CHAPTER 3 MONEY-TIME RELATIONSHIPS AND EQUIVALENCE

INTEREST “Janganlah kamu sakiti sesiapapun agar orang lain tidak menyakiti kamu pula. Ingatlah bahawa sesungguhnya kamu akan menemui Tuhan kamu dan Dia pasti akan membuat perhitungan atas segala amalan kamu. Allah telah mengharamkan riba', oleh itu segala urusan yang melibatkan riba' hendaklah dibatalkan mulai sekarang” “Hurt no one so that no one may hurt you. Remember that you will indeed meet your LORD, and that HE will indeed reckon your deeds. ALLAH has forbidden you to take usury (interest), therefore all interest obligation shall henceforth be waived”. Muhammad (pbuh), The Farewell Sermon, Mount Arafat (0632)

MONEY Medium of Exchange -- Means of payment for goods or services; What sellers accept and buyers pay ; Store of Value -- A way to transport buying power from one time period to another; Unit of Account -- A precise measurement of value or worth; Allows for tabulating debits and credits; Medium of Exchange -- Means of payment for goods or services; What sellers accept and buyers pay ; Store of Value -- A way to transport buying power from one time period to another; Unit of Account -- A precise measurement of value or worth; Allows for tabulating debits and credits;

CAPITAL Wealth in the form of money or property that can be used to produce more wealth.

KINDS OF CAPITAL Equity capital is that owned by individuals who have invested their money or property in a business project or venture in the hope of receiving a profit. Debt capital, often called borrowed capital, is obtained from lenders (e.g., through the sale of bonds) for investment. Equity capital is that owned by individuals who have invested their money or property in a business project or venture in the hope of receiving a profit. Debt capital, often called borrowed capital, is obtained from lenders (e.g., through the sale of bonds) for investment.

Exchange money for shares of stock as proof of partial ownership

INTEREST The fee that a borrower pays to a lender for the use of his or her money. INTEREST RATE The percentage of money being borrowed that is paid to the lender on some time basis. The fee that a borrower pays to a lender for the use of his or her money. INTEREST RATE The percentage of money being borrowed that is paid to the lender on some time basis.

HOW INTEREST RATE IS DETERMINED Interest Rate Quantity of Money

HOW INTEREST RATE IS DETERMINED Interest Rate Quantity of Money ieie Money Demand Money Supply MS 1 MS 2 i2i2 MS 3 i3i3

SIMPLE INTEREST The total interest earned or charged is linearly proportional to the initial amount of the loan (principal), the interest rate and the number of interest periods for which the principal is committed. When applied, total interest “I” may be found by I = ( P ) ( N ) ( i ), where –P = principal amount lent or borrowed –N = number of interest periods ( e.g., years ) –i = interest rate per interest period The total interest earned or charged is linearly proportional to the initial amount of the loan (principal), the interest rate and the number of interest periods for which the principal is committed. When applied, total interest “I” may be found by I = ( P ) ( N ) ( i ), where –P = principal amount lent or borrowed –N = number of interest periods ( e.g., years ) –i = interest rate per interest period

COMPOUND INTEREST Whenever the interest charge for any interest period is based on the remaining principal amount plus any accumulated interest charges up to the beginning of that period. Period Amount Owed Interest Amount Amount Owed Beginning of for Period at end of period 10% ) period 1$1,000$100$1,100 2$1,100$110$1,210 3$1,210$121$1,331 Whenever the interest charge for any interest period is based on the remaining principal amount plus any accumulated interest charges up to the beginning of that period. Period Amount Owed Interest Amount Amount Owed Beginning of for Period at end of period 10% ) period 1$1,000$100$1,100 2$1,100$110$1,210 3$1,210$121$1,331

ECONOMIC EQUIVALENCE Established when we are indifferent between a future payment, or a series of future payments, and a present sum of money. Considers the comparison of alternative options, or proposals, by reducing them to an equivalent basis, depending on: –interest rate; –amounts of money involved; –timing of the affected monetary receipts and/or expenditures; –manner in which the interest, or profit on invested capital is paid and the initial capital is recovered. Established when we are indifferent between a future payment, or a series of future payments, and a present sum of money. Considers the comparison of alternative options, or proposals, by reducing them to an equivalent basis, depending on: –interest rate; –amounts of money involved; –timing of the affected monetary receipts and/or expenditures; –manner in which the interest, or profit on invested capital is paid and the initial capital is recovered.

