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Week 3 Lecture 3 Ross, Westerfield and Jordan 7e Chapter 6 Discounted Cash Flow Valuation 6-0 Last Week.. • Sources and Uses of Cash: • A or L and SE = Source • A or L and SE = Use • Ratios – Most Used in Finance: • Profitability Ratios: PM, ROA(ROI), ROE • Market Value Ratios: P/E, M/B • Leverage Ratios: D/E, EM • Simple vs Compound Interest • Present & Future Values 6-1 Chapter 6 Outline • Future and Present Values of Multiple Cash Flows • Valuing Level Cash Flows: Annuities and Perpetuities • Comparing Rates: The Effect of Compounding 6-2 Multiple Cash Flows –Future Value Example 6.1 • If you have 7000 now to invest plus 4000 each year for 3 years and the rate is 8% p.a. what is the value of your investments in 3 years? In 4 years? • First find the value at year 3 of each cash flow and add them together. • • • • • Year 0 (today): FV = 7000(1.08)3 = 8,817.98 Year 1: FV = 4,000(1.08)2 = 4,665.60 Year 2: FV = 4,000(1.08) = 4,320 Year 3: value = 4,000 Total value in 3 years = 8817.98 + 4665.60 + 4320 + 4000 = 21,803.58 • Value at year 4 = 21,803.58(1.08) = 23,547.87 6-3 Multiple Cash Flows – Future Value Example 2 • 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? • First way: FV = 500(1.09)2 + 600(1.09) = = 594 + 654 = 1248.05 • Second way: FV = [500(1.09) + 600](1.09) = = (545 + 600)(1.09) = 1145(1.09) = 1248.05 • How much will you have in 5 years if you make no further deposits? • First way: FV = 500(1.09)5 + 600(1.09)4 = 1616.26 • Second way: FV = 1248.05(1.09)3 = 1616.26 6-4 Multiple Cash Flows – FV Example 3 • 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%? • FV = 100(1.08)4 + 300(1.08)2 = 136.05 + 349.92 = 485.97 Year 0 1 100 2 3 4 300 5 FV=? 2 years 4 years 6-5 FV of Multiple Cash Flows • Formula: FV = C1(1+r)n-1 + C2(1+r)n-2 + …+Ct(1+r)n-t • where C is the cash flow generated at time t, and r is the discount rate, n = number of periods t = current period. • Note: if there is an immediate cash flow C0 at the start then begin sum with t = 0. • If there is a cash flow in the final year, don’t compound last cash flow (see example 6.1) 6-6 Multiple Cash Flows – Present Value Example 6.3 • You are offered an investment that will pay you $200 in one year, $400 the next year, $600 the next year and $800 at the end of the next year. You can earn 12 percent on very similar investments. What is the most you should pay for this one now? • Find the PV of each cash flows and add them • • • • • Year 1 CF: 200 / (1.12)1 = 178.57 Year 2 CF: 400 / (1.12)2 = 318.88 Year 3 CF: 600 / (1.12)3 = 427.07 Year 4 CF: 800 / (1.12)4 = 508.41 Total PV = 178.57 + 318.88 + 427.07 + 508.41 = 1432.93 6-7 Example 6.3 Timeline 0 1 200 2 3 4 400 600 800 178.57 318.88 427.07 508.41 1432.93 = Total PV 6-8 PV of Multiple Cash Flows • Formula: PV = C1/(1+r) + C2/(1+r)2 + …+Ct/(1+r)t • where Ct is the cash flow generated at time t, and r is the discount rate, t = current period • Note: if there is an immediate cash flow (C0 ) at the start then begin sum with t = 0. Do not discount!! 6-9 Multiple Cash Flows Using a Spreadsheet • You can use the PV or FV functions in Excel to find the present value or future value of a set of cash flows • Setting the data up is half the battle – if it is set up properly, then you can just copy the formulas 6-10 Quick Quiz • Suppose you are looking at the following possible cash flows: Year 1 CF = $100; Years 2 and 3 CFs = $200; Years 4 and 5 CFs = $300. The required discount rate is 7% • What is the value of the cash flows at year 5? • What is the value of the cash flows today? • What is the value of the cash flows at year 3? 