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© 2003 The McGraw-Hill Companies, Inc. All rights reserved. Discounted Cash Flow Valuation (Formulas) Chapter Six

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Key Concepts and Skills Be able to compute the future value of multiple cash flows Be able to compute the present value of multiple cash flows Be able to compute loan payments Be able to find the interest rate on a loan Understand how loans are amortized or paid off Understand how interest rates are quoted

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Chapter Outline Future and Present Values of Multiple Cash Flows Valuing Level Cash Flows: Annuities and Perpetuities Comparing Rates: The Effect of Compounding Periods Loan Types and Loan Amortization

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Future Future Value with Multiple Cash Flows Two ways to calculate future value of multiple cash flows –Compound the accumulated balance forward one period at a time –Calculate the future value of each cash flow and add them up.

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Present Value with Multiple Cash Flows Two ways to calculate the present value of multiple cash flows –Discount the last amount back one period and add them up as you go –Discount each amount to time zero and then add them up.

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A Note on Cash Flow Timing In general, assume that cash flows occur at the end of each time period. This assumption is implicit in the ordinary annuity formulas presented.

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Valuing Level Cash Flows: Annuities and Perpetuities Present Value for Annuity Cash Flows –Ordinary Annuity ‑ multiple, identical cash flows occurring at the end of each period for a fixed number of periods. –PVIFA is just the sum of the PVIFs across the same time period. –The present value of an annuity of $C per period for t periods at r percent interest: PV = C[1 ‑ 1/(1 + r) t ] / r

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Example If you are willing to make 36 monthly payments of $100 at 1.5% per month, what size loan can you obtain? PV = 100[1 ‑ 1/(1.015) 36 ] /.015 = 100(27.6607) = 2766.07

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Example: You wish to purchase a $170,000 home. You are going to put 10% down, so the loan amount will be $153,000 at 7.75% APR (.6458333333% per month), with monthly payments for 30 years. How much will each payment be? –1096.11 How much interest will you pay over the life of the loan? –241,599.60

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Example continued How much is owed at the end of year 20? The outstanding balance of the loan at any time equals the present value of the remaining payments. So, after 240 payments, the outstanding balance equals: PMT = ‑ 1096.11; N = 120; I/Y = 7.75/12; CPT PV = 91,334.41 After making 2/3 of the payments, 60% of the principal remains unpaid.

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Example continued (II) How much interest will be paid in year 20? The interest paid in any year is equal to the sum of the payments made during the year minus the change in principal. After 228 months (19 years), the outstanding loan balance is $97,161.79. –The change in principal is 97,161.79 ‑ 91,334.41 = 5,827.38. –Total interest paid in year 20 = 12(1096.11) ‑ 5,827.38 = $7,325.94.

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Finding the payment, C, given PV, r and t PV = C[1 ‑ 1/(1 + r) t ] / r C = PV {r / [1 ‑ 1/(1 + r) t ]} Example: If you borrow $400, promising to repay in 4 monthly installments at 1% per month, how much are your payments? –102.51

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Finding the number of payments given PV, C and r PV = C [1 ‑ 1/(1 + r) t ] / r t = ln[1 / (1 ‑ rPV/C)] / ln(1 + r) Example: How many $100 payments will pay off a $5,000 loan at 1% per period? –69.66 periods

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Finding the rate given PV, C and t No analytical solution

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Future Value for Annuities FV = C[(1 + r) t ‑ 1] / r Example: If you make 20 payments of $1000 at the end of each period at 10% per period, how much will your account grow to be? –$57,275

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A Note on Annuities Due Annuity due ‑ the first payment occurs at the beginning of the period instead of the end. If the first payment occurs at the beginning of the period, then FV's have one additional period for compounding and PV's have one less period to be discounted. Consequently, multiply both the future value and the present value by (1 + r) to account for the change in timing.

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Perpetuities Perpetuity ‑ series of level cash flows forever Formula: PV = C / r Preferred stock is a good example of a perpetuity.