ECONOMIC EQUIVALENCE FOR FOUR REPAYMENT PLANS OF AN $8,000 LOAN Plan #1: $2,000 of loan principal plus 10% of BOY principal paid at the end of year; interest paid at the end of each year is reduced by $200 (i.e., 10% of remaining principal) YearAmount Owed Interest Accrued Total Principal Total end at beginning for Year Money Payment of Year of Year owed at Payment ( BOY ) end of Year 1$8,000$800 $8,800$2,000 $2,800 2$6,000$600 $6,600$2,000 $2,600 3$4,000$400 $4,400$2,000 $2,400 4$2,000$200 $2,200$2,000 $2,200 Total interest paid ($2,000) is 10% of total dollar-years ($20,000) Plan #1: $2,000 of loan principal plus 10% of BOY principal paid at the end of year; interest paid at the end of each year is reduced by $200 (i.e., 10% of remaining principal) YearAmount Owed Interest Accrued Total Principal Total end at beginning for Year Money Payment of Year of Year owed at Payment ( BOY ) end of Year 1$8,000$800 $8,800$2,000 $2,800 2$6,000$600 $6,600$2,000 $2,600 3$4,000$400 $4,400$2,000 $2,400 4$2,000$200 $2,200$2,000 $2,200 Total interest paid ($2,000) is 10% of total dollar-years ($20,000)

Plan #2: $0 of loan principal paid until end of fourth year; $800 interest paid at the end of each year YearAmount Owed Interest Accrued Total Principal Total end at beginning for Year Money Payment of Year of Year owed at Payment ( BOY ) end of Year 1$8,000$800 $8,800$0 $800 2$8,000$800 $8,800$0 $800 3$8,000$800 $8,800$0 $800 4$8,000$800 $8,800$8,000 $8,800 Total interest paid ($3,200) is 10% of total dollar-years ($32,000) Plan #2: $0 of loan principal paid until end of fourth year; $800 interest paid at the end of each year YearAmount Owed Interest Accrued Total Principal Total end at beginning for Year Money Payment of Year of Year owed at Payment ( BOY ) end of Year 1$8,000$800 $8,800$0 $800 2$8,000$800 $8,800$0 $800 3$8,000$800 $8,800$0 $800 4$8,000$800 $8,800$8,000 $8,800 Total interest paid ($3,200) is 10% of total dollar-years ($32,000) ECONOMIC EQUIVALENCE FOR FOUR REPAYMENT PLANS OF AN $8,000 LOAN

Plan #3: $2,524 paid at the end of each year; interest paid at the end of each year is 10% of amount owed at the beginning of the year. YearAmount Owed Interest Accrued Total Principal Total end at beginning for Year Money Payment of Year of Year owed at Payment ( BOY ) end of Year 1$8,000$800 $8,800$1,724 $2,524 2$6,276$628 $6,904$1,896 $2,524 3$4,380$438 $4,818$2,086 $2,524 4$2,294$230 $2,524$2,294 $2,524 Total interest paid ($2,096) is 10% of total dollar-years ($20,950) Plan #3: $2,524 paid at the end of each year; interest paid at the end of each year is 10% of amount owed at the beginning of the year. YearAmount Owed Interest Accrued Total Principal Total end at beginning for Year Money Payment of Year of Year owed at Payment ( BOY ) end of Year 1$8,000$800 $8,800$1,724 $2,524 2$6,276$628 $6,904$1,896 $2,524 3$4,380$438 $4,818$2,086 $2,524 4$2,294$230 $2,524$2,294 $2,524 Total interest paid ($2,096) is 10% of total dollar-years ($20,950)

ECONOMIC EQUIVALENCE FOR FOUR REPAYMENT PLANS OF AN $8,000 LOAN Plan #4: No interest and no principal paid for first three years. At the end of the fourth year, the original principal plus accumulated (compounded) interest is paid. YearAmount Owed Interest Accrued Total Principal Total end at beginning for Year Money Payment of Year of Year owed at Payment ( BOY ) end of Year 1$8,000$800 $8,800$0 $0 2$8,800$880 $9,680$0 $0 3$9,680$968 $10,648$0 $0 4$10,648$1,065 $11,713$8,000 $11,713 Total interest paid ($3,713) is 10% of total dollar-years ($37,128) Plan #4: No interest and no principal paid for first three years. At the end of the fourth year, the original principal plus accumulated (compounded) interest is paid. YearAmount Owed Interest Accrued Total Principal Total end at beginning for Year Money Payment of Year of Year owed at Payment ( BOY ) end of Year 1$8,000$800 $8,800$0 $0 2$8,800$880 $9,680$0 $0 3$9,680$968 $10,648$0 $0 4$10,648$1,065 $11,713$8,000 $11,713 Total interest paid ($3,713) is 10% of total dollar-years ($37,128)

CASH FLOW DIAGRAMS / TABLE NOTATION i = effective interest rate per interest period N = number of compounding periods (e.g., years) P = present sum of money; the equivalent value of one or more cash flows at the present time reference point F = future sum of money; the equivalent value of one or more cash flows at a future time reference point A = end-of-period cash flows (or equivalent end-of- period values ) in a uniform series continuing for a specified number of periods, starting at the end of the first period and continuing through the last period G = uniform gradient amounts -- used if cash flows increase by a constant amount in each period i = effective interest rate per interest period N = number of compounding periods (e.g., years) P = present sum of money; the equivalent value of one or more cash flows at the present time reference point F = future sum of money; the equivalent value of one or more cash flows at a future time reference point A = end-of-period cash flows (or equivalent end-of- period values ) in a uniform series continuing for a specified number of periods, starting at the end of the first period and continuing through the last period G = uniform gradient amounts -- used if cash flows increase by a constant amount in each period

