The Time Value of Money Chapter 9

The Time Value of Money uWhich would you rather have ? F $100 today - or F $100 one year from today F Sooner is better !

The Time Value of Money uHow about $100 today or $105 one year from today? uWe revalue current dollars and future dollars using the time value of money uCash flow time line graphically shows the timing of cash flows

uTime 0 is today; Time 1 is one period from today Time Cash Flows % Outflow Interest rate 105 Inflow Cash Flow Time Lines

Future Value uCompounding F the process of determining the value of a cash flow or series of cash flows some time in the future when compound interest is applied

Future Value uPV = present value or starting amount, say, $100 ui = interest rate, say, 5% per year would be shown as 0.05 uINT = dollars of interest you earn during the year $100  0.05 = $5 uFV n = future value after n periods or $100 + $5 = $105 after one year u= $100 ( ) = $100(1.05) = $105

Future Value

uThe amount to which a cash flow or series of cash flows will grow over a given period of time when compounded at a given interest rate

Compounded Interest uInterest earned on interest

Cash Flow Time Lines Time Interest % 5.00 Total Value

Future Value Interest Factor for i and n (FVIF i,n ) uThe future value of $1 left on deposit for n periods at a rate of i percent per period uThe multiple by which an initial investment grows because of the interest earned

Future Value Interest Factor for i and n (FVIF i,n ) uFV n = PV(1 + i) n = PV(FVIF i,n ) For $100 at i = 5% and n = 5 periods

uFV n = PV(1 + i) n = PV(FVIF i,n ) For $100 at i = 5% and n = 5 periods $100 (1.2763) = $ Future Value Interest Factor for i and n (FVIF i,n )

Financial Calculator Solution uFive keys for variable input F N = the number of periods F I = interest rate per period may be I, INT, or I/Y F PV = present value F PMT = annuity payment F FV = future value

uFind the future value of $100 at 5% interest per year for five years u1. Numerical Solution: Two Solutions Time Cash Flows % 5.00 Total Value FV 5 = $100(1.05) 5 = $100(1.2763) = $127.63

Two Solutions 2. Financial Calculator Solution: Inputs: N = 5 I = 5 PV = -100 PMT = 0 FV = ? Output: =

Graphic View of the Compounding Process: Growth uRelationship among Future Value, Growth or Interest Rates, and Time i= 15% i= 10% i= 5% i= 0% Periods Future Value of $1 0

Present Value uOpportunity cost F the rate of return on the best available alternative investment of equal risk uIf you can have $100 today or $ at the end of five years, your choice will depend on your opportunity cost

Present Value uThe present value is the value today of a future cash flow or series of cash flows uThe process of finding the present value is discounting, and is the reverse of compounding uOpportunity cost becomes a factor in discounting

PV = ? 5% Cash Flow Time Lines

Present Value uStart with future value: uFV n = PV(1 + i) n

uFind the present value of $ in five years when the opportunity cost rate is 5% u1. Numerical Solution: % PV = ? ÷ Two Solutions

uFind the present value of $ in five years when the opportunity cost rate is 5% u2. Financial Calculator Solution: Inputs: N = 5 I = 5 PMT = 0 FV = PV = ? Output: = -100 Two Solutions

uRelationship among Present Value, Interest Rates, and Time Graphic View of the Discounting Process i= 15% i= 10% i= 5% i= 0% Periods Present Value of $

Solving for Time and Interest Rates uCompounding and discounting are reciprocals uFV n = PV(1 + i) n Four variables: PV, FV, i and n If you know any three, you can solve for the fourth

uFor $78.35 you can buy a security that will pay you $100 after five years uWe know PV, FV, and n, but we do not know i FV n = PV(1 + i) n $100 = $78.35(1 + i) 5 Solve for i Solving for i i = ? 100

uFVn = PV(1 + i) n u$100 = $78.35(1 + i) 5 Numerical Solution

uInputs: N = 5 PV = PMT = 0 FV = 100 I = ? uOutput: = 5 uThis procedure can be used for any rate or value of n, including fractions Financial Calculator Solution

Solving for n uSuppose you know that the security will provide a return of 10 percent per year, that it will cost $68.30, and that you will receive $100 at maturity, but you do not know when the security matures. You know PV, FV, and i, but you do not know n - the number of periods.

Solving for n uFVn = PV(1 + i)n u$100 = $68.30(1.10)n uBy trial and error you could substitute for n and find that n = 4 012n-1n=? % 100

Financial Calculator Solution uInputs: I = 10 PV = PMT = 0 FV = 100 N = ? uOutput: = 4.0

Annuity uAn annuity is a series of payments of an equal amount at fixed intervals for a specified number of periods uOrdinary (deferred) annuity has payments at the end of each period uAnnuity due has payments at the beginning of each period uFVA n is the future value of an annuity over n periods

Future Value of an Annuity uThe future value of an annuity is the amount received over time plus the interest earned on the payments from the time received until the future date being valued uThe future value of each payment can be calculated separately and then the total summed

uIf you deposit $100 at the end of each year for three years in a savings account that pays 5% interest per year, how much will you have at the end of three years? = 100 (1.05) = 100 (1.05) = 100 (1.05) Future Value of an Annuity 01235% 100