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Comparing Rates: The Effect of Compounding Periods Effective Annual Rates and Compounding Stated or quoted interest rate ‑ rate before considering any compounding effects, such as 10% compounded quarterly Effective annual interest rate ‑ rate on an annual basis, that reflects compounding effects, e.g. 10% compounded quarterly has an effective rate of 10.38% EAR is not used directly in time value of money calculations, except when we have annual periods. –Primarily used for comparison purposes, not for calculation purposes.

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Calculating and Comparing Effective Annual Rates (EAR) EAR = [1 + (quoted rate)/m] m ‑ 1 where m is the number of periods per year

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Example 18% compounded monthly is [1 + (.18/12)] 12 ‑ 1 = 19.56%

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Ethics Note: Rent ‑ to ‑ own agreements and tax refund loans Because of the structure of the contracts, they do not have to provide information on interest rates. However, when you work out the rates implied in the contracts, they can be extraordinarily high.

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Example Suppose you are in a hurry to get your income tax refund. If you mail your tax return, you will receive your refund in 3 weeks. If you file the return electronically through a tax service, you can get the estimated refund tomorrow. The service subtracts a $50 fee and pays you the remaining expected refund. The actual refund is then mailed to the preparation service. Assume you expect to get a refund of $978. What is the APR with weekly compounding? What is the EAR? How large does the refund have to be for the APR to be 15%?

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Example (continued) Using a financial calculator to find the APR: PV = 978 ‑ 50 = 928; FV = ‑ 978; N = 3 weeks; CPT I/Y = 1.765% per week; APR = 1.765 (52 weeks per year) = 91.76%!!! Compute the EAR = (1.01765) 52 ‑ 1 = 148.34%!!!! You would be better off taking a cash advance on your credit card and paying it off when the refund check comes, even if you have the most expensive card available. Refund needed for a 15% APR: PV + 50 = PV(1 + (.15/52))3 PV = $5,761.14

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Example Suppose you are going to have $50 deducted from your paycheck every two weeks and have it placed in an account that pays 8% compounded daily. How much will you have in 35 years? You are depositing money every two weeks (26 times per year), but compounding occurs daily. You need a period rate that corresponds to every two weeks, but you can only divide the APR given by 365. What can we do?

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Example Continued Find the EAR for the daily compounded rate. This is the rate we will earn each year after we account for compounding.EAR = (1 +.08/365)365 ‑ 1 =.08327757179 What we need is an APR based on compounding every two weeks that will pay the same effective rate of interest. So we take the EAR computed above and convert to an APR based on 26 compounding periods per year. APR = 26[(1.08327757179)1/26 ‑ 1] =.0801144104 –The small difference in rates can make a difference over long periods of time.

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Example Find the FV –PMT = 50 –N = 35(26) = 910 –I/Y = 8.01144104 / 26 =.308132348 –CPT FV = $250,535.24 –If you just use I/Y = 8/26, you would get a FV = $249,829.21; a difference of $706.03.

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Continuous Compounding Sometimes investments or loans are figured based on continuous compounding EAR = e q – 1 There is a special function on the calculator normally denoted by e x Example: What is the effective annual rate of 7% compounded continuously? EAR = e.07 – 1 =.0725 or 7.25%

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Loan Types and Loan Amortization Pure Discount Loans Interest-only Loans Amortized Loans

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Pure Discount Loans Borrower pays a single lump sum (principal and interest) at maturity. Treasury bills are a common example of pure discount loans.

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Interest ‑ Only Loans Borrower pays interest only each period and the entire principal at maturity. Corporate bonds are a common example of interest ‑ only loans.

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Amortized Loans Borrower repays part or all of principal over the life of the loan. Two methods are –Fixed amount of principal to be repaid each period — results in uneven payments –Fixed payments, which results in uneven principal reduction. Traditional auto and mortgage loans are examples of the second type of amortized loans.

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Example Consider a $200,000, 30 ‑ year loan with monthly payments of $1330.60 (7% APR with monthly compounding). –You would pay a total of $279,016 in interest over the life of the loan. Suppose instead, you cut the payment in half and pay $665.30 every two weeks –(note that this entails paying an extra $1330.60 per year because there are 26 two week periods). –You will cut your loan term to just under 24 years and save almost $70,000 in interest over the life of the loan.

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