CASH FLOW DIAGRAM NOTATION = N 1 1 Time scale with progression of time moving from left to right; the numbers represent time periods (e.g., years, months, quarters, etc...) and may be presented within a time interval or at the end of a time interval. P =$8, Present expense (cash outflow) of $8,000 for lender. A = $2, Annual income (cash inflow) of $2,524 for lender. i = 10% per year 4 4 Interest rate of loan. 5 5 Dashed-arrow line indicates amount to be determined.

INTEREST FORMULAS FOR ALL OCCASIONS relating present and future values of single cash flows; relating a uniform series (annuity) to present and future equivalent values; –for discrete compounding and discrete cash flows; –for deferred annuities (uniform series); equivalence calculations involving multiple interest; relating a uniform gradient of cash flows to annual and present equivalents; relating a geometric sequence of cash flows to present and annual equivalents; relating present and future values of single cash flows; relating a uniform series (annuity) to present and future equivalent values; –for discrete compounding and discrete cash flows; –for deferred annuities (uniform series); equivalence calculations involving multiple interest; relating a uniform gradient of cash flows to annual and present equivalents; relating a geometric sequence of cash flows to present and annual equivalents;

INTEREST FORMULAS FOR ALL OCCASIONS relating nominal and effective interest rates; relating to compounding more frequently than once a year; relating to cash flows occurring less often than compounding periods; for continuous compounding and discrete cash flows; for continuous compounding and continuous cash flows; relating nominal and effective interest rates; relating to compounding more frequently than once a year; relating to cash flows occurring less often than compounding periods; for continuous compounding and discrete cash flows; for continuous compounding and continuous cash flows;

RELATING PRESENT AND FUTURE EQUIVALENT VALUES OF SINGLE CASH FLOWS Finding F when given P: Finding future value when given present value F = P ( 1+i ) N –(1+i) N single payment compound amount factor –functionally expressed as F = ( F / P, i%,N ) –predetermined values of this are presented in Interest Factors Table for Discrete Compounding. Finding F when given P: Finding future value when given present value F = P ( 1+i ) N –(1+i) N single payment compound amount factor –functionally expressed as F = ( F / P, i%,N ) –predetermined values of this are presented in Interest Factors Table for Discrete Compounding. P 0 N = F = ?

Example: Given: i = 10%, N = 8 years, P = 2000 F = 2000(1+0.10) 8 = 4, Given: i = 10%, N = 8 years, P = 2000 F = 2000(1+0.10) 8 = 4,287.18

Finding P when given F: Finding present value when given future value P = F [1 / (1 + i ) ] N – (1+i) -N single payment present worth factor – functionally expressed as P = F ( P / F, i%, N ) –predetermined values of this are presented in Interest Factors Table for Discrete Compounding Finding P when given F: Finding present value when given future value P = F [1 / (1 + i ) ] N – (1+i) -N single payment present worth factor – functionally expressed as P = F ( P / F, i%, N ) –predetermined values of this are presented in Interest Factors Table for Discrete Compounding RELATING PRESENT AND FUTURE EQUIVALENT VALUES OF SINGLE CASH FLOWS P = ? 0N = F

Example: Given: i = 12%, N = 5 years, F = 1000 P = 1000(1+0.12) -5 = Given: i = 12%, N = 5 years, F = 1000 P = 1000(1+0.12) -5 =

RELATING A UNIFORM SERIES (ORDINARY ANNUITY) TO PRESENT AND FUTURE EQUIVALENT VALUES Finding F given A: Finding future equivalent income (inflow) value given a series of uniform equal Payments ( 1 + i ) N - 1 F = A i –uniform series compound amount factor in [ ] –functionally expressed as F = A ( F / A,i%,N ) –predetermined values are in Interest Factors Table for Discrete Compounding Finding F given A: Finding future equivalent income (inflow) value given a series of uniform equal Payments ( 1 + i ) N - 1 F = A i –uniform series compound amount factor in [ ] –functionally expressed as F = A ( F / A,i%,N ) –predetermined values are in Interest Factors Table for Discrete Compounding F = ? A =

Example: Given: i = 6%, N = 5 years, A = 5000 F = 5000 (F/A, 6%, 5) = 5000 (5.6371) = 28, Given: i = 6%, N = 5 years, A = 5000 F = 5000 (F/A, 6%, 5) = 5000 (5.6371) = 28,185.46

