Presentation on theme: "CHAPTER 5 Time Value of Money"— Presentation transcript:
1 CHAPTER 5 Time Value of Money Future valuePresent valueAnnuitiesRates of returnAmortization
2 Last week Objective of the firm Business forms Agency conflicts Capital budgeting decision and capital structure decision
3 The plan of the lecture Time value of money concepts present value (PV)discount rate/interest rate (r)Formulae for calculating PV ofperpetuityannuityInterest compoundingHow to use a financial calculator
4 Financial choices with time Which would you rather receive?$1000 today$1040 in one yearBoth payments have no risk, that is,there is 100% probability that you will be paid
5 Financial choices with time Why is it hard to compare ?$1000 today$1040 in one yearThis is not an “apples to apples” comparison. They have different units$1000 today is different from $1000 in one yearWhy?A cash flow is time-dated money
6 Present value To have an “apple to apple” comparison, we convert future payments to the present valuesor convert present payments to the future valuesThis is like converting money in Canadian $ to money in US $.
7 Some termsFinding the present value of some future cash flows is called discounting.Finding the future value of some current cash flows is called compounding.
8 What is the future value (FV) of an initial $100 after 3 years, if i = 10%? Finding the FV of a cash flow or series of cash flows is called compounding.FV can be solved by using the arithmetic, financial calculator, and spreadsheet methods.FV = ?12310%100
9 Solving for FV: The arithmetic method After 1 year:FV1 = c ( 1 + i ) = $100 (1.10) = $110.00After 2 years:FV2 = c (1+i)(1+i) = $100 (1.10) =$121.00After 3 years:FV3 = c ( 1 + i )3 = $100 (1.10) =$133.10After n years (general case):FVn = C ( 1 + i )n
10 Set up the Texas instrument 2nd, “FORMAT”, set “DEC=9”, ENTER2nd, “FORMAT”, move “↓” several times, make sure you see “AOS”, not “Chn”.2nd, “P/Y”, set to “P/Y=1”2nd, “BGN”, set to “END”P/Y=periods per year,END=cashflow happens end of periods
11 Solving for FV: The calculator method Solves the general FV equation.Requires 4 inputs into calculator, and it will solve for the fifth.310-100INPUTSNI/YRPVPMTFVOUTPUT133.10
12 What is the present value (PV) of $100 received in 3 years, if i = 10%? Finding the PV of a cash flow or series of cash flows is called discounting (the reverse of compounding).The PV shows the value of cash flows in terms of today’s worth.12310%PV = ?100
13 Solving for PV: The arithmetic method i: interest rate, or discount ratePV = C / ( 1 + i )nPV = C / ( 1 + i )3= $100 / ( 1.10 )3= $75.13
14 Solving for PV: The calculator method Exactly like solving for FV, except we have different input information and are solving for a different variable.310100INPUTSNI/YRPVPMTFVOUTPUT-75.13
15 Solving for N: If your investment earns interest of 20% per year, how long before your investments double?20-12INPUTSNI/YRPVPMTFVOUTPUT3.8
16 Solving for i: What interest rate would cause $100 to grow to $125 Solving for i: What interest rate would cause $100 to grow to $ in 3 years?3-100125.97INPUTSNI/YRPVPMTFVOUTPUT8
17 Now let’s study some interesting patterns of cash flows… AnnuityPerpetuity
18 ordinary annuity and annuity due PMT123i%PMT123i%Annuity Due
19 Value an ordinary annuity Here C is each cash paymentn is number of paymentsIf you’d like to know how to get the formula below (not required), see me after class.
20 Solving for FV: 3-year ordinary annuity of $100 at 10% $100 payments occur at the end of each period. Note that PV is set to 0 when you try to get FV.310-100INPUTSNI/YRPVPMTFVOUTPUT331
21 Solving for PV: 3-year ordinary annuity of $100 at 10% $100 payments still occur at the end of each period. FV is now set to 0.310100INPUTSNI/YRPVPMTFVOUTPUT
22 Exampleyou win the $1million dollar lottery! but wait, you will actually get paid $50,000 per year for the next 20 years if the discount rate is a constant 7% and the first payment will be in one year, how much have you actually won?
