# THE TIME VALUE OF MONEY

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1 QUANTITATIVE METHODS THE TIME VALUE OF MONEY Reading 5 1

2 Learning Objective Statements (LOS) a. Interest Rates as Required rate of return, Discount Rate and Opportunity Cost b. Interest rate and its components c. Effective annual interest rates and compounding periods other than annual d. Future value and Present value of different dff streams of cash flow e. Time line and application based questions for Time Value of Money 2

3 What is Time Value of Money? TVM is the basic principle p that money can earn interest, so something that is worth \$1 today will be worth more in the future if invested. Definition: The difference in the value of cash received (expended) now versus its value if received sometime in the future. Application in real life: The value of an asset is determined dby estimating i the worth of the stream of future cash flows. To Remember As investment analysts we evaluate several transactions with present and future cash flows. 3

4 Compounding Consider the following transaction: Time 0 1 Entity You Bank Rs.10,000 Present Value One Year 10% You Rs.11,000 Future Value You receive Rs.11,000 Rs.10,000 = Rs.1,000 more, i.e. time value of money. This is explained as Rs.10,000 today and Rs.11,000 in one year are equivalent in value. This method to compute future values of all the cash flows at the end of a given time horizon for a defined rate of interest is known as compounding. 4

5 Discounting In Discounting we compute the present values of all the future cash inflows at a given rate of interest i.e. the value of money at time 0. Time (years) 0 1 Bank Rs.11,000 Future Value You Rs.10, Present Value 10% One Year Example Compute the present value of \$24,200 to be given at the end of 2 years for a rate of interest of 10%. PV = / (1+ 10%) 2 = 20,

6 Interest Rate Interest Rate Interest Rate Interpretation Required rate of return or Minimum expected rate of return Minimum return an Minimum return an investor must receive to accept an investment Opportunity Cost Is the value that i investors forgo by f b choosing a particular course of action Discount rate Used to compute the Used to compute the present value of a future cash flow 6

7 Components of Interest Rate Mathematically we compute the interest rate as r = Real risk free interest rate + inflation premium + default risk premium + liquidity premium + maturity premium Real risk free interest rate It reflects the time preferences of individuals for current versus future real consumption Inflation Premium Default Risk Premium Liquidity premium Average inflation expected over the maturity of the debt Compensates the investor for default in payment by the borrower Compensates for the risk of conversion of the investment to cash To Remember Nominal Risk Free Rate = Real Risk Free Rate + Inflation Maturity Premium Longer the investment maturity, higher is the maturity premium needed for compensating investor needs 7

8 Future value of single cash flow Formula Where, FV = PV * (1+r) N FV = future value of the investment N periods from today PV = present value o the investment r = rate of interest per period N 1 N PV FV = PV * (1 + r) N Salient Points: All cash flows should be brought in one time frame Future value increase with ihnumber of periods Future value increase with the interest rates 8

9 Future Value of lump sum Example A bank offers interest rate of 9% per year compounded annually. How much will you have at the end of 5 years, given the amount invested in the scheme is Rs.50,000? 9

10 Future Value of lump sum Solution: Given: Using Financial Calculator (TI BAII) ,000 FV = PV * (1 + r) N r = 9% PV = 50,000 r = 9% N = 5 FV = * ( 1 + 9%) 5 = 76, Steps Press Buttons Remarks 1 2ND Selects the second function of the calculator 2 CLR WORK Clears memory Number fed on the screen 4 PV Number appearing on the screen is assigneed to the PV (Present Value) memory 5 9 Number fed on the screen Number appearing on the screen is assigneed to the I/Y 6 I/Y (Interest, remember this is a percent memory location) memory 7 5 Number fed on the screen 8 N Number appearing on the screen is assigneed to the N (Number of years) memory 9 0 Number fed on the screen 10 PMT Number appearing on the screen is assigneed to the PMT (Payment or Annuity) memory 11 CPT Puts the calculator in computation mode 12 FV Computes FV (Future Value) for the given set of numbers 10

11 Future Value of lump sum (different annual time frame) Example A bank offers interest rate of 9% per year compounded annually. How much will you have at the end of 10 years, given the amount invested in the scheme is Rs.50,000 at the end of 5 th year? How much will be the amount, if we remain invested till 15 years, in the above scheme? 11

