Chapter 5, 6-Time Value of Money
Chapter 5, 6-Time Value of Money FIN 302
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This 6 page Class Notes was uploaded by Quinn Shapiro on Wednesday September 30, 2015. The Class Notes belongs to FIN 302 at Arizona State University taught by Dr. Luke Stein in Summer 2015. Since its upload, it has received 16 views. For similar materials see Managerial Finance in Finance at Arizona State University.
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Date Created: 09/30/15
Time Value of Money 0 Start out by focusing on FV and PV 0 FV how much will you be willing to accept in the future instead of xed amount of dollars today 0 PV how much now instead of xed amount in the future 0 Single Payment Security Save now single payment in exchange get in 10 years 1 deposit 1 withdrawal Zero coupon bond 0 Multiple Payment Security Annuitiesequal payments Perpetuatesstock Pays you a couple times a year forever 0 FV1500k 15525 1principa 5interest FV252515 Or 500105105 FV5500105quot5 At the end of N years you multiply the Principal PV by 1nquotn FVnPV1nquotn FV2500105quot2 5001515 5001555quot2 5002525125 125 is compound interest interest on the interest OOOOO From Graph 0 Constant increase in rates of return from blue to green is 5 increase green to red is 5 increase 0 The higher the percent number the more dramatic a 1 increase will be 2D3 is not big difference 2122 is bigger when you start from a higher base line 0 More the dollar comes 0 When interest is compounded more frequently than they are quoted 0 Credit card Interest rates are quoted annually but typically compounded daily and billed monthly 0 One formula but has to be applied at the level at which they compound 0 Suppose we invest 100 at 15 IR and invest for 10 years PV100 i APR15nyear1O Monthly compounded m12 So 1512125 is APR and n1012120 View table as sensitivity table 0 0 Higher the interest rate more money you get 0 As we move down the colum within interest rate the numbers get bigger more frequent compounding generates higher interest income o Compounding frequency matters more at high interest rates than it does at low interest rates If you are comparing two investments that have different compounding frequencies use EAR EAR asks what would the interest rate have to be on this product if it compounded annually to deliver the same kind of returns Compounded daily 0 100115365quot36511618 o 15365 is periodic interest rate Compounded annually 0 1001EAYEAR11618 o EAR is the interest rate you could invest in to get the same amount if it were compounded daily 0 EAR 115365quot365 1 o So1APRmquotm1EARquot1 We can calculate EAR by equotAPR1 Present Value 0 PVFV1iquotn OR PVFV1iquotn 0 PV factor1iquotn Sign Convention calculators and excel treats out ows as negative 0 Will be neg or pos depending on whether the investor receives or pays cash 0 Investing is a negative cash flow PV of 500 and future value is positive 0 Loan is a positive cash flow positive PV 500 and then future value is NegaUve Negative cash ows down arrows investment put money in up front neg CF receive money later pos CF Neg PV and P05 FV Positive cash ow up arrow loan borrow money later have to pay back Pos PV and Neg FV FV PV 1iquotn Solving for Rate of Interest 0 Solve equation for i Need to get n out of exponents Divide by PV FVPV1iquotn quot1n FVPVquot1n1i lFVPVquot1n 1 Either PV or FV has to be negative in calculator Doesn t matter when you solve by hand only matters when you use calculator or excel do you need to obey sign convention OOOOO Solving for number of periods 0 How many periods until it grow to certain level 000000 0000 0000 0 Need to solve for n Take the log FVPV1iquotn LogFVPVnln1i Nnfvpvln1i What if we ask quotHow long to double your money if you have 10 interest rategrowth PV100 FV110011 FV210011quot2 FV610011quot6 17716 Fv710011quot7195 Fv810011quot8214 Pv100 FV200 i10 n Calc says 727 Rule of 72approximate number of periods it takes to double an investment is 72interest rate Can be used to gure out how long til you get to a certain amount Or to gure out the growth rate nd the doubling time and then nd the growth rate Securities that pay more than once Car loan student loan home owners loan 0 O 0 0000000 0000 Called an annuity Perpetuity is an asset that delivers a regular payment forever Keep paying Stock is similar structure You buy a claim on dividends til the end of time Deposit 50 in bank account at t0 At t1 account balance is 60 Withdrawal 10 from bank now you have 50 At t2 60 Withdrawal 10 ends year with 50 Put 50 in the bank and then use the interest to quotbuy