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# FINA 4920 Chapter 2 Textbook Notes FINA 4920E

UGA

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This 15 page Class Notes was uploaded by Morgan Notetaker on Sunday August 28, 2016. The Class Notes belongs to FINA 4920E at University of Georgia taught by Chris Pope in Fall 2016. Since its upload, it has received 6 views. For similar materials see Financial Modeling in Finance at University of Georgia.

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Date Created: 08/28/16

Chapter 2: Time Value of Money (TVM) with Excel Learning Objectives: Find the FV of a lump sum, series of cash flows, and annuity Apply FV function in Excel to find the FV of a cash flow stream Graph the FV of cash flow streams Perform sensitivity analysis using one way and two way tables Excel Skills: Excel Functions: FV, PV, EFFECT, EXP Data table: one way and two way data tables Making a graph Formatting graph I. Introduction Firm’s goal is to maximize shareholder wealth Real World: Facebook acquires WhatsApp for $19 billion. Investors didn’t understand why they paid so much for something with less than $1 billion in revenue. Facebook was broadening their services to maintain subscribers Will reference this example throughout chapter II. Future Value Calculation A. Single Period Investment One-time cash flow= Lump sum Either invest cash flow (CF) today and find its value in one year or find value today of a CF we will get in one year Basic Terms for TVM: PV= present value= value today of an investment FV= future value= value of investment at end of time period N= time period= when a CF occurs (N=0=today, N=1= one year form today, etc.) r= rate of interest= rate of return on investment/ discount rate FV= PV x (1+r) FV of a lump sum; value of initial CF after it earns interest Ex. Traveler invests $1000 today and gets 10% interest annually. What is account worth in one year? PV= 1000, r= 10$, N=1, solve for FV FV= 1000 x (1+.10)= 1100 B. Multiple Period Investing Simple Interest: only earn interest on the principal/ initial amount; principal remains unchanged Ex. Invest $100 today and earn 5% annually. What is account value in two years with simple interest? First year: earn $5 interest ($100x5%). Second year: earn $5 interest again ($100x5%) Balance after two years: $100+ $5+ $5= $110 Compound Interest: interest is re-invested each period and added to the principal; interest earns interest; results in larger FV’s Ex. Invest $100 today and earn 5% annually. What is account value in two years with compound interest? First year: earn $5 interest ($100x5%). Add to principal. Second year: earn $5.25 interest ($105x5%). Balance after two years: $100 + $5 + $5.25= $110.25 Difference between simple and compound increases the more interest is compounded C. Future Value of a Lump Sum Ex. Traveler will take trip in 2 years. Invests $1000 today and earns 10% interest annually. What is account worth in two years? n FVn= PV x (1+r) PV= 1000, N= 2, r=10%, FV=? FV2= 1000 x (1.10) = $1,210 Future Value of a Lump Sum in Excel Absolute Reference= specific cell that you need in each calculation Ex. The absolute references for finding the FV are the cells containing the interest rate and the PV (these don’t change) Put dollar signs on part of the cell reference that wen want to lock Highlight cells in the formula that you want to make unchanged and press F4 key to lock them Relative Reference= when you want the cell to adjust by column or row as we copy the formula To copy a formula down (form year 1- year x), click on the first cell with the formula, put mouse over the small square in the corner of the cell, hold and drag to the end of the column. This copies the formula to each cell you dragged it through The absolute references will stay the same in each cell The relative references will result in “N” increasing as you increase in time going down the column OR double click after putting the cursor on the little square to copy and paste it quicker Graphing in Excel Highlight relevant data (Year and FV in this example) Go to Insert at top of spreadsheet Choose Charts Choose Scatter (with Straight likes and Markers) In Excel, the first column is the x-variable and the second column is the y-variable Click on titles to change them Adjust x-axis to go from year 1 to year X (default graph goes from 