12 Future Value of lump sum Solution: Given: At the end of 5 th year PV = 50,000 r = 9% N = 5 FV = * ( 1 + 9%) 5 = Given At the end of 5 th year PV = 50,000 r = 9% N = 10 FV = * ( 1 + 9%) 10 = FV = PV * (1 + r) N r = 9%, N = FV = PV * (1 + r) N r = 9%, N =

13 Frequency of Compounding Formula Where, FV r s ( 1 ) = PV + m mn FV = future value of the investment N periods from today PV = present value o the investment r s = rate of interest per period m = number of compounding periods per year 13

14 Future Value of lump sum (different time frame) Example A bank offers interest rate of 9% per year. How much will you have at the end of 5 years, given the amount invested in the scheme is Rs.50,000? If compounding is annual? If compounding is semi annual? If compounding is quarterly? If compounding is monthly? If compounding is daily? 14

15 Future Value of lump sum (different time frame) Solution Formulae Used PV = 50,000 N = 5 r = 9% FV = PV * ( 1 + r ) N FV r ( 1 s ) = PV + m mn For Annual Compounding For Semi Annual For Quarterly Compounding Compounding r = 9% r = 9% r = 9% N = 5 N = 5 N = 5 m = 1 m = 2 m = 4 FV = 76, FV = 77, FV = 78, For Monthly Compounding For Daily Compounding r = 9% r = 9% N = 5 N = 5 m = 12 m = 365 FV = 78, FV = 78,

16 Continuous Compounding Formula Where, FV = PV * e (r s N) FV = future value of the investment N periods from today PV = present value o the investment r s = rate of interest per period e = (constant) Example A bank offers interest rate of 9% per year. How much will you have at the end of 5 years, given the amount invested in the scheme is Rs.50,000? If compounding is continuous? 16

17 Continuous Compounding Solution For Continuous Compounding PV r = = 50,000 9% N = 5 (r N) e = FV = 78, Formula FV = PV * e (r s N) 17

18 Effective Rates Formula Effective Annual Rate = ( 1 + Periodic Interest Rate ) m 1 For continuous compounding, r s EAR = e 1 Where, FV = future value of the investment N periods from today PV = present value o the investment r s = rate of interest per period m = number of compounding periods per year Example A bank offers interest rate of 9% per year compounded annually. How much will you have at the end of 2.5 years, given the amount invested in the scheme is Rs.50,000? 18

19 Effective Rates Solution Using Financial Calculator (TI BAII) Steps Press Buttons Remarks PV = 50, ND Selects the second function of the calculator 2 CLR WORK Clears memory r = 9% (Compounded Annually) Number fed on the screen N = y x Selects the raise to (index) function Number fed on the screen m = 2 6 = Carries out the operation = Formula 7 Subtraction sign pressed 8 1 Number fed on the screen Effective Annual Rate = ( 1 + Periodic Interest Rate ) m 1 9 * Multiplication sign pressed Number fed on the screen 9% = ( 1 + Eff.Semi Annual Rate) = Output obtained is Number appearing on the screen is assigneed to the I/Y 12 I/Y (Interest, remember this is a percent memory location) Eff.Semi Annual Rate = (1 + 9%) (1/2) 1 memory 13 5 Number fed on the screen Eff/ Semi Annual Rate = 4.403% 14 N Number appearing on the screen is assigneed to the N (Number of years) memory Number fed on the screen FV = PV * ( 1 + Eff. Semi Annual Rate) (m*n) Number appearing on the screen is assigneed to the PV 16 PV memory FV = 62, Number fed on the screen 18 PMT Number appearing on the screen is assigneed to the PMT (Payment or Annuity) memory 19 CPT Puts the calculator in computation mode 20 FV Computes FV for the given set of numbers 19

20 Future Value of a series of cash flow Equal Cash Flow Ordinary Annuity Formula : Example FV = A [ (1+r) N 1 ] r Future value annuity factor A bank offers interest rate of 9% per year compounded annually. An investor deposits Rs.5000 at equally spaced interval of one year for the next 5 years. What is the future value of this ordinary annuity after the last deposit at t=5? Terms An Annuity is a finite series of equal cash flows An Ordinary Annuity has a first cash flow that occurs one period from now, t =1. An Annuity Due has a first cash flow that occurs immediately, t = 0 A Perpetuity is an infinite series of equal cash flows starting from t =

