FINANC 3000 1nd Edition Lecture 6 Outline of Last Lecture I. Moving Cash FlowsII. Rule of 72III. Finding Interest RatesIV. Solving For TimeOutline of Current Lecture I. Future Value- Multiple Cash FlowsII. Future Value- Level Cash FlowsIII. The Cost of WaitingIV. Present Value- Multiple Cash FlowsV. Present Value- Level Cash FlowsVI. PerpetuityCurrent LectureIntroduction- Time value of money calculationso Can deal with either single cash flows or multiple cash flows over time- Multiple cash flowso Regular, evenly-spaced Car loans and home mortgage loans Saving for retirement Companies paying interest on debt Companies paying dividendsFuture Value- Multiple Cash Flows- Concept: Compoundingo Value in the futureo Different cash flows paid at different times FV = future value of first cash flow + future value of second cash flow + …+future value of last cash flow FVN = [PMTm x (1+i)^(N-m)] + [PMTn x (1+i)^(N-m)] + [PMTp x (1+i)^(N-p)]These notes represent a detailed interpretation of the professor’s lecture. GradeBuddy is best used as a supplement to your own notes, not as a substitute.- Ex. Finding FV – multiple cash flowso Assumptions:  Invest $100 today (compounds for 3 years) Invest $125 at end of year 2 (compounds for 2 years) Invest $150 at end of year 3 (compounds for 1 year) Interest rates: 7%- FV3 = [$100 x (1+0.07)^3] + [$125 x (1+0.07)^2] + [$150 x (1+0.07)^1] = $426.11Future Value- Level Cash Flows- Each cash flow is the same and occurs every year- “annuity”o Occurs at the end of the first year and continues every year until the last yearo Same cash flows paid in every periodo Find value of these cash flows in the future FVAN = future value of first payment + future value of second payment +…+future value of last payment FVAN = [PMT x (1+i)^(N-1)] + [PMT x (1+i)^(N-2)] + [PMT x (1+i)^(N-3)] +…+ [PMT x (1+i)^0]Future Value of Annuity- FVA – Future Value of Annuity- Payment multiplied by compounding factoro Future value of an annuity = payment x annuity compoundingFVAN = PMT x {[(1+i)^N]-1}/i- Ex. Finding FV- Annuityo Assumptions: Invest $100 at the end of each year for 5 years Interest rates: 8%- FVA5 = $100 x {[(1+0.08)^5]-1}/0.08o =$100 x 5.8666 = $586.66Sidebar on Time Lines- Outlines each cash flow and when it occurs- If same point in time = can be added (a dollar today = a dollar today)- If different point in time = can NOT be added (a dollar today is NOT the same as tomorrow or yesterday)- Remember the 5 variables and find at least 4 of themThe Cost of Waiting- Ex. Suppose you just turned 25 and plan to save $2,000 per year for retirement, starting in one year. The account earns 12% compounded annually.- If you make your last payment on your 65th birthday, how much will you have saved for your retirement at age 65?o N=40, I/Y=12, PV=0, PMT=-$2000 FV=$1,543,182.84- Ex. Suppose you just turned 25 and plan to save $2000 per year for retirement, starting today. The account earns 12% compounded annually.- If you make your last payment on your 64th birthday, how much will you have saved for you retirement at age 65?o N=39, I/Y=12, PV=-$2000, PMT=-$2000 FV= $1,543,182.84 * 1.12 = $1,718,285- Starting sooner gained you $200,000Present Value of Multiple Cash Flows- Multiple Cash flows:o Car loans and home mortgage loanso Determining value of business opportunitieso Uneven cash flows- Present value of Multiple cash flowso Opposite of future valueo Different cash flows paid in at different timeso Finding the sum of present values of each cash flow PV = present value of first cash flow + present value of second cash flow +…+ present value of last cash flow PV = {PMTm/[(1+i)^(N-m)]} + {PMTn/[(1+i)^(N-n)]} +…+ {PMTp/[(1+i)^(N-p)]}- Ex. Finding PV – Multiple Cash Flowso Assumptions: Deposit $100 today Deposit $125 next year Deposit $150 at end of year 2 Interest rates: 7%- PV = [$100/(1+0.07)^0] + [$125/(1+0.07)^1] + [$150/(1+0.07)^2]o =$347.84Present Value- Level Cash Flows- Present value of equal cash flowso Value of future payments todayo Level cash flows paid in at different times Most loans set up with even payments throughout life of loan Present value = payment x annuity discount- PVAN = PMT x {1-[1/(1+i)^N]/i}- Ex. Finding PV- Equal cash flowso Assumptions: $100 payments at end of year for 5 years Interest rates: 8% per year- PVA5 = $100 x {1-[1/(1+0.08)^5]/0.08}o =$100 x 3.9927 = $399.27Perpetuity- Special Annuity- Perpetuityo Stream of level cash flows paid forevero Preferred stocks are an exampleo Value of investment is present value of all future annuity payments Present value of a perpetuity = payment/interest rate- PV of a perpetuity = PMT/i- Ex. Perpetuityo Present value of an annual $100 perpetuity discounted at 10% vs. present value $100 annuity for 40 years PVperpetuity = 100/0.1 = 1,000 vs. PVannuity = 977.91Ordinary Annuity vs. Annuity Due- Cash flows at beginning, not at end of period- Five annuity-due cash flows basically same as payment today plus 4-year ordinary annuity- Payments occur one period sooner than ordinary annuity – earn extra period of interestFuture Value of Annuity Due- First cash flow occurs at time 0 or today vs. at the end of the first year- Last cash flow compounds for one year vs. 0 years for ordinary annuity- Main difference: all cash flows compounds for a year moreo Future value of an annuity due = future value of an annuity x one year of compoundingo FVANdue = FVAN x (1+i)- Ex. Future Value of Annuity Dueo Assumptions: 5 annuity-due cash flows of $100 each- First cash flow compounds for 5 years- Last cash flow compounds for 1 year Interest rates: 8%- FVA5 = $100 x {[(1+0.08)^5]-1}/0.08o =$100 x 5.8666 = $586.66 =$586.66 x 1.08 = $633.59Present Value of an Annuity Due- Main difference: all cash flows get discounted for a year lesso PVANdue = PVAN x (1+i)- Ex. PV of an Annuity Dueo Assumptions: 5 annuity-due cash flows of $100 First cash flow paid today – not discounted  Last cash flow discounted 4 years All cash flows discounted for one year less than ordinary annuity Interest rates: 8%- =$399.27 x 1.08 =