Rate ?Years 5Beg / End EndCompounding Periods 12Rate / Period 1.04%Years * Periods 60
Annual Rate of ReturnFuture Value 60,000.00 Present Value (10,000.00) Payment (500.00) Annual Rate 12.50%Years 5Beg / End EndCompounding Periods 12Rate / Period 1.04%Years * Periods 60
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?
Future ValuePresent ValuePaymentAnnual RateYearsBeg / EndCompounding PeriodsRate / PeriodYears * Periods
Number of PeriodsFuture Value 20,000.00 Present ValuePayment (300.00) Annual Rate 9.75%Years ?Beg / End BegCompounding Periods 12Rate / Period 0.81%Years * Periods ?
Number of PeriodsFuture Value 20,000.00 Present ValuePayment (300.00) Annual Rate 9.75%Years ?Beg / End BegCompounding Periods 12Rate / Period 0.81%Years * Periods 53
Number of PeriodsFuture Value 20,000.00 Present ValuePayment (300.00) Annual Rate 9.75%Years 4.43Beg / End BegCompounding Periods 12Rate / Period 0.81%Years * Periods 53
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 1 2,000 Year 2 3,000 Year 5 1,000
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
Uneven Cash Flows
CF 1 CF 2 CF 3FV 2,000 3,000 1,000I/Y 9.00% 9.00% 9.00%N 1 2 5PV (1,834.86) (2,525.04) (649.93)
Total PV (5,009.83)
Uneven Cash Flows
• Alternative Two – Use the CF Register
1. Draw Time Line
2. Input Cash Flows into CF Register
3. Go to NPV Register1. Input Rate of Return
2. Compute NPV
Uneven Cash Flows Example 1 – Alternative Two
1. Draw Time Line
2. Push CF button
3. Clear CF register
2nd CLR Work
4. Input Cash Flows
Cash Flow Register
• Inputs– CF0 = Investment, Price, Cost at Time 0
We are solving for PV so CF0 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
Uneven Cash Flows Example 1
1. Draw Time Line
2. Clear the CF Register
3. Input Cash Flowsa. CF0 = 0, b. C01 = 2,000; F01 = 1, c. C02 = 3,000; F02 =1, d. C03 = 0; F03 = 2, e. C04 = 1,000; F04 = 1,
Uneven Cash Flows Example 1
1. Check Inputs
2. Go To NPV Register
3. Input I 9 ENTER,
4. CPT NPV = 5,009.83
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%.
Uneven Cash Flows Example 2
1. Draw Time Line
2. Input Cash Flowsa. CF0 = 0
b. C01 = 500; F01 = 1c. C02 = 1,000; F02 = 5d. C03 = 1,500; F03 = 4e. C04 = 30,000; F04 = 1
Uneven Cash Flows Example 2
3. Check your Inputs
4. Go to “NPV” Register
1. Enter I = 17.0;
2. Press “CPT”
NPV = ?
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.
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
Effective Interest Rate Calculation
The annual rate of return actually earned when
compounding or payment periods are less than 1 year.
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
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?
Effective Interest Rate with Compounding Periods < 1 Year
100*1m
i1Rate Int. Eff.
m
Where m = the number of compounding periods within the year.