2Interest Rates Why interest rates are positive? People have ‘positive time preference’Behavior of human beingsCurrent resources have productive usesTechnology and natural process
3Simple vs. Compound Interest Simple InterestNo interest is earned on interest money paid in the previous periodsMoney grows at a slower rateCompound InterestInterest is earned on interest money paid in the previous periodsMoney grows at a faster rate
4Simple Interest Example $100 at 8% simple annual interest for 2 yearsFirst year interest100 x (.08) = $8 Total = = $___Second year interest100 x (.08) = $8 Total = = $___Total Interest after 2 years: = $__
5Another exampleYou 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?
6Compound Interest Example Invest $100 at 8% compounded annually for 2 yearsTotal after first year:100 x ( ) = $108Total after second year108 x ( ) = $_____Total Interest = = $______
7Compound Interest Example Year Begin. Amount Interest Earned Ending Amount1 $ $10.00 $110.00Total interest $61.05[What would be the total interest earned in simple interest case? Ans: $_______ ]
8Future Value for a Lump Sum Notice that1. $ = $ ( )2. $ = $ ( ) = $100 * 1.1 * 1.1 = $100 * 1.123. $ = $ ( ) = $100 * 1.1 * 1.1 * 1.1= $ ________In general, the future value, FVt, of $1 invested today at r% for t periods isFVt = $1 * (1 + r)tThe expression (1 + r)t is called the future value factor.
9FV on CalculatorWhat is the FV of $5000 invested at 12% per year for 4 years compounded annually?Clear all memory: CLEAR ALLEnsure # compounding periods is 1:Enter amount invested today: -5000Enter # of years: 4Enter interest rate: 12Find Future Value:Answer: $___________P/YRPVNI/YRFV
10Notice.. You entered $5000 as a negative amount You got FV answer as a positive amountWhy the negative sign?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)
11Another exampleCalculate the future value of $500 invested today at 9% per year for 35 yearsAnswer: ________
12Present Values Here you simply reverse the question You are given Future ValueNumber of PeriodsInterest Rateand need to find the sum (PRESENT VALUE) needed today to achieve that FV
13Present Value for a Lump Sum 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)?From before:FVt = PV x (1 + r)t$20,000 = PV __________Rearranging:PV = $20,000/(1.08)3= $_____________
14In general, the present value, PV, of a $1 to be received in t periods when the rate is r is PV = FVt(1+r)tPresent Value Factor = (1+r)t‘r’ is also called the discount rate
15PV on CalculatorYour 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 ALLEnsure # compounding periods is 1:Enter amount future value:Enter # of years: 3Enter interest rate: 8Find Present Value:Answer: $___________P/YRFVNI/YRPV
16Another PV exampleVincent 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: _____________
18Present Value of $1 for Different Periods and Rates 1.00.90.80.126.96.36.199.30.20.10r = 0%Present value of $1 ($)r = 5%r = 10%r = 15%r = 20%Time (years)
19Notice... As time increases, present value declines As interest rate increases, present value declinesThe rate of decline is not a straight line!
20Notice Four Components Present Value (PV)Future Value at time t (FVt)Interest rate per period (r)Number of periods (t)Given any three, the fourth can be found
21Finding ‘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: __________
22Finding ‘t’How many years does it take to double your $100,000 inheritance if you can invest the money earning 11% compounded annually? Answer: __________
23Note:When calculating future value what you are doing is compounding a sumWhen calculating present value, what you are doing is discounting a sum
24FV - 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.Answer: ____________Draw a time line!!!
25PV - Multiple Cash Flows An investment pays: $200 in year 1 $600 in year $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: __________Draw time line!!!
26Important…You can add cash flows ONLY if they are brought back (or taken forward) to the SAME point in timeAdding cash flows occurring at different points in time is like adding apples and oranges!
27Level Multiple Cash Flows Examples of constant level cash flows for more than one periodAnnuitiesPerpetuitiesMost of the time we assume that the cash flow occurs at the END of the period
28Examples of Annuities Car loan payments Mortgage on a house Most other consumer loansContributions to a retirement planRetirement payments from a pension plan
29Saving a Fixed SumYou 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
30Future Value Calculated Save $2,000 every year for 5 years into an account that pays 10%. What is the accumulated balance after 5 years?Future value calculated by compounding each cash flow separately12345Time (years)$2,000$2,000$2,000$2,000$2, , , , $12,210.20x 1.1x 1.12x 1.13x 1.14Total future value
32Important to understand inputs ‘r’ is the interest rate per period‘t’ is the # of periods.For example,if ‘t’ is # of years, ‘r’ is annual rateif ‘t’ is # of months, ‘r’ is the monthly rate
33FV of Annuity ExampleYou 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!!!
34Time Line Notice: Payment begins at the end of first year 1 2 34 35 123435-5000-5000-5000-5000Notice: Payment begins at the end of first year
35FV of Annuity on Calculator Clear all memory: CLEAR ALLEnsure # compounding periods is 1:Enter payments:Enter # of payments:Enter interest rate: 10Find Future Value:Answer: $___________P/YRPMTNI/YRFV
36FV 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?Note: Payments are monthly, interest quoted is annual!!!
37Two 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 1Set “P/YR” on calculator as 12: 12 and enter the annual interest rateORP/YR
38‘N’ on calculator You can either: 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(notice it appears as 360)ORN
39FV Annuity on Calculator (2) Clear all memory: CLEAR ALLMonthly-> # compounding periods is 12: 12Enter Future Value: 1,000,000Enter # of payments:Enter interest rate: 12Find payments:Answer: $___________P/YRFVNI/YRPMTNote thedifference!