23 Solving for FV: 3-year annuity due of $100 at 10% $100 payments occur at the beginning of each period.FVAdue= FVAord(1+i) = $331(1.10) = $Alternatively, set calculator to “BEGIN” mode and solve for the FV of the annuity:BEGIN310-100INPUTSNI/YRPVPMTFVOUTPUT364.10
24 Solving for PV: 3-year annuity due of $100 at 10% $100 payments occur at the beginning of each period.PVAdue= PVAord(1+I) = $248.69(1.10) = $Alternatively, set calculator to “BEGIN” mode and solve for the PV of the annuity:BEGIN310100INPUTSNI/YRPVPMTFVOUTPUT
25 What is the present value of a 5-year $100 ordinary annuity at 10%? Be sure your financial calculator is set back to END mode and solve for PV:N = 5, I/YR = 10, PMT = 100, FV = 0.PV = $379.08
26 What if it were a 10-year annuity? A 25-year annuity? A perpetuity? N = 10, I/YR = 10, PMT = 100, FV = 0; solve for PV = $25-year annuityN = 25, I/YR = 10, PMT = 100, FV = 0; solve for PV = $Perpetuity (N=infinite)PV = PMT / i = $100/0.1 = $1,000.
27 What is the present value of a four-year annuity of $100 per year that makes its first payment two years from today if the discount rate is 9%?$297.22$323.97$ $ $ $100
28 What is the PV of this uneven cash flow stream? 10013002310%-50490.91247.93225.39-34.15= PV
29 Solving for PV: Uneven cash flow stream Input cash flows in the calculator’s “CF” register:CF0 = 0CF1 = 100CF2 = 300CF3 = 300CF4 = -50Enter I/YR = 10, press NPV button to get NPV = $ (Here NPV = PV.)
30 Detailed steps (Texas Instrument calculator) To clear historical data:CF, 2nd ,CE/CTo get PV:CF , ↓,100 , Enter , ↓,↓ ,300 , Enter, ↓,2,Enter, ↓, 50, +/-,Enter, ↓,NPV,10,Enter, ↓,CPT“NPV= ”
31 The Power of Compound Interest A 20-year-old student wants to start saving for retirement. She plans to save $3 a day. Every day, she puts $3 in her drawer. At the end of the year, she invests the accumulated savings ($1,095=$3*365) in an online stock account. The stock account has an expected annual return of 12%.How much money will she have when she is 65 years old?
32 Solving for FV: Savings problem If she begins saving today, and sticks to her plan, she will have $1,487, when she is 65.4512-1095INPUTSNI/YRPVPMTFVOUTPUT1,487,262
33 Solving for FV: Savings problem, if you wait until you are 40 years old to start If a 40-year-old investor begins saving today, and sticks to the plan, he or she will have $146, at age 65. This is $1.3 million less than if starting at age 20.Lesson: It pays to start saving early.2512-1095INPUTSNI/YRPVPMTFVOUTPUT146,001
34 Will the FV of a lump sum be larger or smaller if compounded more often, holding the stated i% constant?LARGER, as the more frequently compounding occurs, interest is earned on interest more often.12310%100133.10Annually: FV3 = $100(1.10)3 = $133.101235%456134.01100Semiannually: FV6 = $100(1.05)6 = $134.01
35 What is the FV of $100 after 3 years under 10% semiannual compounding What is the FV of $100 after 3 years under 10% semiannual compounding? Quarterly compounding?
36 Classifications of interest rates 1. Nominal rate (iNOM) – also called the APR, quoted rate, or stated rate. An annual rate that ignores compounding effects. Periods must also be given, e.g. 8% Quarterly.2. Periodic rate (iPER) – amount of interest charged each period, e.g. monthly or quarterly.iPER = iNOM / m, where m is the number of compounding periods per year. e.g., m = 12 for monthly compounding.
37 Classifications of interest rates 3. Effective (or equivalent) annual rate (EAR, also called EFF, APY) : the annual rate of interest actually being earned, taking into account compounding.If the interest rate is compounded m times in a year, the effective annual interest rate is
38 Example, EAR for 10% semiannual investment An investor would be indifferent between an investment offering a 10.25% annual return, and one offering a 10% return compounded semiannually.
39 EAR on a Financial Calculator Texas Instruments BAII Pluskeys:description:[2nd] [ICONV]Opens interest rate conversion menu[↑] [C/Y=] 2 [ENTER]Sets 2 payments per year[↓][NOM=] 10 [ENTER]Sets 10 APR.[↓] [EFF=] [CPT]10.25
40 Why is it important to consider effective rates of return? An investment with monthly payments is different from one with quarterly payments.Must use EAR for comparisons.If iNOM=10%, then EAR for different compounding frequency:Annual %Quarterly %Monthly %Daily %
41 If interest is compounded more than once a year EAR (EFF, APY) will be greater than the nominal rate (APR).
44 What’s the FV of a 3-year $100 annuity, if the quoted interest rate is 10%, compounded semiannually? 1100235%456Payments occur annually, but compounding occurs every 6 months.Cannot use normal annuity valuation techniques.