0-20): Click on axes numbers, click on box that forms around the numbers, choose “Format Axis”, change minimum to 1 and maximum to X.; Format y-axis the same way D. Future Value With Annual Deposits Annuity= series of equal payments that last for a specified period of time Ex. Loans, paychecks, rent payments Have maturity date (end date; set number of payments Perpetuity= an annuity with no ending point/ maturity date Includes preferred stock Ordinary perpetuity means payments are made/received at the end of the period Future Value of an Ordinary Annuity You invest series of CF’s and arrive at a future balance TVM: can only add up cash flows if they are valued at the same point in time PMT N F V= r x(1+r )−1 ) Future Value of Annuity Due Payments are paid/ received at beginning of the period You get CF one period sooner than an ordinary annuity, so it’s worth one extra period of interest Seen in lease/ rental agreement, retirement withdrawals F V= PMT x(1+r) −1 )x(1+r) r Ex. You invest $1000 and leave it in the bank. You intend to retire in 15 years and travel the world after retirement. You plan to make 15 annual deposits for $1000, with the first deposit made at time 0 (today) and each succeeding deposit made at the end of the year. Assume an interest rate of 10%. ** This is an annuity due because you make the first contribution to the account today (at the beginning of the period) FV= PMT x(1+r) −1 )x(1+r) r FV= 1000 x 1+0.1 −1 x 1+0.1 =$34,949.73 0.1 (( ) ) ( ) Future Value of Annuity Due in Excel Make a table stating the Annual Investment ($1000), Interest Rate (10%) and Number of Years (15). Beginning balance of Year 1 is 0 because there is no balance in account at day 0. For the deposit at Year 1, use an absolute reference to the Annual Investment cell ($1000). This number doesn’t change annually, so use absolute reference. To find the interest earned in the first year, reference the interest rate cell (10%) and multiply by the total invested for the year (deposit cell and beginning balance cell). Find Year 1’s ending balance by adding the total money invested (Beginning balance and deposit) with the interest earned. Ending balance of year 1 is beginning balance of year 2 Under Beginning Balance of Year 2, make sure to put that it equal the Ending Balance of Year 1. Double click on the bottom right corners of each formula to fill in the rest of the table. It should look like this: Future Value Formula in Excel =FV (rate, nper, pmt, [pb], [type]) Rate= r, nper= N type= 0 for ordinary annuity (paid at end of period) and 1 for annuity due (paid at beginning of period) Same example above would be… = FV(10%,15,1000,0,1) One-Way Table in Excel: What is FV with different interest rates One Way table is good for performing sensitivity analysis when you have one variable that has range of possible values Ex. Assume interest rate might be difference than what we assumed First create column with possible interest rates (0%-15%) Set the cell above the interest rate column equal to the FV function value Highlight entire grid from row above 0% Click on Data, Data Table Put Interest Rate cell reference in the Column Input Cell section; Click OK Should Look Like This: Two-Way Table in Excel: What is FV with different interest and Deposits Create row of all possible indifferent annual deposits. Create column with all possible interest rates. Make the cell above the first interest rate in the same column reference the FV formula value Highlight the entire table, Click Data, Click Data Table Put annual investment cell in row input cell. Put original interest rate in column input cell. Click Ok. Should look like this: **Excel is plugging in every combination of interest rate and annual contribution into the FV function **For data table to work, you must use cell references for the equations and calculations needed to get the results III. Present Value Calculation What is the value today of a future cash flow? How much do I need to invest today to reach a goal in the future? A. Present Value of a Lump Sum Values a single CF in the future and values it in today’s dollars FV N PV= N (1+r) EX. Man wants to get $5000 for his honeymoon two years from today. He earns 9% annually on his investments. How much will he need to invest today to reach his goal? FV= 5000, N= 2, I= 9, Pmt=0, PV=? FV $5,000 PV= NN = 2=$4,208.40 (1+r) (1.09) In other