40Present Value of Annuities 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
41PV of Annuity Example Cash flow per period (CFt) = $500 Number of periods (t) = 4 yearsInterest Rate (r) = 9% per yearWhat is the present value (PV) = ?ALWAYS DRAW A TIME LINE!!!
42PV of Annuity on Calculator Clear all memory: CLEAR ALLEnsure # compounding periods is 1:Enter payments: 500Enter # of payments: 4Enter interest rate: 9Find Present Value:Answer: $___________P/YRPMTNI/YRPV
43PV of AnnuityAgain: ‘r’ and ‘t’ must match – i.e. if t is # of months, r must be monthly rate
44Car Loan Example Car costs $ 20,000 Interest rate per month = 1% 5-year loan ---> number of months = t = 60What is the monthly payment?Answer: ___________
45Mortgage payments House cost $250,000 Mortgage Rate = 7.5% annually Term of loan = 30 yearsPayments made monthlyWhat are your payments?Answer: _____________
46To Reiterate... Be VERY careful about compounding periods 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)
48PerpetuityNote: C and r measured over same interval
49Perpetuity ExamplePreferred stock pays $1.00 dividend per quarter. The required return, r, is 2.5% per quarter.What is the stock value?
50Perpetuity ExampleSteve Forbes’s 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 campaignCharge: He was running for presidency for personal gainDid the charge make sense
51Forbes continued... What should be ‘r’ in the example? At what ‘r’ would Forbes have gained from being a president and steamrolling flat-tax proposal?
52Compounding Periods Interest can be compounded Annually SemiannuallyMonthly Daily ContinuouslySmaller the compounding period, faster is the growth of moneyThe same PV or FV formula can be used: BUT UNDERSTAND THE INPUTS!!
53Compounding example Invest $5,000 in a 5-year CD Quoted Annual Percentage Rate (APR) = 15%Calculate FV5 for annual, semi-annual, monthly and daily compoundingKey: Adjust “P/YR” on calculator
55Continuous compouding Compounded every instant “microsecond”r = interest rate per periodt = number of periodsPrevious example answer: $ 10,585.00
56Continuous compounding example Invest $4,500 in an account paying 9.5% compounded continuouslyWhat is the balance after 4 years? Answer: _________
57Quoted vs. Effective Interest Rates Quoted Rate: Usually stated annually along with compounding period (APR)e.g. 10% compounded quarterlyEffective Annual Rate (EAR): Interest rate actually earned IF the compounding period were one year
59EAR on CalculatorWhat is the EAR for quoted rate of 15% per year compounded quarterly?Set number of periods per year: 4Enter quoted annual rate: 15Compute EAR:Answer: _______P/YRI/YREFF%
60EAR Example Compute EAR for 12% compounded AnnuallyQuarterlyMonthlyDailyAnswers: ____ , ____ , ____ , ____
61EAR for Continuous compounding Example: Quoted rate is 10% compounded continuouslyEAR = _____%
62Complicatons to TVM When payments begin beyond year 1 PV and FV combinedWhen payments begin in year 0 (Annuities Due)
63Payments beyond year 1A 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: ___________
64PV and FV combinedHow 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: ___________
65PV and FV combined 2 You have 2 options: Receive $100 for next 10 years onlyReceive $100 forever beginning in year 11If r = 10% which one would you prefer?At what interest rate are you indifferent between the two options?
66Annuities Due Payments begin in year 0 Trick: Ex. Rent/Lease PaymentsTrick:Adjust BEG/END on calculator to BEGLeave to END, but multiply (1+r) for both PV and FVOR
67Annuity Due ExampleFind PV of a 4-year (5 payment), $400 annuity due. r = 10%Find FV in year 5 of the above annuity dueAnswers:PV = $1,667.95FV5 = $2,686.2412345Time (years)$400$400$400$400$400FV
68Another ExampleYou 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 rateAnswer: $_______
69Tricky but Legal...Add-on Interest Called ‘add-on’ interest because interest is added on to the principal before the payments are calculatedPoints on a Loan: Percentage of loan amount reduced up frontUsed in home mortgages
70Example: 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 = $39What is the effective annual rate (EAR)?Steps:Calculate the APR interest (I/YR)Use answer to calculate the EAR
71Add-on Example (2)Calcuate the EAR on a 6-year, $7,000 loan at 13% ‘add-on’ interest. The payments are monthly.Answer: ________
72Example: 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]What is the EAR?$110 = $98 (1+r) r = 12.24%
73Points 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 monthlyAnswer: _________
74Balloon PaymentsAmount on the loan outstanding after a certain number of payments have been madeSometimes called ‘residual’ on a loane.g. when you want to pay off a loan early
75Balloon ExampleYou 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: $__________
76Two ways to calculate Balloons First calculate paymentsTake the present value of the remaining (unpaid) paymentsUse amortization function on calculatorEnter the period : periodEnter , and thenORINPUTAMORT===
77Another 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.Answer: $____________
78TVM TIPS Draw time line! Check & set BEG/END on calculator Check & set P/YR on calculatorCheck & set # of decimal places to 4
79TVM Tips Continued... Clear all previously stored #’s in memory Especially true when same problem requires multiple TVM calculationsMake sure that for FV and PV calculation, you have correctly signed (+/-) the cash flows