21 Future Value of a series of cash flow Solution UsingFinancial Calculator (TIBAII) Future Value N =0 N = 1, r = 9% N = 2, r = 9% N = 3, r = 9% 5,000 5,450 5,941 6,475 N = 4, r = 9% 7,058 Total = 29,924 Steps Press Buttons Remarks 1 9 Number fed on the screen 2 I/Y Number appearing on the screen is assigned to the I/Y (Interest, remember this is a percent memory location) memory 3 5 Number fed on the screen Number appearing on the screen is 4 N assigneed to the N (Number of years) memory 5 0 Number fed on the screen 6 PV Number appearing on the screen is assigneed to the PV memory Number fed on the screen Number appearing on the screen is 8 PMT assigneed to the PMT (Payment or Annuity) memory 9 CPT Puts the calculator in computation mode 10 FV Computes FV for the given set of numbers 21

22 Future Value of a series of cash flow Unequal Cash Flow Ordinary Annuity At times cash flow streams are unequal In such cases the FVIFA formula cannot be used Example r = 9% Time Future Value Cash Flow at year 5 t = 1 2,000? t = 2 4,000? t = ,000? t = 4 8,000? t = 5 10,000? Sum =? What is the future value after 5 years for the given cash flow stream? 22

23 Future Value of a series of cash flow Solution Given r = 9% Compounding Future Value Cash Flow Period at year 5 t = ,823 t = ,180 t = ,129 t = ,720 t = ,000 Sum = 33,852 Formula FV = PV * ( 1 + r ) N 23

24 Present value of single cash flow Formula FV = PV * (1+r) N PV = FV / (1+r) N Where, FV = future value of the investment N periods from today PV = present value o the investment r = rate of interest per period N 1 N FV PV = FV / (1 + r) N 24

25 Present Value of lump sum Example A bank offers interest rate of 9% per year compounded annually. How much will you invest today to earn Rs.50,000 at the end of 5 years? 25

26 Present Value of lump sum Solution Given: ,000 PV = FV / (1 + r) N r = 9% FV = 50, R = 9% N = 5 PV = / (1+9%) 5 = 32, Using Financial Calculator (TI BAII) Steps Press Buttons Remarks 1 2ND Selects the second function of the calculator 2 CLR WORK Clears memory Number fed on the screen 4 FV Number appearing on the screen is assigneed to the FV memory 5 9 Number fed on the screen Number appearing on the screen is assigneed to the I/Y 6 I/Y (Interest, remember this is a percent memory location) memory 7 5 Number fed on the screen 8 N Number appearing on the screen is assigneed to the N (Number of years) memory 9 0 Number fed on the screen 10 PMT Number appearing on the screen is assigneed to the PMT (Payment or Annuity) memory 11 CPT Puts the calculator in computation mode 12 PV Computes PV for the given set of numbers 26

27 Present value of single cash flow (different annual time frame) Example Bank fixed deposit will give Rs.5,00,000 after 10 years for the investment you make today. How much will this investment grow to at the end of 4 years from now? Interest rate offered by bank is 9% per year compounded annually. 27

28 Present Value of lump sum Solution: PV = FV / ( 1 + r )N r = 9%, N = 6 At the end of 4 th year FV = 5,00,000 r = 9% N = 6 PV = / ( 1 + 9%) 6 = 2,98,

29 Present valueof single cash flow (different timeframe) Example A bank offers interest rate of 9% per year. How much will you invest today to earn Rs.50,000 at the end of 5 years? If compounding is annual? If compounding is semi annual? If compounding is quarterly? If compounding is monthly? If compounding is daily? If compounding is continuous? 29

30 Present valueof single cash flow (different timeframe) Solution Formulae Used FV = 50,000 N = 5 r = 9% PV = FV / ( 1 + r ) N PV = FV / ( 1 r + s ) m mn PV = FV / e (r sn) For Annual Compounding For Semi Annual For Quarterly Compounding Compounding r = 9% r = 9% r = 9% N = 5 N = 5 N = 5 m = 1 m = 2 m = 4 PV = 32, PV = 32, PV = 32, For Monthly For Continuous For Daily Compounding Compounding Compounding r = 9% r = 9% r = 9% N = 5 N = 5 N = 5 (r N) m = 12 m = 365 e = 1.57 PV = 31, PV = 31, PV = 31,

