Discounted Cash, Flow Valuation

Key Concepts and Skills Chapter 5 Discounted Cash Flow Valuation 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 • 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 Multiple Cash Flows – FV Example 5.1 • Find the value at year 3 of each cash flow and add them together. – Today (year 0): FV = $7,000(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 = $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 Multiple Cash Flows – FV 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? FV = $500(1.09)2 + $600(1.09) = $1,248.05 Example 2 Continued • 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 = $1,616.26 • Second way – use value at year 2: FV = $1,248.05(1.09)3 = $1,616.26 1-1 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%? Example 3 Time Line 0 1 2 3 100 4 5 300 136.05 FV = $100(1.08)4 + $300(1.08)2 = $136.05 + $349.92 = $485.97 349.92 485.97 Multiple Cash Flows – PV Example 5.3 • Find the PV of each cash flow 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 Multiple Cash Flows – PV Another Example • 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? PV = $1,000 / (1.1)1 = $909.09 PV = $2,000 / (1.1)2 = $1,652.89 PV = $3,000 / (1.1)3 = $2,253.94 PV = $909.09 + 1,652.89 + 2,253.94 = $4,815.92 Example 5.3 Time Line 0 1 200 2 3 400 600 4 800 178.57 318.88 427.07 508.41 1,432.93 Decisions, Decisions • Your broker calls you and tells you that he has this great investment opportunity. If you invest $100 today, you will receive $40 in one year and $75 in two years. If you require a 15% return on investments of this risk, should you take the investment? PV = $40/(1.15)1 + $75/(1.15)2 = $91.49 No! The broker is charging more than you would be willing to pay. 1-2 Saving For Retirement • You are offered the opportunity to put some money away for retirement. You will receive five annual payments of $25,000 each beginning in 40 years. How much would you be willing to invest today if you desire an interest rate of 12%? Saving For Retirement Time Line 0 1 2 … 39 0 0 0 … 0 40 41 42 25K 25K 25K 43 44 25K 25K PV = $25,000/(1.12)40 + $25,000/(1.12)41 + $25,000/(1.12)42 + $25,000/(1.12)43 + $25,000/(1.12)44 = $1,084.71 Quick Quiz: Part 1 • 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? Annuities and Perpetuities – Basic Formulas • Perpetuity: PV = C / r • Annuities: 1 PV = C 1 (1 + r ) r 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 Annuity – Example 5.5 • You borrow money TODAY so you need to compute the present value. • Formula: t 1PV = 632 FV = C (1 + r ) r t - 1 1 (1 .01) 48 .01 = 23,999 .54 1-3 Annuity – Sweepstakes Example • Suppose you win the Publishers Clearinghouse $10 million sweepstakes. The money is paid in equal annual installments of $333,333.33 over 30 years. If the appropriate discount rate is 5%, how much is the sweepstakes actually worth today? PV = $333,333.33[1 – 1/1.0530] / .05 = $5,124,150.29 Buying a House - Continued • Bank loan Monthly income = $36,000 / 12 = $3,000 Maximum payment = .28($3,000) = $840 PV = $840[1 – 1/1.005360] / .005 = $140,105 • Total Price Closing costs = .04($140,105) = $5,604 Down payment = $20,000 – 5,604 = $14,396 Total Price = $140,105 + 14,396 = $154,501 Buying a House • You are ready to buy a house and you have $20,000 for a down payment and closing costs. Closing costs 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 (.5% per month) for a 30-year fixed rate loan. How much money will the bank loan you? How much can you offer for the house? Quick Quiz: Part 2 • 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 want to receive $5,000 per month in retirement. If you can earn .75% per month and you expect to need the income for 25 years, how much do you need to have in your account at retirement? 1-4