RELATING A UNIFORM SERIES (ORDINARY ANNUITY) TO PRESENT AND FUTURE EQUIVALENT VALUES Finding P given A: Finding present equivalent value given a series of uniform equal receipts ( 1 + i ) N - 1 P = A i ( 1 + i ) N –uniform series present worth factor in [ ] –functionally expressed as P = A ( P / A,i%,N ) –predetermined values are in Interest Factors Table for Discrete Compounding Finding P given A: Finding present equivalent value given a series of uniform equal receipts ( 1 + i ) N - 1 P = A i ( 1 + i ) N –uniform series present worth factor in [ ] –functionally expressed as P = A ( P / A,i%,N ) –predetermined values are in Interest Factors Table for Discrete Compounding P = ? A =

Example: Lump Sum $104 m or 7.92 m annually in 25 years total 198 m Given: i = 8%, N = 25 years, A = 7.92m P = 7.92 (P/A, 8%, 25) m =7.92( ) = m Lump Sum $104 m or 7.92 m annually in 25 years total 198 m Given: i = 8%, N = 25 years, A = 7.92m P = 7.92 (P/A, 8%, 25) m =7.92( ) = m

RELATING A UNIFORM SERIES (ORDINARY ANNUITY) TO PRESENT AND FUTURE EQUIVALENT VALUES Finding A given F: Finding amount A of a uniform series when given the equivalent future value i A = F ( 1 + i ) N -1 –sinking fund factor in [ ] –functionally expressed as A = F ( A / F,i%,N ) –predetermined values are in Interest Factors Table for Discrete Compounding Finding A given F: Finding amount A of a uniform series when given the equivalent future value i A = F ( 1 + i ) N -1 –sinking fund factor in [ ] –functionally expressed as A = F ( A / F,i%,N ) –predetermined values are in Interest Factors Table for Discrete Compounding F = A =?

Example: Given: i = 7%, N = 8 years, F = 100,000 A = 100,000 (A/F, 7%, 8) = ( ) = 9,747 Given: i = 7%, N = 8 years, F = 100,000 A = 100,000 (A/F, 7%, 8) = ( ) = 9,747

RELATING A UNIFORM SERIES (ORDINARY ANNUITY) TO PRESENT AND FUTURE EQUIVALENT VALUES Finding A given P: Finding amount A of a uniform series when given the equivalent present value i ( 1+i ) N A = P ( 1 + i ) N -1 –capital recovery factor in [ ] –functionally expressed as A = P ( A / P,i%,N ) –predetermined values are in Interest Factors Table for Discrete Compounding Finding A given P: Finding amount A of a uniform series when given the equivalent present value i ( 1+i ) N A = P ( 1 + i ) N -1 –capital recovery factor in [ ] –functionally expressed as A = P ( A / P,i%,N ) –predetermined values are in Interest Factors Table for Discrete Compounding P = A =?

Example: You borrowed RM21, to finance the educational expenses for your diploma course. The loan carries an interest rate of 6% per year and is to be repaid in equal annual installments over the next five years i = 6%, N = 5 years, P = 21, A = P (A/P, 6%, 5) = 21, (0.2374) = 5,000 i = 6%, N = 5 years, P = 21, A = P (A/P, 6%, 5) = 21, (0.2374) = 5,000

RELATING A UNIFORM SERIES (DEFERRED ANNUITY) TO PRESENT / FUTURE EQUIVALENT VALUES If an annuity is deferred j periods, where j < N And finding P given A for an ordinary annuity is expressed by: P = A ( P / A, i%,N ) This is expressed for a deferred annuity by: A ( P / A, i%,N - j ) at end of period j This is expressed for a deferred annuity by: A ( P / A, i%,N - j ) ( P / F, i%, j ) as of time 0 (time present) If an annuity is deferred j periods, where j < N And finding P given A for an ordinary annuity is expressed by: P = A ( P / A, i%,N ) This is expressed for a deferred annuity by: A ( P / A, i%,N - j ) at end of period j This is expressed for a deferred annuity by: A ( P / A, i%,N - j ) ( P / F, i%, j ) as of time 0 (time present)

EQUIVALENCE CALCULATIONS INVOLVING MULTIPLE INTEREST All compounding of interest takes place once per time period (e.g., a year), and to this point cash flows also occur once per time period. Consider an example where a series of cash outflows occur over a number of years. Consider that the value of the outflows is unique for each of a number (i.e., first three) years. Consider that the value of outflows is the same for the last four years. Find a) the present equivalent expenditure; b) the future equivalent expenditure; and c) the annual equivalent expenditure All compounding of interest takes place once per time period (e.g., a year), and to this point cash flows also occur once per time period. Consider an example where a series of cash outflows occur over a number of years. Consider that the value of the outflows is unique for each of a number (i.e., first three) years. Consider that the value of outflows is the same for the last four years. Find a) the present equivalent expenditure; b) the future equivalent expenditure; and c) the annual equivalent expenditure

