1-0 5-0 5-0 McGraw-Hill/Irwin Copyright © 2011 by the McGraw-Hill Companies, Inc. All rights reserved. 1-1 5-1 Multiple cash flows • Deposit an amount in one year, then deposit an amount in two years, then deposit an amount in three years 1 1-2 5-2 Future Value of Multiple Cash Flows 1. Compound accumulated balance forward one year at a time Deposit x (1+ r ) + New Deposit = New Balance New Balance x (1+ r ) + New Deposit = New Balance 2. Calculate FV of each cash flow, and add cash flows together FV = Principal x (1 + r)t 2 1-3 5-3 Future Value of Multiple Cash Flows Example Bank account has $7,000 and pays 8% interest. You will deposit $4,000 at the end of each year for the next 3 years. How much will you have in 3 years? 1.Compound accumulated balance forward one year at a time 2. Calculate FV of each cash flow, and add cash 3 Multiple Cash Flows – FV Example 2 1-4 5-4 • Suppose you invest $500 in a mutual fund today and $600 in one year. If the fund pays 9% annually, how much will you have in two years? 4 1-5 5-5 Example 2 Continued • How much will you have in 5 years if you make no further deposits? 5 Multiple Cash Flows – FV Example 3 1-6 5-6 • Suppose you plan to deposit $100 into an account in one year and $300 into the account in three years. How much will be in the account in five years if the interest rate is 8%? 6 1-7 5-7 Present Value of Multiple Cash Flows • How much do you need today in order to make a payment in one year and another payment in two years? 7 1-8 5-8 Present Value of Multiple Cash Flows 1. Discount amount back one period at a time 2. Calculate the PV of each cash flow and add together 8 1-9 5-9 PV of Multiple Cash Flows Examples You are offered an investment that will pay $200 in one year, $400 the next year, and $800 at the end of the next year. Assume an interest rate of 12%. What is the most you should pay for this investment? 1. Discount amount back one period at a time 2. Calculate the PV of each cash flow and add together 9 Multiple Cash Flows – PV Another Example 1-10 5-10 • You are considering an investment that will pay you $1,000 in one year, $2,000 in two years and $3,000 in three years. If you want to earn 10% on your money, how much would you be willing to pay? 10 Multiple Uneven Cash Flows – Using the Calculator 1-11 5-11 • Another way to use the financial calculator for uneven cash flows is to use the cash flow keys – Texas Instruments BA-II Plus • Clear the cash flow keys by pressing CF and then 2nd CLR Work • Press CF and enter the cash flows beginning with year 0. • You have to press the “Enter” key for each cash flow • Use the down arrow key to move to the next cash flow • The “F” is the number of times a given cash flow occurs in consecutive years • Use the NPV key to compute the present value by [ENTER]ing the interest rate for I, pressing the down arrow, and then computing NPV 11 1-13 5-13 Annuities • Multiple cash flows in the same amount • Ex.: Loan with equal payments 13 1-14 5-14 Ordinary Annuity • Series of constant cash flows that occur at the end of the period for a fixed # of periods Ex.: Cash flow from asset: $500 at the end of the next years for 3 years, 10% rate What is the present value? 14 1-15 5-15 Ordinary Annuity • Need a formula when there are a large number of payments Annuity PV = Cash flow X (1 – PV factor r,t r ) Remember, PV factor = 1 (1 + r)t 15 1-16 5-16 Find the Cash Flow Cash flow = PV of annuity / PV annuity factor 16 1-17 5-17 Find the Cash Flow Bath’s Bank offers you a $60,000, 7 year term loan at 9% annual interest. What will your annual loan payment be? 17 1-18 5-18 Find the # of payments Find the PV of annuity factor Annuity PV / Cash Flow = PV annuity factor Look on table under the rate column for the factor, see what # of periods corresponds 18 1-19 5-19 Find the # of payments You are investing $30,000 in order to receive payment at the end of each year of $10,000. If the rate is 20%, how many payments will you receive? 19 1-20 5-20 Find the rate Find PV annuity factor Annuity PV / Cash Flow = PV annuity factor Go to # of periods on the table and go across until you find the factor, then look to see what % column it falls in 20 1-21 5-21 Find the rate When you see the factor amount falls in between 2 % columns, can plug #s into formula to pinpoint the rate Cash flow x [1 – [1/(1 + r)t ]] r 21 1-22 5-22 Find the rate You are investing $32,000 in order to receive payment at the end of each year for 5 years of $8,000. What is the rate on this investment? 22 1-23 5-23 Annuity Due • Series of equal payments where the payment occurs at the beginning of the period • Ex.: When you lease an apartment, the first payment is due immediately, subsequent payments are due at the beginning of the month 23 1-24 5-24 Annuity Due Calculate ordinary annuity and multiply by 1 + r 24 1-25 5-25 Annuity Due • You are saving for a new house and you put $10,000 per year in an account paying 8%. The first payment is made today. How much will you have at the end of 3 years? 