6-11 Annuities and Perpetuities Defined • Annuity – finite series of equal payments that occur at regular intervals • If the first payment occurs at the end of the period, it is called an ordinary annuity • If the first payment occurs at the beginning of the period, it is called an annuity due • Perpetuity – infinite series of equal payments at regular intervals 6-12 Annuities and Perpetuities – Basic Formulas Ordinary Annuities: 1 1 (1 r)t P V C r Annuity interest factor (1 r) 1 FV C r C Perpetuity: PV r t C 1 PV 1 r (1 r)t or C FV (1 r)t 1 r Annuity Due Value = Ordinary Annuity Value x (1+r) Other way of calculating FVannuity = PVannuity x (1+r)t 6-13 Annuity – Example 6.5 p158 • You can afford to pay $632 per month towards a new car. You go to the bank and find out that the rate is 1% per month for 48 months. How much can you borrow? • You borrow money TODAY so you need to compute the present value. • t = 48, r = 1%, C = 632, PV = ? • Formula: 1 1 (1 r)t P V C r 1 1 (1.01) 48 PV 632 .01 23,999.54 6-14 Buying a House • You are ready to buy a house and you have $20,000 saved for a deposit and other fees. You enquire at the bank for a possible loan and they tell you the following: • Loan fees are estimated to be 4% of the loan value. • You have an annual salary of $36,000 and the bank is willing to allow your monthly mortgage payment to be equal to 28% of your monthly income. • The interest rate on the loan is 6% per year with monthly compounding (0.5% per month) for a 30-year fixed rate loan. • 1)How much money will the bank loan you? • 2)How much can you offer for a house? 6-15 Buying a House - Continued • 1)How much is the bank loan? or.. what is the present value of the annuity? • Monthly income = 36,000 / 12 = 3,000 • Maximum payment = .28(3,000) = 840 • PV = 840[1 – 1/1.005360] / .005 = 140,105 • 2) How much can you offer for the house? • Loan fees = .04(140,105) = 5,604 • Deposit = 20,000 – 5604 = 14,396 • Offer Price = 140,105 + 14,396 = 154,501 6-16 Quick Quiz • You know the payment amount for a loan and you want to know how much was borrowed. Do you compute a present value or a future value? • You are 21 and want to receive 5000 per month in retirement. In you’re retirement you expect the interest to be 0.75% per month and you think you’ll need the income for 25 years. How much do you need to have in your account at retirement? 6-17 Finding the Payment • Suppose you want to borrow 1 $20,000 to renovate your 1 (1 r)t house. You can borrow at 8% P V C r per year. If you take a 4 year loan, what is your monthly payment? PV C 1 • PV = 20000, r = 0.67%, t = 48 1 48 20,000 = C[1 – 1 / (1.0067) ] / .0067 (1 r)t C = 20000/{[1-1/(1.0067)48]/.0067} r C = 488.64 6-18 Finding the Number of Payments – Example 6.6 p.161 • You spend $1000 on your credit card and can afford to pay only the minimum payments of $20 per month. Knowing the interest rate is 1.5% per month, how long will it take you to pay $1000? • Start with the equation to isolate t and use logarithms. • 1000 = 20[(1 – 1/1.015t) / 0.015] • • • • • • 1 1000/20 = (1 – 1/1.015t) / 0.015 1 (1 r)t t 50 x 0.015 = 1 – 1/1.015 PV C r 0.75 = 1 – 1 / 1.015t t 0.25 = 1 / 1.015 (negative signs cancel out) 1 / 0.25 = 1.015t t = LN(1/0.25) / LN(1.015) = 93.111 months/12 = 7.76 years • And this is only if you don’t charge anything more on the card! 6-19 Finding “t” - the Number of Payments • Suppose you borrow $2000 at 5% and you are going to make annual payments of $734.42. How long before you pay off the loan? • • • • • • • • • • • 2000 = 734.42 (1 – 1/1.05t) / 0.05 2000/734.42 = (1 – 1/1.05t) / 0.05 2.723237 = (1 – 1/1.05t) / 0.05 2.723237 x 0.05 = (1 – 1/1.05t) 0.136161869 = 1 – 1/1.05t 0.863838131 = 1/1.05t (negative signs cancel out) 1/ 0.863838131 = 1.05t 1.157624287 = 1.05t LN(1.157624287) = LN(1.05t ) LN(1.157624287) = t x LN(1.05) t = LN(1.157624287) / LN(1.05) = 3 years 6-20 Annuity – Finding the Rate • Trial and Error Process • Choose an interest rate and compute the PV of the payments based on this rate • Compare the computed PV with the actual given amount • If the computed PV > given amount, then the interest rate is too low. Action: increase rate • If the computed PV < given amount, then the interest rate is too high. Action: lower rate • Adjust the rate and repeat the process until the computed PV and the given amount are equal 6-21 Example – Finding “r” • An insurance company offers to pay $1000 per year for 10 years if you pay $6710 upfront. What is the implied rate of