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Time Value of Money Money Value of Time???

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Interest Rates Why interest rates are positive? Why interest rates are positive? –People have positive time preference Behavior of human beings Behavior of human beings –Current resources have productive uses Technology and natural process Technology and natural process

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Simple vs. Compound Interest Simple Interest Simple Interest –No interest is earned on interest money paid in the previous periods –Money grows at a slower rate Compound Interest Compound Interest –Interest is earned on interest money paid in the previous periods –Money grows at a faster rate

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Simple Interest Example $100 at 8% simple annual interest for 2 years $100 at 8% simple annual interest for 2 years –First year interest 100 x (.08) = $8Total = = $___ 100 x (.08) = $8Total = = $___ –Second year interest 100 x (.08) = $8Total = = $___ 100 x (.08) = $8Total = = $___ –Total Interest after 2 years: = $__

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Another example You deposit $5000 into a savings account that earns 13% simple annual interest. What is the amount in the account after 6 years? You deposit $5000 into a savings account that earns 13% simple annual interest. What is the amount in the account after 6 years?Answer:_________ What is the total amount of interest earned? What is the total amount of interest earned?Answer:_________

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Compound Interest Example Invest $100 at 8% compounded annually for 2 years Invest $100 at 8% compounded annually for 2 years –Total after first year: 100 x (1 +.08) = $ x (1 +.08) = $108 –Total after second year 108 x (1 +.08) = $_____ 108 x (1 +.08) = $_____ –Total Interest = = $______

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Compound Interest Example YearBegin. Amount Interest Earned Ending Amount 1 $100.00$10.00$ $100.00$10.00$ Total interest $61.05 Total interest $61.05 [What would be the total interest earned in simple interest case? Ans: $_______ ]

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Notice that Notice that –1.$110 = $100 (1 +.10) –2.$121 = $110 (1 +.10) = $100 * 1.1 * 1.1 = $100 * –3.$ = $121 (1 +.10) = $100 * 1.1 * 1.1 * 1.1 =$100 ________ =$100 ________ In general, the future value, FV t, of $1 invested today at r% for t periods is In general, the future value, FV t, of $1 invested today at r% for t periods is FV t = $1 * (1 + r) t The expression (1 + r) t is called the future value factor. The expression (1 + r) t is called the future value factor. Future Value for a Lump Sum

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FV on Calculator What is the FV of $5000 invested at 12% per year for 4 years compounded annually? What is the FV of $5000 invested at 12% per year for 4 years compounded annually? Clear all memory: CLEAR ALL Clear all memory: CLEAR ALL Ensure # compounding periods is 1: 1 Ensure # compounding periods is 1: 1 Enter amount invested today: Enter amount invested today: Enter # of years: 4 Enter # of years: 4 Enter interest rate: 12 Enter interest rate: 12 Find Future Value: Find Future Value: Answer: $___________ Answer: $___________ PV N I/YR FV P/YR

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Notice.. You entered $5000 as a negative amount You entered $5000 as a negative amount You got FV answer as a positive amount You got FV answer as a positive amount Why the negative sign? Why the negative sign? It turns out that the calculator follows cash flow convention It turns out that the calculator follows cash flow convention –Cash outflow is negative (i.e. money going out) –Cash inflow is positive (i.e. money coming in)

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Another example Calculate the future value of $500 invested today at 9% per year for 35 years Calculate the future value of $500 invested today at 9% per year for 35 years Answer: ________ Answer: ________

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Present Values Here you simply reverse the question Here you simply reverse the question You are given You are given –Future Value –Number of Periods –Interest Rate and need to find the sum (PRESENT VALUE) needed today to achieve that FV and need to find the sum (PRESENT VALUE) needed today to achieve that FV

