TVOM

Slide 1: 

Time is more value than money. You can get more money, but you cannot get more time…….Jim Rohn RUSHNA AWAN & SAJID MIR SHAIKH

Slide 2: 

2 Future value Present Value Ordinary Annuity and Annuity Due Nominal Interest rate. Effective Annual rate EAR) Amortization 2/8/2009 11:30:18 AM

What is The Time Value of Money? : 

3 What is The Time Value of Money? A dollar received today is worth more than a dollar received tomorrow This is because a dollar received today can be invested to earn interest The amount of interest earned depends on the rate of return that can be earned on the investment. Time value of money quantifies the value of a dollar through time 2/8/2009 11:30:18 AM

Uses of Time Value of Money : 

4 Uses of Time Value of Money Time Value of Money, or TVM, is a concept that is used in all aspects of finance including: Bond valuation Stock valuation Accept/reject decisions for project management Financial analysis of firms And many others! 2/8/2009 11:30:18 AM

Basic Time Value Concepts : 

Notes Leases Pensions and Other Postretirement Benefits Long-Term Assets Basic Time Value Concepts Sinking Funds Business Combinations Disclosures Installment Contracts 2/8/2009 11:30:18 AM 5 Applications to Topics:

Basic Rules : 

6 Basic Rules The following are simple rules that you should always use no matter what type of TVM problem you are trying to solve: Stop and think: Make sure you understand what the problem is asking. You will get the wrong answer if you are answering the wrong question. Draw a representative timeline and label the cash flows and time periods appropriately. Write out the complete formula using symbols first and then substitute the actual numbers to solve. Check your answers using a calculator. 2/8/2009 11:30:18 AM

Why TIME? : 

TIME allows one the opportunity to postpone consumption and earn INTEREST. NOT having the opportunity to earn interest on money is called OPPORTUNITY COST. Why TIME? 2/8/2009 11:30:18 AM 7

The Interest Rate : 

Obviously, $10,000 today. You already recognize that there is TIME VALUE TO MONEY!! The Interest Rate Which would you prefer -- $10,000 today or $10,000 in 5 years? 2/8/2009 11:30:18 AM 8

Types of Interest : 

Types of Interest Compound Interest Interest paid (earned) on any previous interest earned, as well as on the principal borrowed (lent). Simple Interest Interest paid (earned) on only the original amount, or principal, borrowed (lent). 2/8/2009 11:30:18 AM 9

Simple Interest Formula : 

Simple Interest Formula Formula SI = P0(i)(n) SI: Simple Interest P0: Deposit today (t=0) i: Interest Rate per Period n: Number of Time Periods 2/8/2009 11:30:18 AM 10

Simple Interest Example : 

SI = P0(i)(n) = $1,000(.07)(2) = $140 Simple Interest Example Assume that you deposit $1,000 in an account earning 7% simple interest for 2 years. What is the accumulated interest at the end of the 2nd year? 2/8/2009 11:30:18 AM 11

Frequency of Compounding : 

General Formula: FVn = PV0(1 + [i/m])mn n: Number of Years m: Compounding Periods per Year i: Annual Interest Rate FVn,m: FV at the end of Year n PV0: PV of the Cash Flow today Frequency of Compounding 2/8/2009 11:30:18 AM 12

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13 The Millionaire Game: A Perfect Metaphor For Compound Interest 2/8/2009 11:30:18 AM

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14 A: Answer A C: Answer C B: Answer B D: Answer D 50:50 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1 $1 Million $500,000 $250,000 $125,000 $64,000 $32,000 $16,000 $8,000 $4,000 $2,000 $1,000 $500 $300 $200 $100 2/8/2009 11:30:18 AM

Impact of Frequency : 

Julie Miller has $1,000 to invest for 2 Years at an annual interest rate of 12%. Annual FV2 = 1,000(1+ [.12/1])(1)(2) = 1,254.40 Semi FV2 = 1,000(1+ [.12/2])(2)(2) = 1,262.48 Impact of Frequency 2/8/2009 11:30:18 AM 15

Impact of Frequency : 