PRESENT EQUIVALENT EXPENDITURE Use P 0 = F( P / F, i%, N ) for each of the unique years: -- F is a series of unique outflow for year 1 through year 3; -- i is common interest for each calculation; -- N is the year in which the outflow occurred; -- Multiply the outflow times the associated table value; -- Add the three products together; Use A ( P / A,i%,N - j ) ( P / F, i%, j ) -- deferred annuity -- for the remaining (common outflow) years: -- A is common for years 4 through 7; -- i remains the same; -- N is the final year; -- j is the last year a unique outflow occurred; -- multiply the common outflow value times table values; -- add this to the previous total for the present equivalent expenditure. Use P 0 = F( P / F, i%, N ) for each of the unique years: -- F is a series of unique outflow for year 1 through year 3; -- i is common interest for each calculation; -- N is the year in which the outflow occurred; -- Multiply the outflow times the associated table value; -- Add the three products together; Use A ( P / A,i%,N - j ) ( P / F, i%, j ) -- deferred annuity -- for the remaining (common outflow) years: -- A is common for years 4 through 7; -- i remains the same; -- N is the final year; -- j is the last year a unique outflow occurred; -- multiply the common outflow value times table values; -- add this to the previous total for the present equivalent expenditure.

RELATING A UNIFORM GRADIENT OF CASH FLOWS TO FUTURE EQUIVALENTS Find F when given G: Find the future equivalent value when given the uniform gradient amount (1+i) N-1 -1 (1+i) N-2 -1 (1+i) 1 -1 F = G Functionally represented as (G/ i) (F/A,i%,N) - (NG/ i) Usually more practical to deal with annual and present equivalents, rather than future equivalent values Find F when given G: Find the future equivalent value when given the uniform gradient amount (1+i) N-1 -1 (1+i) N-2 -1 (1+i) 1 -1 F = G Functionally represented as (G/ i) (F/A,i%,N) - (NG/ i) Usually more practical to deal with annual and present equivalents, rather than future equivalent values i i i

Cash Flow Diagram for a Uniform Gradient Increasing by G Dollars per period 1234N-2N-1N G 2G 3G (N-3)G (N-2)G (N-1)G i = effective interest rate per period End of Period

RELATING A UNIFORM GRADIENT OF CASH FLOWS TO ANNUAL AND PRESENT EQUIVALENTS Find A when given G: Find the annual equivalent value when given the uniform gradient amount 1 N A = G - i(1 + i ) N - 1 Functionally represented as A = G ( A / G, i%,N ) The value shown in [ ] is the gradient to uniform series conversion factor and is presented in column 9 of Compound Interest Factor Tables Find A when given G: Find the annual equivalent value when given the uniform gradient amount 1 N A = G - i(1 + i ) N - 1 Functionally represented as A = G ( A / G, i%,N ) The value shown in [ ] is the gradient to uniform series conversion factor and is presented in column 9 of Compound Interest Factor Tables

RELATING A UNIFORM GRADIENT OF CASH FLOWS TO ANNUAL AND PRESENT EQUIVALENTS Find P when given G: Find the present equivalent value when given the uniform gradient amount 1(1 + i ) N -1N P = G- ii (1 + i ) N (1 + i ) N Functionally represented as P = G ( P / G, i%,N ) The value shown in{ } is the gradient to present equivalent conversion factor and is presented in column 8 of Compound Interest Factor Tables Find P when given G: Find the present equivalent value when given the uniform gradient amount 1(1 + i ) N -1N P = G- ii (1 + i ) N (1 + i ) N Functionally represented as P = G ( P / G, i%,N ) The value shown in{ } is the gradient to present equivalent conversion factor and is presented in column 8 of Compound Interest Factor Tables

RELATING GE0METRIC SEQUENCE OF CASH FLOWS TO PRESENT AND ANNUAL EQUIVALENTS Projected cash flow patterns changing at an average rate of f each period; Resultant end-of-period cash-flow pattern is referred to as a geometric gradient series; A 1 is cash flow at end of period 1 A k = (A k-1 ) ( 1 +f ),2 < k < N A N = A 1 ( 1 + f ) N-1 f = (A k - A k-1 ) / A k-1 f may be either positive or negative Projected cash flow patterns changing at an average rate of f each period; Resultant end-of-period cash-flow pattern is referred to as a geometric gradient series; A 1 is cash flow at end of period 1 A k = (A k-1 ) ( 1 +f ),2 < k < N A N = A 1 ( 1 + f ) N-1 f = (A k - A k-1 ) / A k-1 f may be either positive or negative

01234N A1A1 A 2 =A 1 (1+f ) A 3 =A 1 (1+f ) 2 A N =A 1 (1+f ) N - 1 End of Period Cash-flow diagram for a Geometric Sequence of Cash Flows