ice creamquot By withdrawing the interest every year the year ends with exactly how much you had when you started How much does it cost to get 10 a year in CF from the bank50 PVPMTi Depositpayment receivedinterest rate How much is perpetuity worth payment made each period divided by interest rate periodic IR IR per period PV1i PMT PV PVPViPMTPV Subtract PV PViPMT Divide by PV lPMTPV British console is a bond that is going to pay interest rates forever Suppose you are paying perpetuities that have different interest rates the one with higher interest rate costs less The bigger the payments the bigger the PV will be As interest rates rise the PV of perpetuity falls Interest rate represents the opportunity cost 0 When interest rate of govt bonds rises it makes them more attractive Growing perpetuitydelivers a growing return each year rather than constant Key insight whatever the price of the perpetuity is this year how much is it going to be worth next year X growth rate Going to deposit PV gains interest so PV1i PMT PV1g 20 PMT1000 1g103 PVPMTperiod1ig where I is interest rate and g is growth rate of payouts 1000203 5882 Higher growth rates are going to increase the value of the growing perpetuity However if ggt it doesn39t work Formula blows up 0 Only works if glti Annuity OOOO Equal dollar amount payment Similar to at perpetuity but nite Not in nite nite number of payments Whether payments are received at beginning or end of period Ordinary Annuityreceived at END of period Involves series of cash ows don39t need to con rm which period just payment If we want to know how much the annuity is worth right now PV the amount of money you would be willing to pay to collect the future stream of CF Typical example bond 5 10 or 30 year bond pay dividend or coupons when come from a bond PV of bond or any annuity can split it up into separate single pmt securities If 5 year bond its 5 separate single security payments and add them 0 Can think of ordinary annuity as perpetuity that you sell after X years 0 0 Value perpetuity at beginning when we buy it And value perpetuity at the end when you sell it o For Calculator 0 0 We know PV N and PMT and we leave out FV We leave out FV bc PV of Annuity We kow PMT500 I6 N5 years O Solve for PV210618 Or Borrow Money Up front and repay FV of Annuity O O O 0 Make investments like deposits and make big withdrawal at the end Same 5 payments invest 5k each year for the next 5 years All annuities for this class all pmts coming at the end of period Comes at end of year so only accrues for 4 years not 5 starting one year from now means at end of year Last deposit comes at the end of year 5 so it doesn39t gain interest For Calculator Exclude PV We know N PMT and FV N5 Pmt5000 6 negative pm tbc its deposit so negative CF Solving for payments of aunnities O 0 If it says at end of year future vaue On Caucuator PVO bc we exclude N18 12 FV100000 Solve for PMT179373 each year needed What if you want to make paymentsdeposits more frequently 0 We change N from 18 year to the months in 18 years 216 o For interest rate 12divide by 12 give you 1 monthly IR Fnding Interest rate 0 We know to solve using FV bc we want money in the future exclude pv to PVO o N20 o o Pmt2500 o FV100k o 677 0 Cannot be done for more than 3 Solving for number of periods 0 iy5 o pv0 o pmt6000 o fv50000 0 NUMBER OF PERIODS714 Amortized Loan 0 Loan paid off with equal pmts o Borrow money up front PVamount you borrow and then repay the loan with individual pmts over time like car loan or mortgage 0 Typical loan always the PV of an Annuity Bal y06000 During 4 years Bal y40 4 equal pmts so PV so zero out FV Pv6000 N4 years l15 Pmtsolve negative 210959 Borrow 6000 year 1900 6900 owed pay 210159 subtract to get wht you owe in year 2 Owe479841 Accrue 71976 in interest pay 210159 0 Each period you are building equity interest is getting smaller so more of your money is getting paid to cover payments not interest 0 The amount you ower outstanding loan balance is always to PV of all remaining pmts 000000 0 Types of Securities Summary 0 1 Single pmt security Pays FV n periods from now 0 PVFV1iquotn Investment put in money now and get money later Loan get money now have to pay later 0 2 Pays pmt next period growing at g forever to value stocks have growing dividends forever 0 PVPMTin period 1ig o If constant growth g0 Perpetuities put in money now and continue to get money in the future Larger payments growing over time at or growing 0 3 PV of Ordinary Annuities like bonds have constant payments 0 4 FV of Ordinary Annuity
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