25 1-26 5-26 Perpetuities • Infinite series of equal payments • Value = Cash flow / r 26 Annuities and Perpetuities – Basic Formulas 1-27 5-27 • Perpetuity: PV = C / r • Annuities: 1 1 (1 r ) t PV C r (1 r ) t 1 FV C r 27 1-28 5-28 Annuities and the Calculator • You can use the PMT key on the calculator for the equal payment • The sign convention still holds • Ordinary annuity versus annuity due – You can switch your calculator between the two types by using the 2nd BGN 2nd Set on the TI BA-II Plus – If you see “BGN” or “Begin” in the display of your calculator, you have it set for an annuity due – Most problems are ordinary annuities 28 1-29 5-29 Example: Spreadsheet Strategies – Annuity PV • The present value and future value formulas in a spreadsheet include a place for annuity payments • Click on the Excel icon to see an example 29 Effective Annual Rate (EAR) 1-30 5-30 • This is the actual rate paid (or received) after accounting for compounding that occurs during the year • If you want to compare two alternative investments with different compounding periods you need to compute the EAR and use that for comparison. 30 1-31 5-31 EAR • Stated or quoted interest rate – May be compounded • Annually – Then quoted rate = EAR • Semiannually m = 2 • Quarterly m = 4 • Monthly m = 12 m = # of times interest compounds during the year 31 1-32 5-32 Calculate EAR 1. Divide quoted rate by # of times interest is being compounded during the year (m) 2. Add 1 to the result in step 1 3. Raise the result in step 2 to the power of m 4. Subtract 1 from the result in step 3 EAR = (1+ Quoted rate / m)m - 1 32 1-33 5-33 Calculate EAR EAR = FV of investment / PV of investment 33 1-34 5-34 Annual Percentage Rate • This is the annual rate that is quoted by law • By definition APR = period rate times the number of periods per year • Consequently, to get the period rate we rearrange the APR equation: – Period rate = APR / number of periods per year 34 1-35 5-35 Computing APRs • What is the APR if the monthly rate is .5%? • What is the APR if the semiannual rate is .5%? • What is the monthly rate if the APR is 12% with monthly compounding? 35 Things to Remember 1-36 5-36 • You ALWAYS need to make sure that the interest rate and the time period match. – If you are looking at annual periods, you need an annual rate. – If you are looking at monthly periods, you need a monthly rate. • If you have an APR based on monthly compounding, you have to use monthly periods for lump sums, or adjust the interest rate appropriately if you have payments other than monthly 36 Computing EARs - Example 1-37 5-37 • Suppose you can earn 1% per month on $1 invested today. – What is the APR? – How much are you effectively earning? • Suppose if you put it in another account, you earn 3% per quarter. – What is the APR? – How much are you effectively earning? 37 1-38 5-38 EAR - Formula m APR 1 EAR 1 m Remember that the APR is the quoted rate, and m is the number of compounds per year 38 1-39 5-39 Computing APRs from EARs • If you have an effective rate, how can you compute the APR? Rearrange the EAR equation and you get: 1 m APR m (1 EAR) -1 39 1-40 5-40 Loan Types • Pure Discount • Interest Only • Amortized 40 1-41 5-41 Pure Discount Loans • • • • Short term loans Simplest loan form Pay a lump sum at the end Determine how much would you be willing to lend in order to get a FV given r,t – Lender collect the present value from the borrower 41 Pure Discount Loans – Example 5.11 1-42 5-42 • Treasury bills are excellent examples of pure discount loans. The principal amount is repaid at some future date, without any periodic interest payments. • If a T-bill promises to repay $10,000 in one year, and the market interest rate is 7 percent, how much will the bill sell for in the market? 42 1-43 5-43 Interest-Only Loans • Example: Bonds • Repay the original loan amount at some point in the future • Interest is paid each period (PxRxT) • At the end of the last period, pay principal in addition to last interest payment 43 1-44 5-44 Interest-Only Loan - Example • Consider a 5-year, interest-only loan with a 7% interest rate. The principal amount is $10,000. Interest is paid annually. – What would the stream of cash flows be? 44 1-45 5-45 Amortized loans • Used for medium term loans • Lender requires borrower to make payments that include a portion of the principal and interest • Amortizing a loan: make regular principal payments; pay interest and a fixed amount of principal – Total payment will go down each period because as principal goes down, interest amount each period will go down 45 1-46 5-46 Amortized loans: Fixed principal repayments Amortization Schedule Beg. Total Interest Year Bal. Pmt Paid 1 5,000 1,450 450 2 4,000 1,360 360 3 3,000 1,270 270 4 2,000 1,180 180 5 1,000 1,090 90 Princ. Paid 1,000 1,000 1,000 1,000 1,000 End. Bal. 4,000 3,000 2,000 1,000 0 46 Amortized Loan with Fixed Payment 1-47 5-47 • Borrower makes a fixed payment every month • Each payment covers the interest expense; plus, it reduces principal • used for car loans, mortgages 47 Amortized Loan with Fixed Payment 1-48 5-48 To determine the fixed payment Loan amount = Payment x [1- 1/(1+r)t ] r or Loan amount = payment x PV annuity factor Payment = loan amount / PV annuity factor 48 Amortized Loan with Fixed Payment 1-49 5-49 Interest = Beg of period balance x r Principal = Payment amount – Interest End of period balance = Beg. of period balance – principal repaid Interest portion goes down each period Principal portion goes up each period 49 Amortized Loan with Fixed Payment - Example 1-50 5-50 • Each payment covers the interest expense; plus, it reduces principal • Consider a 4-year loan with annual payments. The interest rate is 8% and the principal amount is $5,000. – What is the annual payment? 50 1-51 5-51 Amortization Table for Example Year 1 Beg. Balance 5,000.00 Total Payment Interest Paid Principal Paid End. Balance 1,509.60 2 3 4 Totals 51