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Q. Suppose you need $20,000 in three years to pay tuition at SU. If you can earn 8% on your money, how much do you need today? Q. Suppose you need $20,000 in three years to pay tuition at SU. If you can earn 8% on your money, how much do you need today? A.Here we know the future value is $20,000, the rate (8%), and the number of periods (3). What is the unknown present amount (called the present value)? A.Here we know the future value is $20,000, the rate (8%), and the number of periods (3). What is the unknown present amount (called the present value)? From before: From before: FV t = PV x (1 + r) t FV t = PV x (1 + r) t $20,000= PV __________ $20,000= PV __________ Rearranging: Rearranging: PV= $20,000/(1.08) 3 PV= $20,000/(1.08) 3 = $_____________ = $_____________ Present Value for a Lump Sum

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In general, the present value, PV, of a $1 to be received in t periods when the rate is r is PV = FV t PV = FV t (1+r) t Present Value Factor = 1 (1+r) t r is also called the discount rate

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PV on Calculator Your friend promises to pay you $5,000 after 3 years. How much are you willing to pay her today? You can earn 8% compounded annually elsewhere. Your friend promises to pay you $5,000 after 3 years. How much are you willing to pay her today? You can earn 8% compounded annually elsewhere. Clear all memory: CLEAR ALL Clear all memory: CLEAR ALL Ensure # compounding periods is 1: 1 Ensure # compounding periods is 1: 1 Enter amount future value: 5000 Enter amount future value: 5000 Enter # of years: 3 Enter # of years: 3 Enter interest rate: 8 Enter interest rate: 8 Find Present Value: Find Present Value: Answer: $___________ Answer: $___________ FV N I/YR PV P/YR

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Another PV example Vincent van Gogh painted Portrait of Dr. Gachet in It sold in 1987 for $82.5 million. How much should he have sold it in 1889 if annual interest rate over the period was 9%? Vincent van Gogh painted Portrait of Dr. Gachet in It sold in 1987 for $82.5 million. How much should he have sold it in 1889 if annual interest rate over the period was 9%? Answer: _____________ Answer: _____________

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Vincent Van Gogh The Portrait of Dr Gachet

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Present Value of $1 for Different Periods and Rates Present value of $1 ($) Time (years) r = 0% r = 15% r = 10% r = 20% r = 5%

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Notice... As time increases, present value declines As time increases, present value declines As interest rate increases, present value declines As interest rate increases, present value declines The rate of decline is not a straight line! The rate of decline is not a straight line!

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Notice Four Components Present Value (PV) Present Value (PV) Future Value at time t (FV t ) Future Value at time t (FV t ) Interest rate per period (r) Interest rate per period (r) Number of periods (t) Number of periods (t) Given any three, the fourth can be found Given any three, the fourth can be found

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Finding r You need $8,000 after four years. You have $7,000 today. What annual interest rate must you earn to have that sum in the future? Answer: __________ You need $8,000 after four years. You have $7,000 today. What annual interest rate must you earn to have that sum in the future? Answer: __________

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Finding t How many years does it take to double your $100,000 inheritance if you can invest the money earning 11% compounded annually? Answer: __________ How many years does it take to double your $100,000 inheritance if you can invest the money earning 11% compounded annually? Answer: __________

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Note: When calculating future value what you are doing is compounding a sum When calculating future value what you are doing is compounding a sum When calculating present value, what you are doing is discounting a sum When calculating present value, what you are doing is discounting a sum

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FV - Multiple Cash Flows You deposit $100 in one year $200 in two years $300 in three years How much will you have in three years? r = 7% per year. You deposit $100 in one year $200 in two years $300 in three years How much will you have in three years? r = 7% per year. Answer: ____________ Answer: ____________ Draw a time line!!! Draw a time line!!!

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PV - Multiple Cash Flows An investment pays: $200 in year 1$600 in year 3 $400 in year 2 $800 in year 4 You can earn 12% per year on similar investments. What is the most you are willing to pay now for this investment? An investment pays: $200 in year 1$600 in year 3 $400 in year 2 $800 in year 4 You can earn 12% per year on similar investments. What is the most you are willing to pay now for this investment? Answer: __________ Answer: __________ Draw time line!!! Draw time line!!!

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Important… You can add cash flows ONLY if they are brought back (or taken forward) to the SAME point in time You can add cash flows ONLY if they are brought back (or taken forward) to the SAME point in time Adding cash flows occurring at different points in time is like adding apples and oranges! Adding cash flows occurring at different points in time is like adding apples and oranges!