Qrtly FV2 = 1,000(1+ [.12/4])(4)(2) = 1,266.77 Monthly FV2 = 1,000(1+ [.12/12])(12)(2) = 1,269.73 Daily FV2 = 1,000(1+[.12/365])(365)(2) = 1,271.20 Impact of Frequency 2/8/2009 11:30:18 AM 16

Slide 17: 

17 Time lines show timing of cash flows. Tick marks at ends of periods. Time 0 is today; Time 1 is the end of Period 1; or the beginning of Period 2. 90% of getting a Time Value problem correct is setting up the timeline correctly!!! Interest Rate Cash Flows 2/8/2009 11:30:18 AM

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18 100 0 1 2 Year i% Time line for a $100 lump sum due at the end of Year 2. 100 100 100 0 1 2 3 i% Time line for an ordinary annuity of $100 for 3 years. 100 50 75 0 1 2 3 i% -50 Time line for uneven CFs: -$50 at t = 0 and $100, $75, and $50 at the end of Years 1 through 3. 2/8/2009 11:30:18 AM

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Translate Rs.1 today into its equivalent in the future (compounding). Translate Rs.1 in the future into its equivalent today (discounting). 2/8/2009 11:30:18 AM 19

FUTURE VALUE : 

20 FUTURE VALUE 2/8/2009 11:30:18 AM

Single Sum - Future & Present Value : 

21 Single Sum - Future & Present Value Assume can invest PV at interest rate i to receive future sum, FV Similar reasoning leads to Present Value of a Future sum today. 2/8/2009 11:30:18 AM

What’s the FV of an initial Rs.1000 after 1 years if i = 10%? : 

What’s the FV of an initial Rs.1000 after 1 years if i = 10%? FV = ? 0 1 10% Finding FVs (moving to the right on a time line) is called compounding. + INTREST 2/8/2009 11:30:18 AM 22

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After 1 year: FV1 = PV + INT1 = PV + PV (i) = PV(1 + i) = Rs.1000(1.10) = Rs.1100.00. After 2 years: FV2 = FV1(1+i) = PV(1 + i)(1+i) = PV(1+i)2 = Rs.1000(1.10)2 = Rs.1210.00. 2/8/2009 11:30:18 AM 23

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After 3 years: FV3 = FV2(1+i)=PV(1 + i)2(1+i) = PV(1+i)3 = Rs.1000(1.10)3 = Rs.1330.10. In general, FVn = PV(1 + i)n. 2/8/2009 11:30:18 AM 24

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Spreadsheet Solution Use the FV function = FV(Rate, Nper, Pmt, PV) = FV(0.10, 3, 0, -1000) = 1330.10 2/8/2009 11:30:18 AM 25

What’s the PV of $100 due in 3 years if i = 10%? : 

10% What’s the PV of $100 due in 3 years if i = 10%? Finding PVs is discounting, and it’s the reverse of compounding. 100 0 1 2 3 PV = ? 2/8/2009 11:30:18 AM 26

Slide 27: 

Solve FVn = PV(1 + i )n for PV: ? ? PV = $100 1 1.10 = $100 0.7513 = $75.13. ? ? ? ? ? ? 3 2/8/2009 11:30:18 AM 27

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Spreadsheet Solution Use the PV function: = PV(Rate, Nper, Pmt, FV) = PV(0.10, 3, 0, 100) = -75.13 2/8/2009 11:30:18 AM 28

Slide 29: 

Finding the Time to Double 20% 2 0 1 2 ? -1 FV = PV(1 + i)n $2 = $1(1 + 0.20)n (1.2)n = $2/$1 = 2 nLN(1.2) = LN(2) n = LN(2)/LN(1.2) n = 0.693/0.182 = 3.8. 2/8/2009 11:30:18 AM 29

Slide 30: 

Spreadsheet Solution Use the NPER function: see spreadsheet. = NPER(Rate, Pmt, PV, FV) = NPER(0.10, 0, -1, 2) = 3.8 2/8/2009 11:30:18 AM 30

Slide 31: 

Finding the Interest Rate ?% 2 0 1 2 3 -1 FV = PV(1 + i)n $2 = $1(1 + i)3 (2)(1/3) = (1 + i) 1.2599 = (1 + i) i = 0.2599 = 25.99%. 2/8/2009 11:30:18 AM 31