Future Value of an Annuity

uFinancial calculator solution: uInputs: N = 3 I = 5 PV = 0 PMT = FV = ? uOutput: = uTo solve the same problem, but for the present value instead of the future value, change the final input from FV to PV

Annuities Due uIf the three $100 payments had been made at the beginning of each year, the annuity would have been an annuity due. uEach payment would shift to the left one year and each payment would earn interest for an additional year (period).

u$100 at the end of each year 01235% = 100 (1.05) = 100 (1.05) = 100 (1.05) Future Value of an Annuity

u$100 at the start of each year 01235% = 100 (1.05) = 100 (1.05) = 100 (1.05) 3 Future Value of an Annuity Due

uNumerical solution: Future Value of an Annuity Due

uNumerical solution: Future Value of an Annuity Due

uFinancial calculator solution: uInputs: N = 3 I = 5 PV = 0 PMT = FV = ? uOutput: =

Present Value of an Annuity uIf you were offered a three-year annuity with payments of $100 at the end of each year uOr a lump sum payment today that you could put in a savings account paying 5% interest per year uHow large must the lump sum payment be to make it equivalent to the annuity?

01235% Present Value of an Annuity

uNumerical solution:

Present Value of an Annuity

uFinancial calculator solution: uInputs: N = 3 I = 5 PMT = -100 FV = 0 PV = ? uOutput: =

Present Value of an Annuity Due uPayments at the beginning of each year uPayments all come one year sooner uEach payment would be discounted for one less year uPresent value of annuity due will exceed the value of the ordinary annuity by one year’s interest on the present value of the ordinary annuity

01235% Present Value of an Annuity Due

uNumerical solution: Present Value of an Annuity Due

uFinancial calculator solution: uSwitch to the beginning-of-period mode, then enter uInputs: N = 3 I = 5 PMT = -100 FV = 0 PV = ? uOutput: = uThen switch back to the END mode

i = ? Solving for Interest Rates with Annuities uSuppose you pay $ for an investment that promises to pay you $250 per year for the next four years, with payments made at the end of each year

Solving for Interest Rates with Annuities uNumerical solution: uTrial and error using different values for i using until you find i where the present value of the four-year, $250 annuity equals $ The solution is 7%.

Solving for Interest Rates with Annuities uFinancial calculator solution: uInputs: N = 4 PV = PMT = 250 FV = 0 I = ? uOutput: = 7.0

Perpetuities uPerpetuity - a stream of equal payments expected to continue forever uConsol - a perpetual bond issued by the British government to consolidate past debts; in general, and perpetual bond

Uneven Cash Flow Streams uUneven cash flow stream is a series of cash flows in which the amount varies from one period to the next uPayment (PMT) designates constant cash flows uCash Flow (CF) designates cash flows in general, including uneven cash flows

Present Value of Uneven Cash Flow Streams uPV of uneven cash flow stream is the sum of the PVs of the individual cash flows of the stream

Future Value of Uneven Cash Flow Streams uTerminal value is the future value of an uneven cash flow stream

Solving for i with Uneven Cash Flow Streams uUsing a financial calculator, input the CF values into the cash flow register and then press the IRR key for the Internal Rate of Return, which is the return on the investment.

Compounding Periods uAnnual compounding F interest is added once a year uSemiannual compounding F interest is added twice a year F 10% annual interest compounded semiannually would pay 5% every six months F adjust the periodic rate and number of periods before calculating

Interest Rates uSimple (Quoted) Interest Rate F rate used to compute the interest payment paid per period uEffective Annual Rate (EAR) F annual rate of interest actually being earned, considering the compounding of interest

Interest Rates uAnnual Percentage Rate (APR) F the periodic rate multiplied by the number of periods per year F this is not adjusted for compounding uMore frequent compounding:

Amortized Loans uLoans that are repaid in equal payments over its life uBorrow $15,000 to repay in three equal payments at the end of the next three years, with 8% interest due on the outstanding loan balance at the beginning of each year

01238% PMT 15,000 Amortized Loans

uNumerical Solution: Amortized Loans

uFinancial calculator solution: uInputs: N = 3 I = 8 PV = FV = 0 PMT = ? uOutput: =

Amortized Loans uAmortization Schedule shows how a loan will be repaid with a breakdown of interest and principle on each payment date a Interest is calculated by multiplying the loan balance at the beginning of the year by the interest rate. Therefor, interest in Year 1 is $15,000(0.08) = $1,200; in Year 2, it is $10,379.50(0.08)=$830.36; and in Year 3, it is $5,389.36(0.08) = $ (rounded). b Repayment of principal is equal to the payment of $5, minus the interest charge for each year. c The $0.01 remaining balance at the end of Year 3 results from rounding differences.

Comparing Interest Rates u1. Simple, or quoted, rate, (i simple ) F rates compare only if instruments have the same number of compounding periods per year u2. Periodic rate (i PER ) F APR represents the periodic rate on an annual basis without considering interest compounding F APR is never used in actual calculations

Comparing Interest Rates u3. Effective annual rate, EAR F the rate that with annual compounding (m=1) would obtain the same results as if we had used the periodic rate with m compounding periods per year

End of Chapter 9 The Time Value of Money