RELATING A GEOMETRIC SEQUENCE OF CASH FLOWS TO ANNUAL AND PRESENT EQUIVALENTS Find P when given A: Find the present equivalent value when given the annual equivalent value ( i = f ) A i P = ( P / A, -1, N ) ( 1 + f ) 1 + f Find P when given A: Find the present equivalent value when given the annual equivalent value ( i = f ) A i P = ( P / A, -1, N ) ( 1 + f ) 1 + f

RELATING A GEOMETRIC SEQUENCE OF CASH FLOWS TO ANNUAL AND PRESENT EQUIVALENTS Find P when given A: Find the present equivalent value when given the annual equivalent value ( i = f ) A 1 [1 – (1+i) –N (1+f) N ] P = 1 - f which may also be written as A 1 [1 - (P/F,i%,N) (F/P,f%,N)] P = i - f Find P when given A: Find the present equivalent value when given the annual equivalent value ( i = f ) A 1 [1 – (1+i) –N (1+f) N ] P = 1 - f which may also be written as A 1 [1 - (P/F,i%,N) (F/P,f%,N)] P = i - f

RELATING A GEOMETRIC SEQUENCE OF CASH FLOWS TO ANNUAL AND PRESENT EQUIVALENTS Note that the foregoing is mathematically equivalent to the following (i = f ): A i P = ( P / A -1, N )` 1 + f 1 + f Note that the foregoing is mathematically equivalent to the following (i = f ): A i P = ( P / A -1, N )` 1 + f 1 + f

RELATING A GEOMETRIC SEQUENCE OF CASH FLOWS TO ANNUAL AND PRESENT EQUIVALENTS The foregoing may be functionally represented as A = P (A / P, i%,N ) The year zero “base” of annuity, increasing at constant rate f % is A 0 = P ( A / P, f %, N ) The future equivalent of this geometric gradient is F = P ( F / P, i%, N ) The foregoing may be functionally represented as A = P (A / P, i%,N ) The year zero “base” of annuity, increasing at constant rate f % is A 0 = P ( A / P, f %, N ) The future equivalent of this geometric gradient is F = P ( F / P, i%, N )

RELATING A GEOMETRIC SEQUENCE OF CASH FLOWS TO ANNUAL AND PRESENT EQUIVALENTS Find P when given A: Find the present equivalent value when given the annual equivalent value ( i = f ) P = A 1 N (1+i) -1 which may be written as Find P when given A: Find the present equivalent value when given the annual equivalent value ( i = f ) P = A 1 N (1+i) -1 which may be written as

RELATING A GEOMETRIC SEQUENCE OF CASH FLOWS TO ANNUAL AND PRESENT EQUIVALENTS Find P when given A: Find the present equivalent value when given the annual equivalent value ( i = f ) P = A 1 N (P/F,i%,1) Functionally represented as A = P (A / P, i%,N ) Find P when given A: Find the present equivalent value when given the annual equivalent value ( i = f ) P = A 1 N (P/F,i%,1) Functionally represented as A = P (A / P, i%,N )

RELATING A GEOMETRIC SEQUENCE OF CASH FLOWS TO ANNUAL AND PRESENT EQUIVALENTS Find P when given A: Find the present equivalent value when given the annual equivalent value ( i = f ) P = A 1 N (i+i)-1 which may be written as P = A 1 N (P/F,i%,1) Functionally represented as A = P (A / P, i%,N ) The year zero “base” of annuity, increasing at constant rate f % is A 0 = P ( A / P, f %, N ) The future equivalent of this geometric gradient is F = P ( F / P, i%, N ) Find P when given A: Find the present equivalent value when given the annual equivalent value ( i = f ) P = A 1 N (i+i)-1 which may be written as P = A 1 N (P/F,i%,1) Functionally represented as A = P (A / P, i%,N ) The year zero “base” of annuity, increasing at constant rate f % is A 0 = P ( A / P, f %, N ) The future equivalent of this geometric gradient is F = P ( F / P, i%, N )

NOMINAL AND EFFECTIVE INTEREST RATES Nominal Interest Rate - r - For rates compounded more frequently than one year, the stated annual interest rate. Effective Interest Rate - i - For rates compounded more frequently than one year, the actual amount of interest paid. i = ( 1 + r / M ) M - 1 = ( F / P, r / M, M ) -1 – M - the number of compounding periods per year Annual Percentage Rate - APR - percentage rate per period times number of periods. –APR = r x M Nominal Interest Rate - r - For rates compounded more frequently than one year, the stated annual interest rate. Effective Interest Rate - i - For rates compounded more frequently than one year, the actual amount of interest paid. i = ( 1 + r / M ) M - 1 = ( F / P, r / M, M ) -1 – M - the number of compounding periods per year Annual Percentage Rate - APR - percentage rate per period times number of periods. –APR = r x M

COMPOUNDING MORE OFTEN THAN ONCE A YEAR Single Amounts Given nominal interest rate and total number of compounding periods, P, F or A can be determined by F = P ( F / P, i%, N ) i% = ( 1 + r / M ) M - 1 Uniform and / or Gradient Series Given nominal interest rate, total number of compounding periods, and existence of a cash flow at the end of each period, P, F or A may be determined by the formulas and tables for uniform annual series and uniform gradient series. Single Amounts Given nominal interest rate and total number of compounding periods, P, F or A can be determined by F = P ( F / P, i%, N ) i% = ( 1 + r / M ) M - 1 Uniform and / or Gradient Series Given nominal interest rate, total number of compounding periods, and existence of a cash flow at the end of each period, P, F or A may be determined by the formulas and tables for uniform annual series and uniform gradient series.

