Presentation on theme: "CHAPTER 5 Time Value of Money"— Presentation transcript:
1CHAPTER 5 Time Value of Money Future valuePresent valueAnnuitiesRates of returnAmortization
2Last week Objective of the firm Business forms Agency conflicts Capital budgeting decision and capital structure decision
3The 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
4Financial 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
5Financial 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
6Present 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 $.
7Some termsFinding the present value of some future cash flows is called discounting.Finding the future value of some current cash flows is called compounding.
8What 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
9Solving 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
10Set 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
11Solving 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
12What 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
13Solving 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
14Solving 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
15Solving for N: If your investment earns interest of 20% per year, how long before your investments double?20-12INPUTSNI/YRPVPMTFVOUTPUT3.8
16Solving 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
17Now let’s study some interesting patterns of cash flows… AnnuityPerpetuity
18ordinary annuity and annuity due PMT123i%PMT123i%Annuity Due
19Value 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.
20Solving 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
21Solving 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
22Exampleyou 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?
23Solving 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
24Solving 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
25What 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
26What 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.
27What 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
28What is the PV of this uneven cash flow stream? 10013002310%-50490.91247.93225.39-34.15= PV
29Solving 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.)
30Detailed 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= ”
31The 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?
32Solving 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
33Solving 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
34Will 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
35What 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?
36Classifications 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.
37Classifications 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
38Example, 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.
39EAR 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
40Why 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 %
41If interest is compounded more than once a year EAR (EFF, APY) will be greater than the nominal rate (APR).
44What’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.