Slide 32: 

Spreadsheet Solution Use the RATE function: = RATE(Nper, Pmt, PV, FV) = RATE(3, 0, -1, 2) = 0.2599 2/8/2009 11:30:18 AM 32

Types of Annuities : 

Types of Annuities Ordinary Annuity: Payments or receipts occur at the end of each period. Annuity Due: Payments or receipts occur at the beginning of each period. An Annuity represents a series of equal payments (or receipts) occurring over a specified number of equidistant periods. 2/8/2009 11:30:18 AM 33

Examples of Annuities : 

Examples of Annuities Student Loan Payments Car Loan Payments Insurance Premiums Mortgage Payments Retirement Savings 2/8/2009 11:30:18 AM 34

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Ordinary Annuity PMT PMT PMT 0 1 2 3 i% PMT PMT 0 1 2 3 i% PMT Annuity Due What’s the difference between an ordinary annuity and an annuity due? PV FV 2/8/2009 11:30:18 AM 35

What is the future value of an Ordinary Annuity? : 

What is the future value of an Ordinary Annuity? 2/8/2009 11:30:18 AM 36

Slide 37: 

FV Annuity Formula The future value of an annuity with n periods and an interest rate of i can be found with the following formula: 2/8/2009 11:30:18 AM 37

Slide 38: 

Spreadsheet Solution Use the FV function = FV(Rate, Nper, Pmt, Pv) = FV(0.12, 3, -100, 0) = 337.44 2/8/2009 11:30:18 AM 38

Present Value of an Annuity : 

Present Value of an Annuity 2/8/2009 11:30:18 AM 39

Slide 40: 

PV Annuity Formula The present value of an annuity with n periods and an interest rate of i can be found with the following formula: 2/8/2009 11:30:18 AM 40

Slide 41: 

Spreadsheet Solution Use the PV function: see spreadsheet. = PV(Rate, Nper, Pmt, Fv) = PV(0.12, 3, 100, 0) = -240.183 2/8/2009 11:30:18 AM 41

Perpetuities : 

Perpetuities Suppose you will receive a fixed payment every period (month, year, etc.) forever. This is an example of a perpetuity. You can think of a perpetuity as an annuity that goes on forever. 2/8/2009 11:30:18 AM 42

Slide 43: 

What should you be willing to pay in order to receive $10,000 annually forever, if you require 8% per year on the investment? 2/8/2009 11:30:18 AM 43

Find the FV and PV if theannuity were an annuity due. : 

Find the FV and PV if theannuity were an annuity due. 100 100 0 1 2 3 10% 100 2/8/2009 11:30:18 AM 44

Slide 45: 

PV and FV of Annuity Due vs. Ordinary Annuity PV of annuity due: = (PV of ordinary annuity) (1+i) = (248.69) (1+ 0.10) = 273.56 FV of annuity due: = (FV of ordinary annuity) (1+i) = (331.00) (1+ 0.10) = 364.1 2/8/2009 11:30:18 AM 45

Excel Function for Annuities Due : 

Excel Function for Annuities Due Change the formula to: =PV(10%,3,-100,0,1) The fourth term, 0, tells the function there are no other cash flows. The fifth term tells the function that it is an annuity due. A similar function gives the future value of an annuity due: =FV(10%,3,-100,0,1) 2/8/2009 11:30:18 AM 46

What is the PV of this uneven cashflow stream? : 

What is the PV of this uneven cashflow stream? 0 100 1 300 2 300 3 10% -50 4 90.91 247.93 225.39 -34.15 530.08 = PV 2/8/2009 11:30:18 AM 47

Nominal rate (iNom) : 

Nominal rate (iNom) Stated in contracts, and quoted by banks and brokers. Not used in calculations or shown on time lines Periods per year (m) must be given. Examples: 8%; Quarterly 8%, Daily interest (365 days) 2/8/2009 11:30:18 AM 48

Periodic rate (iPer ) : 

Periodic rate (iPer ) iPer = iNom/m, where m is number of compounding periods per year. m = 4 for quarterly, 12 for monthly, and 360 or 365 for daily compounding. Used in calculations, shown on time lines. Examples: 8% quarterly: iPer = 8%/4 = 2%. 8% daily (365): iPer = 8%/365 = 0.021918%. 2/8/2009 11:30:18 AM 49