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Level Multiple Cash Flows Examples of constant level cash flows for more than one period Examples of constant level cash flows for more than one period –Annuities –Perpetuities Most of the time we assume that the cash flow occurs at the END of the period Most of the time we assume that the cash flow occurs at the END of the period

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Examples of Annuities Car loan payments Car loan payments Mortgage on a house Mortgage on a house Most other consumer loans Most other consumer loans Contributions to a retirement plan Contributions to a retirement plan Retirement payments from a pension plan Retirement payments from a pension plan

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Saving a Fixed Sum You save $450 in a retirement fund every month for the next 30 years. The interest rate earned is 10%. What is the accumulated balance at the end of 30 years? You save $450 in a retirement fund every month for the next 30 years. The interest rate earned is 10%. What is the accumulated balance at the end of 30 years? This is Future Value of an Annuity This is Future Value of an Annuity

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Future Value Calculated Future value calculated by compounding each cash flow separately Time (years) $2,000 $2, , , , $12, x x x x 1.1 Total future value Save $2,000 every year for 5 years into an account that pays 10%. What is the accumulated balance after 5 years?

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FV of Annuity

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Important to understand inputs r is the interest rate per period r is the interest rate per period t is the # of periods. t is the # of periods. For example, For example, –if t is # of years, r is annual rate –if t is # of months, r is the monthly rate

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FV of Annuity Example You will contribute $5,000 per year for the next 35 years into a retirement savings plan. If your money earns 10% interest per year, how much will you have accumulated at retirement? You will contribute $5,000 per year for the next 35 years into a retirement savings plan. If your money earns 10% interest per year, how much will you have accumulated at retirement? Draw a time line!!! Draw a time line!!!

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Time Line Notice: Payment begins at the end of first year Notice: Payment begins at the end of first year

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FV of Annuity on Calculator Clear all memory: CLEAR ALL Clear all memory: CLEAR ALL Ensure # compounding periods is 1: 1 Ensure # compounding periods is 1: 1 Enter payments: Enter payments: Enter # of payments: 35 Enter # of payments: 35 Enter interest rate: 10 Enter interest rate: 10 Find Future Value: Find Future Value: Answer: $___________ Answer: $___________ N I/YR FV P/YR PMT

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FV Annuity - A Twist.. You estimate you will need $1 million to live comfortably in retirement in 30 years. How much must you save monthly if your money earns 12% interest per year? You estimate you will need $1 million to live comfortably in retirement in 30 years. How much must you save monthly if your money earns 12% interest per year? Note: Payments are monthly, interest quoted is annual!!! Note: Payments are monthly, interest quoted is annual!!!

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Two ways to adjust for compounding periods Divide annual interest rate by 12 and enter interest rate per month into calculator as the interest rate and leave P/YR as 1 Divide annual interest rate by 12 and enter interest rate per month into calculator as the interest rate and leave P/YR as 1 Set P/YR on calculator as 12: 12 and enter the annual interest rate Set P/YR on calculator as 12: 12 and enter the annual interest rate OR P/YR

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N on calculator You can either: Enter # of periods directly (360 in the example) Enter # of periods directly (360 in the example) If you have set 12 as the P/YR then you can also enter it as 30 If you have set 12 as the P/YR then you can also enter it as 30 –(notice it appears as 360) N OR

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FV Annuity on Calculator (2) Clear all memory: CLEAR ALL Clear all memory: CLEAR ALL Monthly-> # compounding periods is 12: 12 Monthly-> # compounding periods is 12: 12 Enter Future Value: 1,000,000 Enter Future Value: 1,000,000 Enter # of payments: 30 Enter # of payments: 30 Enter interest rate: 12 Enter interest rate: 12 Find payments: Find payments: Answer: $___________ Answer: $___________ N I/YR PMT P/YR FV Note the difference!