annuity? • Solution: Trial and Error calculation • Step 1: pick a rate, say 10%, calculate PV which is $6144.. Too low.. We need $6710 • Step 2: start again with a different rate until what you calculate is the same as the given value 6-22 Future Values for Annuities • Suppose you begin saving for your retirement by depositing $2000 per year in an investment account. If the interest rate is 7.5%, how much will you have in 40 years? (1 r)t 1 FV C r • FV = 2000(1.07540 – 1)/.075 = 454,513.04 • Alternative Method: • FV Annuity: (PV Ordinary Annuity) x (1+r)t PV = 2000[1-1/(1.075)40/0.075] = 25,188.82 • FV Annuity: 25,188.82 x (1.075)40 = 454,513.04 6-23 Annuities on the Spreadsheet • The present value and future value formulas for annuities in excel - fill in payment variable • Finding the Annuity Payment: PMT(rate,nper,pv,fv) • The same sign convention holds as for the PV and FV formulas 6-24 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? (1 r)t 1 FV C (1 r) r • FV = 10,000[(1.083 – 1) / .08](1.08) = 32,464(1.08) = 35,061.12 • General relationship between Ordinary and Annuity Due: • Annuity Due = Ordinary Annuity x (1+r) 6-25 Perpetuity – Example 6.7 • A company wants to sell preferred stock at $100 per share. Similar shares sell for $40 with $1 dividend per quarter. What dividend will the new shares offer? • Perpetuity formula: PV = C / r, C = ? • Current required return: • 40 = 1 / r , r = 1/40 • r = .025 or 2.5% per quarter • Dividend for new preferred: • 100 = C / .025 • C = 2.50 per quarter 6-26 Table 6.2 - p167 6-27 Effective Annual Rate (EAR) • This is the actual rate paid (or received) after accounting for compounding that occurs during the year • Is 10% p.a. = 10% compounded semiannually? • If you want to compare two alternative investments with different compounding periods you need to compute the EAR and use that for comparison. • 10% p.a. ≠ 10.25% p.a. comp. semiannually 6-28 Annual Percentage Rate (APR) • This is the annual rate that is quoted by law • By definition APR = period rate x number of periods per year • Period rate = APR / number of periods per year • You should NEVER divide the EAR by the number of periods per year – it will NOT give you the period rate • What is the APR if the monthly rate is 0.5%? • 0.005(12) = 6% • What is the APR if the semiannual rate is 0.5%? • 0.005(2) = 1% • What is the monthly rate if the APR is 12% with monthly compounding? • 0.12 /12 = 1% • Can you divide the above 12% APR by 2 to get the semiannual rate? NO!!! You need an APR based on semiannual compounding to find the semiannual rate. 6-29 APR & EAR - Example • 1) Suppose you can earn 1% per month on $1 invested today. • What is the APR? 1% x (12) = 12% • How much are you effectively earning? • FV = $1x(1.01)12 = $1.1268 • EAR = ($1.1268 – $1) / $1 = .1268 = 12.68% • 2) Suppose you put another $1 in another account, that earns 3% per quarter. • What is the APR? 3% x (4) = 12% • How much are you effectively earning? • FV = $1x(1.03)4 = 1.1255 • EAR = ($1.1255 – $1) / $1 = .1255 = 12.55% 6-30 EAR - Formula m APR EAR 1 1 m Remember that: - APR is the quoted rate -m is the number of compounding periods per year Computing APRs from EARs: 1 APR m (1 EAR) m - 1 -Rearrange the formula - Example: Suppose you want to earn an effective rate of 12% and you are looking at an account that compounds on a monthly basis. What APR must they pay? APR 12 (1 .12)1/12 1 .1138655152 or 11.39% 6-31 Continuous Compounding • Sometimes investments or loans are figured based on continuous compounding • EAR = eq – 1 • q = rate • e = 2.71828, a mathematical constant • There is a special function on the calculator normally denoted by ex • Example: What is the effective annual rate of an investment which pays 7% compounded continuously? • EAR = e.07 – 1 = .0725 or 7.25% 6-32 Lecture 3 - Summary • PV and FV of multiple cash flows • Uneven cash flows • Cash flows occurring at different time periods • Annuities and Perpetuities • Equal cash flows • Equal time periods between the cash flows • Finding C, t, r • EAR and APR 6-33 End Lecture 3 6-34