CHAPTER 6 Time Value of Money,Future value Present value Annuities Rates of return Amortization,Time lines,Show the timing of cash flows. Tick marks occur at the end of periods, so Time 0 is today; Time 1 is the end of the first period (year, month, etc.) or the beginning of the second period.,CF0,CF1,CF3,CF2,0,1,2,3,i%,Drawing time lines: $100 lump sum due in 2 years; 3-year $100 ordinary annuity,Drawing time lines: Uneven cash flow stream; CF0 = -$50, CF1 = $100, CF2 = $75, and CF3 = $50,What is the future value (FV) of an initial $100 after 3 years, if I/YR = 10%?,Finding the FV of a cash flow or series of cash flows when compound interest is applied is called compounding. FV can be solved by using the arithmetic, financial calculator, and spreadsheet methods.,Solving for FV: The arithmetic method,After 1 year: FV1 = PV ( 1 + i ) = $100 (1.10) = $110.00 After 2 years: FV2 = PV ( 1 + i )2 = $100 (1.10)2 =$121.00 After 3 years: FV3 = PV ( 1 + i )3 = $100 (1.10)3 =$133.10 After n years (general case): FVn = PV ( 1 + i )n,Solving for FV: The calculator method,Solves the general FV equation. Requires 4 inputs into calculator, and will solve for the fifth. (Set to P/YR = 1 and END mode.),INPUTS,OUTPUT,N,I/YR,PMT,PV,FV,3,10,0,133.10,-100,PV = ?,100,What is the present value (PV) of $100 due in 3 years, if I/YR = 10%?,Finding the PV of a cash flow or series of cash flows when compound interest is applied is called discounting (the reverse of compounding). The PV shows the value of cash flows in terms of todays purchasing power.,0,1,2,3,10%,Solving for PV: The arithmetic method,Solve the general FV equation for PV: PV = FVn / ( 1 + i )nPV = FV3 / ( 1 + i )3= $100 / ( 1.10 )3= $75.13,Solving for PV: The calculator method,Solves the general FV equation for PV. Exactly like solving for FV, except we have different input information and are solving for a different variable.,INPUTS,OUTPUT,N,I/YR,PMT,PV,FV,3,10,0,100,-75.13,Solving for N: If sales grow at 20% per year, how long before sales double?,Solves the general FV equation for N. Same as previous problems, but now solving for N.,INPUTS,OUTPUT,N,I/YR,PMT,PV,FV,3.8,20,0,2,-1,What is the difference between an ordinary annuity and an annuity due?,Solving for FV: 3-year ordinary annuity of $100 at 10%,$100 payments occur at the end of each period, but there is no PV.,INPUTS,OUTPUT,N,I/YR,PMT,PV,FV,3,10,-100,331,0,Solving for PV: 3-year ordinary annuity of $100 at 10%,$100 payments still occur at the end of each period, but now there is no FV.,INPUTS,OUTPUT,N,I/YR,PMT,PV,FV,3,10,100,0,-248.69,Solving for FV: 3-year annuity due of $100 at 10%,Now, $100 payments occur at the beginning of each period. Set calculator to “BEGIN” mode.,INPUTS,OUTPUT,N,I/YR,PMT,PV,FV,3,10,-100,364.10,0,Solving for PV: 3 year annuity due of $100 at 10%,Again, $100 payments occur at the beginning of each period. Set calculator to “BEGIN” mode.,INPUTS,OUTPUT,N,I/YR,PMT,PV,FV,3,10,100,0,-273.55,What is the PV of this uneven cash flow stream?,Solving for PV: Uneven cash flow stream,Input cash flows in the calculators “CFLO” register: CF0 = 0 CF1 = 100 CF2 = 300 CF3 = 300 CF4 = -50 Enter I/YR = 10, press NPV button to get NPV = $530.09. (Here NPV = PV.),Solving for I: What interest rate would cause $100 to grow to $125.97 in 3 years?,Solves the general FV equation for I.,INPUTS,OUTPUT,N,I/YR,PMT,PV,FV,3,8,0,125.97,-100,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) 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?,Solving for FV: Savings problem,If she begins saving today, and sticks to her plan, she will have $1,487,261.89 when she is 65.,INPUTS,OUTPUT,N,I/YR,PMT,PV,FV,45,12,-1095,1,487,262,0,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 $146,000.59 at age 65. This is $1.3 million less than if starting at age 20. Lesson: It pays to start saving early.,INPUTS,OUTPUT,N,I/YR,PMT,PV,FV,25,12,-1095,146,001,0,Solving for PMT: How much must the 40-year old deposit annually to catch the 20-year old?,To find the required annual contribution, enter the number of years until retirement and the final goal of $1,487,261.89, and solve for PMT.,INPUTS,OUTPUT,N,I/YR,PMT,PV,FV,25,12,-11,154.42,1,487,262,0,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.,Annually: FV3 = $100(1.10)3 = $133.10,Semiannually: FV6 = $100(1.05)6 = $134.01,Classifications of interest rates,Nominal rate (iNOM) also called the quoted or state rate. An annual rate that ignores compounding effects. iNOM is stated in contracts. Periods must also be given, e.g. 8% Quarterly or 8% Daily interest. 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. m = 4 for quarterly and m = 12 for monthly compounding.,