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DISCOUNTED CASH FLOW VALUATION CHAPTER 6 DR.LAKSHMI KALYANARAMAN 1 Future value with multiple cash flows Example 6.1 • You think you will be able to deposit $4,000 at the end of each of the next 3 years in a bank account paying 8% interest. You currently have $7,000 in the account how much will you have in 3 years? In 4 years? DR.LAKSHMI KALYANARAMAN 2 Multiple Cash Flows –Future Value Example 6.1 • Find the value at year 3 of each cash flow and add them together. • $7,000 in the account today will earn interest for 3 years – Today (year 0): FV = 7000(1.08)3 = 8,817.98 – $4,000 you invest at the end of year 1 will earn interest for 2 years – Year 1: FV = 4,000(1.08)2 = 4,665.60 – $4,000 that you invest at the end of year 2 will earn interest for 1 year – Year 2: FV = 4,000(1.08) = 4,320 – $4,000 you invest at the end of year 3 cannot earn any interest – Year 3: value = 4,000 – Total value in 3 years = 8,817.98 + 4,665.60 + 4,320 + 4,000 = 21,803.58 • Value at year 4 = 21,803.58(1.08) = 23,547.87 3 Example 6.2 • If you deposit $100 in one year, $200 in 2 years, and $300 in 3 years, how much will you have in 3 years? How much of this is interest? How much will you have in 5 years if you don’t add additional amounts? Assume a 7% interest rate throughout. DR.LAKSHMI KALYANARAMAN 4 • $100 deposited at the end of year 1 earns interest for 2 years • Year 1: FV = 100(1.07)2 = 114.49 • $200 deposited at the end of year 2 earns interest for 1 year • Year 2: FV = 200(1.07) = 214.00 • $300 deposited at the end of year 3 does not earn any interest • Year 3: value = 300 • Total value in 3 years = 114.49+ 214 + 300 = 628.49 • Interest earned = Future value of your deposit – your deposit • $628.49 – (100+200+300) = $28.49 • How much will you have in 5 years • You do not deposit any additional amount. Leave $628.49 for 2 more years • $628.49 ×(1.07)2=$719.56 DR.LAKSHMI KALYANARAMAN 5 Present value with multiple cash flows Example 6.3 • You are offered an investment that will pay you $200 in one year, $400 in the next year, $600 the next year and $800 at the end of the fourth year. You can earn 12% on very similar investments. What is the most you should pay for this one? DR.LAKSHMI KALYANARAMAN 6 Multiple Cash Flows – Present Value Example 6.3 • 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 = 1,432.93 7 Example 6.3 Timeline 0 1 2 3 4 200 400 600 800 178.57 318.88 427.07 508.41 1,432.93 8 A note about cash flow timing • It is always assumed that cash flows occur at the end of each period. • If it occurs at the beginning of the period you will be told explicitly. DR.LAKSHMI KALYANARAMAN 9 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 10 Annuities and Perpetuities – Basic Formulas • Perpetuity: PV = C / r • Annuities: 1 1 (1 r ) t PV C r (1 r ) t 1 FV C r C= dollars per period t = periods r = interest or rate of return 11 Example 6.5 • After carefully going through your budget, you have determined you can afford to pay $632 per month toward a new sports car. You call up your local bank and find out that the going rate is 1% per month for 48 months. How much can you borrow? DR.LAKSHMI KALYANARAMAN 12 Annuity – Example 6.5 • You borrow money TODAY so you need to compute the present value. • C = $632 t = 48 r = 1% 1 1 (1.01) 48 PV 632 23,999 .54 .01 13 Finding the payment • Suppose you wish to start up a new business that specializes in the latest of health food trends, frozen yak milk. To produce and market your product, you need to borrow $100,000. Because it strikes you unlikely that this particular fad will be long-lived, you propose to pay off the loan quickly in 5 equal annual payments. If the interest rate is 18%, what will the payment be? DR.LAKSHMI KALYANARAMAN 14 Finding the payment 1 (1 / 1 r)^ t Annuity present value C r $100,000 = C×{(1-(1/1.185)/.18} C = $31,978 DR.LAKSHMI KALYANARAMAN 15 Future value of annuity • Future value of C per period for t periods at r per percent per period (1 r )^ t 1 FVA C r DR.LAKSHMI KALYANARAMAN 16 Annuity due • An annuity for which the cash flows occur at the beginning of the period • For both present and future value of an annuity due • Annuity due = Ordinary annuity × (1+r) DR.LAKSHMI KALYANARAMAN 17 Perpetuity – Example 6.7 • Perpetuity formula: PV = C / r • Current required return: 40 = 1 / r r = .025 or 2.5% per quarter • Dividend for new preferred: 100 = C / .025 C = 2.50 per quarter 18 Table 6.2 19