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Present Value of Annuities Here we bring multiple, level cash flows back to the present (year 0) Here we bring multiple, level cash flows back to the present (year 0) Typical examples are consumer loans where the loan amount is the PV and the fixed payments are the cash flows Typical examples are consumer loans where the loan amount is the PV and the fixed payments are the cash flows

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PV of Annuity Example Cash flow per period (CFt) = $500 Cash flow per period (CFt) = $500 Number of periods (t) = 4 years Number of periods (t) = 4 years Interest Rate (r) = 9% per year Interest Rate (r) = 9% per year What is the present value (PV) = ? What is the present value (PV) = ? ALWAYS DRAW A TIME LINE!!! ALWAYS DRAW A TIME LINE!!!

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PV of Annuity on Calculator Clear all memory: CLEAR ALL Clear all memory: CLEAR ALL Ensure # compounding periods is 1: 1 Ensure # compounding periods is 1: 1 Enter payments: 500 Enter payments: 500 Enter # of payments: 4 Enter # of payments: 4 Enter interest rate: 9 Enter interest rate: 9 Find Present Value: Find Present Value: Answer: $___________ Answer: $___________ N I/YR PV P/YR PMT

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PV of Annuity Again: r and t must match – i.e. if t is # of months, r must be monthly rate Again: r and t must match – i.e. if t is # of months, r must be monthly rate

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Car Loan Example Car costs $ 20,000 Car costs $ 20,000 Interest rate per month = 1% Interest rate per month = 1% 5-year loan ---> number of months = t = 60 5-year loan ---> number of months = t = 60 What is the monthly payment? What is the monthly payment? Answer: ___________ Answer: ___________

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Mortgage payments House cost $250,000 House cost $250,000 Mortgage Rate = 7.5% annually Mortgage Rate = 7.5% annually Term of loan = 30 years Term of loan = 30 years Payments made monthly Payments made monthly What are your payments? What are your payments? Answer: _____________ Answer: _____________

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To Reiterate... Be VERY careful about compounding periods Be VERY careful about compounding periods Problem can state annual interest rate, but the cash flows can be monthly, quarterly… Problem can state annual interest rate, but the cash flows can be monthly, quarterly… The convention is to state interest rate annually (Annual Percentage Rate) The convention is to state interest rate annually (Annual Percentage Rate)

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Perpetuity Annuity forever Annuity forever Examples: Preferred Stock, Consols Examples: Preferred Stock, Consols

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Perpetuity Note: C and r measured over same interval Note: C and r measured over same interval

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Perpetuity Example Preferred stock pays $1.00 dividend per quarter. The required return, r, is 2.5% per quarter. Preferred stock pays $1.00 dividend per quarter. The required return, r, is 2.5% per quarter. What is the stock value? What is the stock value?

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Perpetuity Example Steve Forbess flat-tax proposal was expected to save him $500,000 a year forever if passed. He spent $40,000,000 of his own money for campaign Steve Forbess flat-tax proposal was expected to save him $500,000 a year forever if passed. He spent $40,000,000 of his own money for campaign Charge: He was running for presidency for personal gain Charge: He was running for presidency for personal gain Did the charge make sense Did the charge make sense

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Forbes continued... What should be r in the example? What should be r in the example? At what r would Forbes have gained from being a president and steamrolling flat-tax proposal? At what r would Forbes have gained from being a president and steamrolling flat-tax proposal?

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Compounding Periods Interest can be compounded Interest can be compounded –Annually - Semiannually –Monthly - Daily - Continuously Smaller the compounding period, faster is the growth of money Smaller the compounding period, faster is the growth of money The same PV or FV formula can be used: BUT UNDERSTAND THE INPUTS!! The same PV or FV formula can be used: BUT UNDERSTAND THE INPUTS!!

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Compounding example Invest $5,000 in a 5-year CD Invest $5,000 in a 5-year CD Quoted Annual Percentage Rate (APR) = 15% Quoted Annual Percentage Rate (APR) = 15% Calculate FV 5 for annual, semi-annual, monthly and daily compounding Calculate FV 5 for annual, semi-annual, monthly and daily compounding Key: Adjust P/YR on calculator Key: Adjust P/YR on calculator

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Answers: Annual:$10, Annual:$10, Semi-annual$10, Semi-annual$10, Monthly:$10, Monthly:$10, Daily:$10, Daily:$10, Continuous Compounding??? Continuous Compounding???