31 Present Value of a series of cash flows Formula PV = A A A A A (1+r) 1 + (1+r) 2 + (1+r) 3 + (1+r) (1+r) N OR 1 1/(1+r) N PV = A [ r ] Where, A = annuity amount r = interest rate per period N = number of annuity payments 31

32 Present Value of ordinary Annuity Example A bank offers interest rate of 9% per year compounded annually. An investor deposits Rs.5000 at equally spaced interval of one year for the next 5 years, payment happens at the start of year. What is the present value of this annuity? 32

33 Present Value of ordinary Annuity Solution Set the Financial Calculator (TI BAII) into (or out of) beginning mode Steps Press Buttons Remarks 1 2ND Selects the second function of the calculator Present Value ,000 N = 0, r = 9% 4,587 N = 1, r = 9% 4,208 N = 2, r = 9% 3,861 N = 3, r = 9% 2 CLR WORK Clears memory 3 2ND Selects the second function of the calculator 4 PMT Selects the second funtion "BGN" Beginning mode of calculator 5 2ND Selects the second function of the calculator 6 ENTER Sets the calculator into the beginning mode "SET" 3,542 Total = 21,199 N = 4, r = 9% 7 2ND Selects the second function of the calculator 8 CPT Selects the second funtion "QUIT" to complete the selection After setting the calculator in the beginning mode solve the question as per previously discussed steps. PMT = 5000, N = 5, I/Y =9, FV = 0, CPT PV 33

34 Projected Present Value of ordinary Annuity Example A bank offers interest rate of 9% per year compounded annually. An investor deposits Rs.5000 at equally spaced interval of one year for the 5 years, payment happens at the end of year. What is the present value of this ordinary annuity? 34

35 Projected Present Value of ordinary Annuity Solution Present Value ,587 4,208 3, ,542 3,250 Total = 19, N = 1, r = 9% N = 2, r = 9% N = 3, r = 9% N = 4, r = 9% N = 5, r = 9%

36 Present Value of Perpetuity Formula PV = A t = 1 1 (1+r) t For r >0, Example PV = A / r A bank offers interest rate of 9% per year compounded annually. An investor deposits Rs.5000 at equally spaced interval of one year till perpetuity. What is the present value of this ordinary annuity? Solution PV = 5000 / = 55,

37 Present Value of Perpetuity Example A bank offers interest rate of 9% per year compounded annually. An investor deposits Rs.5000 at equally spaced interval of one year till perpetuity. First payment starting 5 years from now. What is the present value of this ordinary annuity? Solution A A A A PV 1 = 5000 / 0.09 = PV = PV 1 / ( ) 4 = / = 39, PV 1 = A / r PV = FV / ( 1 + r ) N r = 9%, N =

38 Present Value of a series of unequal cash flow Example r = 9% Time Cash Flow Present Value t = 1 2,000? t = 2 4,000? t = 3 6,000? t = 4 8,000? t = 5 10,000? Sum =? What is the present value for the given cash flows? 38

39 Present Value of a series of unequal cash flow Solution r = 9% Time Cash Flow Present Vl Value t = 1 2,000 1,835 t = 2 4,000 3,367 t = 3 6,000 4,633 t = 4 8,000 5,667 t = 5 10,000 6,499 Sum = 22,001 Formula Used PV = FV / ( 1 + r ) N 39

40 Solving for interest rates Formula Example FV = PV * (1+r) N r = (FV / PV) (1/N) 1 To Remember In financial calculator add the PV and FV values with opposite signs ( + & ) How much is the interest rate offered by a bank, if an investment of Rs.50,000 becomes equal to Rs.1,00,000 at the end of 5 years? Solution PV = 50000, FV = , N = 5, PMT = 0 Compute I/Y = 14.87% 40

41 Solving for number of periods Example In how many years, an amount of Rs.50,000 will double, given the rate of interest is 9%? Solution PV = 50000, FV = , I/Y = 9, PMT = 0 Compute N = 8.04 years 41

42 Solving or annuity Example For a given loan of Rs.2,00,000, what will be the equated monthly installment, given an interest rate of 9% per year. Tenure of loan is 5 years? a. If compounding is annual? b. If compoundingisis monthly? Solution For Annual Compounding For Monthly Compounding PV = 200,000 PV = 200,000 FV = 0 FV = 0 N = 60 N = 60 Eff. Monthly Rate = 1.09 (1/12) 1 r = 9 / 12 = 0.721% = 0.75 PMT = 4, PMT = 4,

43 Thank You 43

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