CONTINUOUS COMPOUNDING AND DISCRETE CASH FLOWS Continuous compounding assumes cash flows occur at discrete intervals, but compounding is continuous throughout the interval. Given nominal per year interest rate -- r, compounding per year -- M one unit of principal = [ 1 + (r / M ) ] M Given M / r = p, [ 1 + (r / M ) ] M = [1 + (1/p) ] rp Given lim [ 1 + (1 / p) ] p = e 1 = ( F / P, r%, N ) = e rN i = e r - 1 Continuous compounding assumes cash flows occur at discrete intervals, but compounding is continuous throughout the interval. Given nominal per year interest rate -- r, compounding per year -- M one unit of principal = [ 1 + (r / M ) ] M Given M / r = p, [ 1 + (r / M ) ] M = [1 + (1/p) ] rp Given lim [ 1 + (1 / p) ] p = e 1 = ( F / P, r%, N ) = e rN i = e r - 1 p 

CONTINUOUS COMPOUNDING AND DISCRETE CASH FLOWS Single Cash Flow Finding F given P Finding future equivalent value given present value F = P (e rN ) Functionally expressed as ( F / P, r%, N ) e rN is continuous compounding compound amount Predetermined values are in Continuous Compounding Table Finding F given P Finding future equivalent value given present value F = P (e rN ) Functionally expressed as ( F / P, r%, N ) e rN is continuous compounding compound amount Predetermined values are in Continuous Compounding Table

CONTINUOUS COMPOUNDING AND DISCRETE CASH FLOWS Single Cash Flow Finding P given F Finding present equivalent value given future value P = F (e -rN ) Functionally expressed as ( P / F, r%, N ) e -rN is continuous compounding present equivalent Predetermined values are in Continuous Compounding Table Finding P given F Finding present equivalent value given future value P = F (e -rN ) Functionally expressed as ( P / F, r%, N ) e -rN is continuous compounding present equivalent Predetermined values are in Continuous Compounding Table

CONTINUOUS COMPOUNDING AND DISCRETE CASH FLOWS Uniform Series Finding F given A Finding future equivalent value given a series of uniform equal receipts F = A (e rN - 1)/(e r - 1) Functionally expressed as ( F / A, r%, N ) (e rN - 1)/(e r - 1) is continuous compounding compound amount Predetermined values are in Continuous Compounding Table Finding F given A Finding future equivalent value given a series of uniform equal receipts F = A (e rN - 1)/(e r - 1) Functionally expressed as ( F / A, r%, N ) (e rN - 1)/(e r - 1) is continuous compounding compound amount Predetermined values are in Continuous Compounding Table

CONTINUOUS COMPOUNDING AND DISCRETE CASH FLOWS Uniform Series Finding P given A Finding present equivalent value given a series of uniform equal receipts P = A (e rN - 1) / (e rN ) (e r - 1) Functionally expressed as ( P / A, r%, N ) (e rN - 1) / (e rN ) (e r - 1) is continuous compounding present equivalent Predetermined values are in Continuous Compounding Table Finding P given A Finding present equivalent value given a series of uniform equal receipts P = A (e rN - 1) / (e rN ) (e r - 1) Functionally expressed as ( P / A, r%, N ) (e rN - 1) / (e rN ) (e r - 1) is continuous compounding present equivalent Predetermined values are in Continuous Compounding Table

CONTINUOUS COMPOUNDING AND DISCRETE CASH FLOWS Uniform Series Finding A given F Finding a uniform series given a future value A = F (e r - 1) / (e rN - 1) Functionally expressed as ( A / F, r%, N ) (e r - 1) / (e rN - 1) is continuous compounding sinking fund Predetermined values are in Continuous Compounding Table Finding A given F Finding a uniform series given a future value A = F (e r - 1) / (e rN - 1) Functionally expressed as ( A / F, r%, N ) (e r - 1) / (e rN - 1) is continuous compounding sinking fund Predetermined values are in Continuous Compounding Table