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Continuous compouding Compounded every instant microsecond Compounded every instant microsecond r = interest rate per period r = interest rate per period t = number of periods t = number of periods Previous example answer: $ 10, Previous example answer: $ 10,585.00

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Continuous compounding example Invest $4,500 in an account paying 9.5% compounded continuously Invest $4,500 in an account paying 9.5% compounded continuously What is the balance after 4 years? Answer: _________ What is the balance after 4 years? Answer: _________

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Quoted vs. Effective Interest Rates Quoted Rate: Usually stated annually along with compounding period (APR) Quoted Rate: Usually stated annually along with compounding period (APR) –e.g. 10% compounded quarterly Effective Annual Rate (EAR): Interest rate actually earned IF the compounding period were one year Effective Annual Rate (EAR): Interest rate actually earned IF the compounding period were one year

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EAR m = number of compounding periods in a year

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EAR on Calculator What is the EAR for quoted rate of 15% per year compounded quarterly? What is the EAR for quoted rate of 15% per year compounded quarterly? Set number of periods per year: 4 Set number of periods per year: 4 Enter quoted annual rate: 15 Enter quoted annual rate: 15 Compute EAR: Compute EAR: Answer: _______ Answer: _______ P/YR I/YR EFF%

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EAR Example Compute EAR for 12% compounded Compute EAR for 12% compounded –Annually –Quarterly –Monthly –Daily Answers: ____, ____, ____, ____ Answers: ____, ____, ____, ____

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EAR for Continuous compounding Example: Quoted rate is 10% compounded continuously Example: Quoted rate is 10% compounded continuously EAR = _____% EAR = _____%

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Complicatons to TVM When payments begin beyond year 1 When payments begin beyond year 1 PV and FV combined PV and FV combined When payments begin in year 0 (Annuities Due) When payments begin in year 0 (Annuities Due)

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Payments beyond year 1 A car dealer offers no payments for next 12 months deal on a $15,000 car. After that, you will pay monthly payments for the next 4 years. r = 10% APR. What are your monthly payments? A car dealer offers no payments for next 12 months deal on a $15,000 car. After that, you will pay monthly payments for the next 4 years. r = 10% APR. What are your monthly payments? Answer: ___________ Answer: ___________

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PV and FV combined How much must you invest per year to have an amount in 20 years that will provide an annual income of $12,000 per year for 5 years? r = 8% annually. How much must you invest per year to have an amount in 20 years that will provide an annual income of $12,000 per year for 5 years? r = 8% annually. Answer: ___________ Answer: ___________

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PV and FV combined 2 You have 2 options: You have 2 options: –Receive $100 for next 10 years only –Receive $100 forever beginning in year 11 If r = 10% which one would you prefer? If r = 10% which one would you prefer? At what interest rate are you indifferent between the two options? At what interest rate are you indifferent between the two options?

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Annuities Due Payments begin in year 0 Payments begin in year 0 –Ex. Rent/Lease Payments Trick: Trick: Adjust BEG/END on calculator to BEG Adjust BEG/END on calculator to BEG Leave to END, but multiply (1+r) for both PV and FV Leave to END, but multiply (1+r) for both PV and FV OR

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Annuity Due Example Find PV of a 4-year (5 payment), $400 annuity due. r = 10% Find PV of a 4-year (5 payment), $400 annuity due. r = 10% Find FV in year 5 of the above annuity due Find FV in year 5 of the above annuity due Answers: Answers: –PV = $1, –FV 5 = $2, Time (years) $400 FV

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Another Example You start to contribute $500 every month to your IRA account beginning immediately. How much will you accumulate at the end of first year? The return on your investment is 20% per year. You start to contribute $500 every month to your IRA account beginning immediately. How much will you accumulate at the end of first year? The return on your investment is 20% per year. Note: Return here is just another term for the interest rate Note: Return here is just another term for the interest rate Answer: $_______ Answer: $_______