Will the FV of a lump sum be larger or smaller if we compound more often, holding the stated I% constant? Why? : 

Will the FV of a lump sum be larger or smaller if we compound more often, holding the stated I% constant? Why? LARGER! If compounding is more frequent than once a year--for example, semiannually, quarterly, or daily--interest is earned on interest more often. 2/8/2009 11:30:18 AM 50

FV Formula with Different Compounding Periods (e.g., $100 at a 12% nominal rate with semiannual compounding for 5 years) : 

FV Formula with Different Compounding Periods (e.g., $100 at a 12% nominal rate with semiannual compounding for 5 years) = $100(1.06)10 = $179.08. FV = PV 1 . + i m n Nom mn ? ? ? ? ? ? FV = $100 1 + 0.12 2 5S 2x5 ? ? ? ? ? ? 2/8/2009 11:30:18 AM 51

FV of $100 at a 12% nominal rate for 5 years with different compounding : 

FV of $100 at a 12% nominal rate for 5 years with different compounding FV(Annual)= $100(1.12)5 = $176.23. FV(Semiannual)= $100(1.06)10=$179.08. FV(Quarterly)= $100(1.03)20 = $180.61. FV(Monthly)= $100(1.01)60 = $181.67. FV(Daily) = $100(1+(0.12/365))(5x365) = $182.19. 2/8/2009 11:30:18 AM 52

Annual Effective Rate : 

Annual Effective Rate 2/8/2009 11:30:18 AM 53

Annual Effective Rate : 

Annual Effective Rate Interest rates quoted by three banks: Bank X: 15%, compounded daily Bank Y: 15.5%, compounded quarterly Bank Z: 16%, compounded annually 2/8/2009 11:30:18 AM 54

Annual Effective Rate : 

Annual Effective Rate 2/8/2009 11:30:18 AM 55

Can the effective rate ever be equal to the nominal rate? : 

Can the effective rate ever be equal to the nominal rate? Yes, but only if annual compounding is used, i.e., if m = 1. If m > 1, EFF% will always be greater than the nominal rate. 2/8/2009 11:30:18 AM 56

When is each rate used? : 

When is each rate used? iNom: Written into contracts, quoted by banks and brokers. Not used in calculations or shown on time lines. 2/8/2009 11:30:18 AM 57

Slide 58: 

iPer: Used in calculations, shown on time lines. If iNom has annual compounding, then iPer = iNom/1 = iNom. 2/8/2009 11:30:18 AM 58

Slide 59: 

(Used for calculations if and only if dealing with annuities where payments don’t match interest compounding periods.) EAR = EFF%: Used to compare returns on investments with different payments per year. 2/8/2009 11:30:18 AM 59

Steps to Amortizing a Loan : 

1. Calculate the payment per period. 2. Determine the interest in Period t. (Loan Balance at t-1) x (i% / m) 3. Compute principal payment in Period t. (Payment - Interest from Step 2) 4. Determine ending balance in Period t. (Balance - principal payment from Step 3) 5. Start again at Step 2 and repeat. Steps to Amortizing a Loan 2/8/2009 11:30:18 AM 60

Amortizing a Loan Example : 

Julie Miller is borrowing $10,000 at a compound annual interest rate of 12%. Amortize the loan if annual payments are made for 5 years. Step 1: Payment PV0 = R (PVIFA i%,n) $10,000 = R (PVIFA 12%,5) $10,000 = R (3.605) R = $10,000 / 3.605 = $2,774 Amortizing a Loan Example 2/8/2009 11:30:18 AM 61

Amortizing a Loan Example : 

Amortizing a Loan Example [Last Payment Slightly Higher Due to Rounding]

Slide 63: 

iPer = 11.33463%/365 = 0.031054% per day. FV=? 0 1 2 273 0.031054% -100 Note: % in calculator, decimal in equation. ( ) ( ) FV = $100 1.00031054 = $100 1.08846 = $108.85. 273 273 2/8/2009 11:30:18 AM 63

Slide 64: 

64 ANY QUESTION 2/8/2009 11:30:18 AM