CONTINUOUS COMPOUNDING AND DISCRETE CASH FLOWS Uniform Series Finding A given P Finding a series of uniform equal receipts given present equivalent value A = P [e rN (e r - 1) / (e rN - 1) ] Functionally expressed as ( A / P, r%, N ) [e rN (e r - 1) / (e rN - 1) ] is continuous compounding capital recovery Predetermined values are in Continuous Compounding Table Finding A given P Finding a series of uniform equal receipts given present equivalent value A = P [e rN (e r - 1) / (e rN - 1) ] Functionally expressed as ( A / P, r%, N ) [e rN (e r - 1) / (e rN - 1) ] is continuous compounding capital recovery Predetermined values are in Continuous Compounding Table

CONTINUOUS COMPOUNDING AND CONTINUOUS CASH FLOWS Continuous flow of funds suggests a series of cash flows occurring at infinitesimally short intervals of time Given: – a nominal interest rate or r –p is payments per year [ 1 + (r / p ) ] p - 1 P = r [ 1 + ( r / p ) ] p Given Lim [ 1 + ( r / p ) ] p = e r For one year ( P / A, r%, 1 ) = ( e r - 1 ) / re r Continuous flow of funds suggests a series of cash flows occurring at infinitesimally short intervals of time Given: – a nominal interest rate or r –p is payments per year [ 1 + (r / p ) ] p - 1 P = r [ 1 + ( r / p ) ] p Given Lim [ 1 + ( r / p ) ] p = e r For one year ( P / A, r%, 1 ) = ( e r - 1 ) / re r p --> oo

CONTINUOUS COMPOUNDING AND CONTINUOUS CASH FLOWS Finding F given A Finding the future equivalent given the continuous funds flow F = A [ ( e rN - 1 ) / r ] Functionally expressed as ( F / A, r%, N ) ( e rN - 1 ) / r is continuous compounding compound amount Predetermined values are found in Continuous Compounding Table Finding F given A Finding the future equivalent given the continuous funds flow F = A [ ( e rN - 1 ) / r ] Functionally expressed as ( F / A, r%, N ) ( e rN - 1 ) / r is continuous compounding compound amount Predetermined values are found in Continuous Compounding Table

CONTINUOUS COMPOUNDING AND CONTINUOUS CASH FLOWS Finding P given A Finding the present equivalent given the continuous funds flow P = A [ ( e rN - 1 ) / re rN ] Functionally expressed as ( P / A, r%, N ) ( e rN - 1 ) / re rN is continuous compounding present equivalent Predetermined values are found in Continuous Compounding Table Finding P given A Finding the present equivalent given the continuous funds flow P = A [ ( e rN - 1 ) / re rN ] Functionally expressed as ( P / A, r%, N ) ( e rN - 1 ) / re rN is continuous compounding present equivalent Predetermined values are found in Continuous Compounding Table

CONTINUOUS COMPOUNDING AND CONTINUOUS CASH FLOWS Finding A given F Finding the continuous funds flow given the future equivalent A = F [ r / ( e rN - 1 )] Functionally expressed as ( A / F, r%, N ) r / ( e rN - 1 ) is continuous compounding sinking fund Finding A given F Finding the continuous funds flow given the future equivalent A = F [ r / ( e rN - 1 )] Functionally expressed as ( A / F, r%, N ) r / ( e rN - 1 ) is continuous compounding sinking fund

CONTINUOUS COMPOUNDING AND CONTINUOUS CASH FLOWS Finding A given P Finding the continuous funds flow given the present equivalent A = P [ r / ( e rN - 1 )] Functionally expressed as ( A / P, r%, N ) Finding A given P Finding the continuous funds flow given the present equivalent A = P [ r / ( e rN - 1 )] Functionally expressed as ( A / P, r%, N )

CONTINUOUS COMPOUNDING AND CONTINUOUS CASH FLOWS Finding A given P Finding the continuous funds flow given the present equivalent A = F [ re rN / ( e rN - 1 )] Functionally expressed as ( A / P, r%, N ) re rN / ( e rN - 1 ) is continuous compounding capital recovery Finding A given P Finding the continuous funds flow given the present equivalent A = F [ re rN / ( e rN - 1 )] Functionally expressed as ( A / P, r%, N ) re rN / ( e rN - 1 ) is continuous compounding capital recovery

QUIZ 2 1.George obtained a new credit facilities from ABC Bank, with a stated rate 8% compounded quarterly. For a RM12,000 balance at the beginning of the year, find the effective annual rate and total owned to ABC Bank after a year, provided no payments are made during the year. 2.Tasha plans to place money in a Koperasi Unimap capital fund that currently returns 24% per year, compounded daily. What effective rate is this (a) yearly and (b) quarterly 1.George obtained a new credit facilities from ABC Bank, with a stated rate 8% compounded quarterly. For a RM12,000 balance at the beginning of the year, find the effective annual rate and total owned to ABC Bank after a year, provided no payments are made during the year. 2.Tasha plans to place money in a Koperasi Unimap capital fund that currently returns 24% per year, compounded daily. What effective rate is this (a) yearly and (b) quarterly