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Tricky but Legal... Add-on Interest Called add-on interest because interest is added on to the principal before the payments are calculated Add-on Interest Called add-on interest because interest is added on to the principal before the payments are calculated Points on a Loan: Percentage of loan amount reduced up front Points on a Loan: Percentage of loan amount reduced up front –Used in home mortgages

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Example: Add-on Interest You are offered the opportunity to borrow $1,000 for 3 years at 12% add-on interest. The lender calculates the payment as follows: Amt. owed in 3 years: $1000 x (1+.12) 3 = 1,405 Monthly Payment = $1,405 / 36 = $39 You are offered the opportunity to borrow $1,000 for 3 years at 12% add-on interest. The lender calculates the payment as follows: Amt. owed in 3 years: $1000 x (1+.12) 3 = 1,405 Monthly Payment = $1,405 / 36 = $39 What is the effective annual rate (EAR)? What is the effective annual rate (EAR)? Steps: Steps: –Calculate the APR interest (I/YR) –Use answer to calculate the EAR

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Add-on Example (2) Calcuate the EAR on a 6-year, $7,000 loan at 13% add-on interest. The payments are monthly. Calcuate the EAR on a 6-year, $7,000 loan at 13% add-on interest. The payments are monthly. Answer: ________ Answer: ________

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Example: Points on a Loan 1-year loan of $100. r = 10% + 2 points [Note: 1 point = 1% of loan amount. Hence you pay upfront $2 to lender. Hence you are actually getting only $98, not $100] 1-year loan of $100. r = 10% + 2 points [Note: 1 point = 1% of loan amount. Hence you pay upfront $2 to lender. Hence you are actually getting only $98, not $100] What is the EAR? What is the EAR? $110 = $98 (1+r) r = 12.24% $110 = $98 (1+r) r = 12.24%

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Points on a loan (2) Calculate the EAR on a 10-year, $110,000 mortgage when interest rate quoted is 7.75% + 1 point. The payments are monthly Calculate the EAR on a 10-year, $110,000 mortgage when interest rate quoted is 7.75% + 1 point. The payments are monthly Answer: _________ Answer: _________

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Balloon Payments Balloon Payments Amount on the loan outstanding after a certain number of payments have been made Amount on the loan outstanding after a certain number of payments have been made –Sometimes called residual on a loan e.g. when you want to pay off a loan early e.g. when you want to pay off a loan early

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Balloon Example You borrowed $90,000 on a house for 30 years 10 years ago. The annual interest rate then was 17%. The payments are monthly. Since interest rate has fallen, you want to payoff the remaining amount on the loan and refinance it. What is the outstanding amount to be paid off? (Note: Payments are $1,283.11) You borrowed $90,000 on a house for 30 years 10 years ago. The annual interest rate then was 17%. The payments are monthly. Since interest rate has fallen, you want to payoff the remaining amount on the loan and refinance it. What is the outstanding amount to be paid off? (Note: Payments are $1,283.11) Answer: $__________ Answer: $__________

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Two ways to calculate Balloons First calculate payments First calculate payments Take the present value of the remaining (unpaid) payments Take the present value of the remaining (unpaid) payments Use amortization function on calculator Use amortization function on calculator Enter the period : period Enter the period : period Enter, and then Enter, and then INPUT AMORT === OR

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Another Example.. What is the outstanding balance on a 5 year $19,000 car loan at 11% interest after 2-1/2 years have passed? The payments are monthly. What is the outstanding balance on a 5 year $19,000 car loan at 11% interest after 2-1/2 years have passed? The payments are monthly. Answer: $____________ Answer: $____________

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TVM TIPS Draw time line! Draw time line! Check & set BEG/END on calculator Check & set BEG/END on calculator Check & set P/YR on calculator Check & set P/YR on calculator Check & set # of decimal places to 4 Check & set # of decimal places to 4

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TVM Tips Continued... Clear all previously stored #s in memory Clear all previously stored #s in memory –Especially true when same problem requires multiple TVM calculations Make sure that for FV and PV calculation, you have correctly signed (+/-) the cash flows Make sure that for FV and PV calculation, you have correctly signed (+/-) the cash flows

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