Management 3 Quantitative Methods. The Time Value of Money Part 1A

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Geraldine Hawkins
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1 Management 3 Quantitative Methods The Time Value of Money Part 1A

2 Money: Today? Tomorrow? money now is not the same as money tomorrow Money now is better: It can be used now; It can invested now; There is no waiting, no uncertainty. the difference b/w monry now and money tomorrow is the time value of money.

3 The TVM is a Pricing Problem what price do we put on Time? How much more does a buyer of today s dollars pay the lender with future dollars? Obviously, the price for $1 present dollar is going to be at least $1 future dollar plus a few cents each year. You can think of this as renting dollars.

4 The Formulation $ Future Dollar = $ Present Dollar x (Something > 1) invested because the Present Dollar can be lent or $ Future Dollar = $ Present Dollar x (1 + the rate of return) $ Future Dollar = $ Present Dollar x (1 + r) for each period $ Future Dollar = $ Present Dollar x (1 + r) t years

5 The Basic Relationship The earning power of a dollar in hand today is: [1] FV = PV x (1+r) t where there are two $dollar values: 1. FV = future value in $dollars 2. PV = present value, $ dollars in hand today Which are connected by: Time t and A rate of interest r.

6 The Basic Relationship This is the Compounding process: [1] FV = PV x (1+r) t notice that there are four parameters: 1. FV = future value in $dollars 2. PV = present value, the value of $ dollars in hand today 3. r = an annual rate of return or an interest rate 4. t = number of years between today and the future

7 Calculator Keys Exponent y x x ^ Inverse 1/x Change sign +/ ( ) Natural Log ln

8 What is the FV of $100 in PV? Ask yourself, if I borrowed $100 today at 10% interest, how much would I need to pay back in 3 years?

9 The answer is that it depends on how you actually calculate interest simple or compound? Let s say simple. The answer is: 1. $ $ $

10 The answer is $ $ is your Principal. $ is simple interest = 10% x $100 x 3.

11 Now, let s say compound The answer is: 1. $ $ $

12 The answer is $ $ is your Principal. $ is simple interest = 10% x $100 3 times. $ 3.10 is compound interest = 1) $ 2.00 on the first $10 of simple interest 2) $ 1.00 on the second $10 of simple interest 3) $ 0.10 on the first $1.00 of interest.

13 I have $100 (PV) and I lend it for 3 years at 10 percent interest. I n t e r e s t Principal Simple Compound Total Value time 10% 0 $ $ $ $ $ $ $ $ 1.00 $ $ $ $ $ 2.10 $ $ Note that the FVF for t = 3 and r = 0.10 is = 1.331

14 Simple versus compound interest Simple interest is interest only on the original principal, not on any accrued interest. Compound interest is interest on simple interest called interest oninterest thus compounding.

15 Find these on the FVF Tables Notation Formula Factor FVF (10%, 7) (1.10)^7? FVF ( 7%, 10) (1.07)^10? FVF (5%, 7) (1.05)^7? FVF (7%, 5) 1 / (1.07)^5?

16 Find these on the FVF Tables Notation Formula Factor FVF (10%, 7) (1.10)^ FVF ( 7%, 10) (1.07)^ FVF ( 5%, 7) 1 / (1.05)^ FVF ( 7%, 5) 1 / (1.07)^

17 Notice that FVF s are > 1 The Table runs from smallest at the NW corner short term & low rate to largest at the southeast corner long term & high rate.

18 Doubling Your Money Can you find a few combinations if time & rate that will double your money? 1) 2) 3) 4)

19 The Rule of 72 Doubling Time approximately = 72 / annual interest rate Years Rate % % % % % % % % % % % % % % % % % % % % % % % % % % % %

20 Derive the PV from the basic, the FV, equation [1] FV = PV x (1+r) t where (1+r) t is a FVF [2] PV = FV x 1/(1+r) t = / FV x (1+r) t The second term on the RHS of the PV equation gives is the PVF

21 Symmetry We have just reversed the Compounding process by inverting the equation [1] to get: [2] PV = FV x 1/(1+r) t This is called the Discounting process. The PVF s are called discount factors.

22 Find these PVF s on the Tables Notation Formula Factor PVF (10%, 7) PVF ( 7%, 10) PVF (5%, 7) PVF (7%, 5) (1.10)^7 (1.07)^10 1 / (1.05)^7 1 / (1.07)^5

23 Notice that PVF s are < 1 The PVF Table runs from largest at the NW corner short term & low rate to largest at the southeast corner long term & high rate. The PVF Table is the inverse of the FVF Table.

24 What are Factors Factors are magic numbers that convert: Present dollars into future dollars, or Future dollars into present dollars.

25 Verifying the Symmetry b/w FV and PV What is the FV of 5% for 5years? FV = PV x (1+r) t FV = $ 100 x (1+5%) 5 FV = $ 100 x FV = $ What is the PV of $ % in 5years?

26 Verifying the Symmetry b/w FV and PV What is the PV of $ % in 5years? PV = FV x 1/(1+r) t PV = $ x 1/(1+5%) 5 PV = $ x PV = $ This is because the PVF(5%, 5) = 1/FVF(5%,5)

27 Time Dollars $100 $ We are moving single amounts over several periods of time. Here are the intermediate steps: Time Dollars $100 $105 $ $ $

28 Intra Period Compounding: more than once per year. Quarterly compounding. use t = 4x and r = r/4 Monthly compounding. use t = 12x and r = r/12 Daily compounding. use t = 360x and r = r/360

29 One year FVF s using 12 percent Quarterly compounding. (1 +.12/4) ^4 = 1.03^4 or 12.55% Monthly compounding. (1 +.12/12) ^12 = 1.01^12 or 12.68% Daily compounding. (1 +.12/360) ^360 = ^360 or 12.75%

30 If you put $100 in the bank at 4% percent interest compounded (1) annually, (2) quarterly, (3) monthly, (4) daily then how much will you have after one year?

31 Notation for FVF FVF Formula FVF Factor FVF ( 4%, 1) ( )^ FVF ( 1%, 4 ) ( )^ FVF (0.33%,12) ( )^ FVF (0.011%, 360) ( )^

32 If you put $100 in the bank at 5 percent interest compounded daily, then how much will you have your will have after one year? Use ( /360)^360 or $ This is 13 cents more than with annual compounding

33 A.P.R. versus A.P.Y. Annual Percentage Rate is the basic r % published for the compounding process. Annual Percentage Yield is the r % resulting from the actual daily compounding therefore, If APR = 5% then APY = {(1+APR/360)^360 } 1 = {( )^360} 1 = = = 5.13%

Time Value of Money. Ex: How much a bond, which can be cashed out in 2 years, is worth today

Time Value of Money. Ex: How much a bond, which can be cashed out in 2 years, is worth today Time Value of Money The time value of money is the idea that money available now is worth more than the same amount in the future - this is essentially why interest exists. Present value is the current