words, you can either give me $5000 in two years or $4208.40 today. They are equivalent. Present Value of Lump Sum in Excel One approach: Use Excel as calculator = 5000/1.09 2 Use Data Tables to figure out how different inputs affect the value. How much money do I need to invest to in order to get $5000 in two years if the interest rate is different? Enter range of interest rates from 1% to 15%. Set the cell one column to the right and one row up from the interest rate column equal to the present value from B5 Highlight the whole table, click Data, Data Table Refer the original interest rate cell to the column input cell. Press OK. It should look like: Present Value and Graphing Higher interest rates make for lower present values Highlight relevant data, click Insert tab, click Charts, Scatter, Straight line and Markers NOTE: You may notice that your graph does not look like the one in the book Double click the y- axis and format like below: Double click the x-axis and format like below: B. Present Value of an Ordinary Annuity Cash flows that begin one period from today Mortgages, loans, etc. PV= PMT x 1− 1 N r [ 1+r ) ] Ex. You want to spend $5000 at the end of each year for the following 5 years. How much should you invest today with an interest rate of 9%? 5000 1 PV= x 1− 5=$19,448.26 0.09 [ 1+0.09) ] Present Value of Ordinary Annuity in Excel Make one column of years 1-5, one column for cash flows of $5000 each year (make absolute reference to the withdrawal cell), and one column for calculating the present value of these cash flows Use formula to calculate the present value of the cash flows for each year Add up all of these present values to find the overall PV of the annuity Present Value Function in Excel = PV (rate, nper, pmt, [fv], [type]) rate= r= interest rate, nper= N= number of periods, pmt= payment pv= value of investment toady, fv= value of investment at end of last period type= when annuity payments are received (1 for beginning of period and 0 for end of period) C. Present Value of Annuity Due When the first payment or withdrawal begins today or at the start of the period Apartment leases Every cash flow is one period earlier than a regular annuity PMT 1 N PV= x 1− x(1+r) r [ (1+r ] D. Present Value of a Perpetuity Like an annuity, but the value of N (number of periods) is infinite PMT CF 1 PV= = r r IV. Compounding Interest More Than Once Per Year If compounded more than once a year, interest is reinvested sooner, so future value grows quicker Annual Percentage Rate- stated annual interest rate m= number of times we compound interest per year Ex. Monthly compounding means m=12; quarterly means m=4, semi-annually means m=2 r = APR =thesub−annualdiscountrate m N =N xm=thenumber of time periods −annual Ex. Your account pays 8% APR with quarterly compounding. If you invest for 5 years, what is the quarterly interest rate, and what is the number of quarters of the investment? APR 8 r’= = =2 per quarter m 4 N’= N x m= 5x4= 20 quarters V. Future Value of Lump Sum with Sub-Annual Compounding FV =PV x(1+r') N' N' Ex. You deposit $5000 in an account that pays 10% a year, compounded semiannually. You plan to leave it in the account or 5 years. What will be the account balance in 5 years? r’ = APR = 10 =5 per half −year m 2 N’= N x m= 5 x 2= 10 half years N' FV NPV x (1+r')=5000x 1(05 )8144.47 Sub-Annual Compounding in Excel Same Example as above would be… ¿FV (5 ,10,−5000,0,0) VI. Continuous Compounding FV=PV xe rN Ex. What is the future value of the $5000 in an account that pays 105 % per year in 5 years when it is compounded continuously? FV=5000xe .1x=8243.61 Continuous Compounding in Excel Use the EXP function: = EXP(number) Same example as above would be… = 5000 * (.1*5) V. Effective Annual Rate APR does not reflect compounding throughout the year Effective annual rate (EAR) is the true yearly rate/ compounded return; accounts for compounding APR m EAR= (+ ) −1 m EAR allows you to compare accounts on the same time basis EX. You deposit $5000 in an account that pays 10% per year, compounded semiannually. What is the EAR? 0.10 2 EAR= (+ )−1= (.05)−1=0.1025=10.25 2 Finding EAR Using Excel Use the EFFECT formula… = EFFECT (nominal_rate, npery) Nominal_rate= APR and npery= number of compound perios per year (m) Same example above would be… =EFFECT(10%,2)

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