FINANCE 351 Corporate Finance John Graham Things to do before the first class meeting C Read the Gifford and Brealey and Myers material (see class schedule) C Read over the syllabus and class schedule. C REVIEW the following: C 1) Perpetuity formula: where C is the cash flow and r is the discount rate. r Q: What is the present value of an investment that pays $100 each year forever, starting next year, assuming the interest rate is 10%? A: $100/.1=$1000 Note: the first payment is one year from today, not today. Q: Displaced Perpetuity. What is the present value of an investment that pays $100 each year forever, starting in three years, assuming that the interest rate is 10%? A: The present value as of year 2 is $1000, like in the previous problem. This is the value in year 2 (NOT year 3) because the perpetuity formula is designed to value cash flows that start one year hence. That is, the formula calculates the value as of t=2 of cash flows that start in t=3. To answer the question, we need to value the lump sum of $1000 as of today. That is, we need to 1000 discount the $1000 back two periods: = $826.44. 11. 2 C 2) Growing perpetuity formula: where g is the rate of growth of the cash flow. r g Q: What is the present value of an investment that pays $100 next year, $110 in year 2, $121 in year 3, growing at 10% each year forever, assuming that the interest rate is 15%? 100 A: = $2000. 15. 10 1 3) Present value of an annuity formula: 1 where n is the number of years the annuity r x 1 (1+ r) n payments continue. Q: What is the present value of an investment that pays $200 each year for the next 12 years, starting next year, assuming that the interest rate is 15%?
A: $200 x (annuity factor, n=12 years, r=15%) = $200(5.4206) = $1084.12 NOTE: An annuity is an investment that pays the same cash flow (like $200) for a specific number of periods (like 12 years). During the second class session you will learn about deferred annuities which are investment vehicles that allow an investor to defer paying tax on income/interest/dividends. Although there are some similarities, it would be best if you thought of an annuity as being a completely different concept/formula than a deferred annuity; they just happen to have similar names. The next item, a displaced annuity is simply an annuity with a first payment that occurs more than one year in the future. Don t confuse displaced with deferred. Q: Displaced Annuity. What is the present value of an investment that pays $200 each year for 12 years, with the first payment starting in year 5, assuming that the interest rate is 15%? A: From the previous question we know that, as of year 4, the annuity is worth $1084.12. This is the value in year 4 (NOT year 5) because the annuity formula is designed to value cash flows that start one year hence. To answer this question, we need to discount the lump-sum back to year 0. The present value of this 108412. deferred annuity is: =$619.85 115. 4 [ ] 1 4) Future value of an annuity formula: x ( 1+ r) n 1 where n is the number of years the annuity r payments continue. In this case, this is the value of the annuity as of year n. Notice that this is just the present value annuity formula times (1+r) n. You might need this formula to do problem 2.4 in the homework set. This formula is sometimes called the sum of an annuity. Q: What is the future value (as of year 12) of an annuity that pays $200 each year for the next 12 years, starting next year, assuming that the interest rate is 15%? A: $200 x (future value of an annuity factor, 12 years, 15%) = $200(29.002) = $5800.40 Note: this is the answer to the first question in 3) times (1+r) n : 1084.12(1.15) 12 = $5800.4 That is, the FV of an annuity formula is (1+r) n times the PV of an annuity formula. 5) Cash flows: The introductory accounting course teaches that cash flows are not the same thing as accounting income. Here, we review some basics related to calculating cash flows. These cash flows in turn can be used in project valuation. There is nothing magic going on here. You can always figure out the additional cash flows using accounting-like statements (as in the tables) or using the short-cut formula.
a) Depreciation short-cut: ($depreciation)x(tax rate) Q: How much cash flow is added if a firm gets an extra dollar of depreciation and the tax rate is 40% Extra $1 depreciation Oper. Income $20 $20 -Depreciation -10-11 Taxable Income $10 $9 - taxes paid -4-3.6 Net Income $6 $5.4 + Depreciation +10 +11 Cash Flow $16 $16.4 A: An extra dollar of depreciation adds $1(tax rate) = $1(.40) = $0.40 extra cash flow. b) Reducing expenses short-cut: ($expense reduction)x(1 - tax rate) Q: How much cash flow is added if a firm reduces expenses by $1 and the tax rate is 40%? Expenses reduced by $1 Sales Revenue $40 $40 -Expenses $20 -$19 Oper. Income $20 $21 -Depreciation -10-10 Taxable Income $10 $11 - taxes paid -4-4.4 Net Income $6 $6.6 + Depreciation +10 +10 Cash Flow $16 $16.6 A: Reducing expenses by $1 adds $1(1-tax rate) = $1(1-0.40) = $0.60 extra cash flow. Note: The depreciation is not needed in this example. It is included only for completeness.
c) Revenue short-cut: ($revenue) x (1-tax rate) Q: How much cash flow is added by increasing revenue by $1 if the tax rate is 40%? Sales Revenue $40 $41 -Expenses $20 -$20 Oper. Income $20 $21 -Depreciation -10-10 Taxable Income $10 $11 - taxes paid -4-4.4 Net Income $6 $6.6 + Depreciation +10 +10 Cash Flow $16 $16.6 Extra $1 sales revenue A: Increasing revenue by $1 adds $1(1-tax rate) = $1(1-0.40) = $0.60 extra cash flow. Note: this is the exact same concept as reducing expenses by $1. d) Interest deduction short-cut: ($interest)x(tax rate) Q: How much extra cash flow do investors get if a firm takes an extra dollar of interest deduction (relative to paying out the dollar to equity investors out of net income)? Tax rate is 40% Oper. Income $30 $30 - Depreciation -10-10 -Interest expense -0-1 Taxable Income $20 $19 - taxes paid -8-7.6 Net Income $12 $11.4 Extra $1 interest CF to stockholders CF to bondholders $22 =$10 dep+ $12 NI $0 Total Cash Flow $22 $22.4 $21.4= $10 dep + $11.4 NI $1 A: An extra dollar of interest adds $1(tax rate) = $1(.40) = $0.40 extra cash flow to investors.
Note: depreciation included in this example for completeness only. Note: we usually use the short-cut to account for interest benefits in the cash flows, rather than using this long income-statement approach. This example might look fairly simple. However, if I told you that you can either account for the tax benefits of debt in the cash flows or in the discount rate, not both you might become bewildered. Accounting for the tax benefits of debt in the cash flows amounts to knowing that the cash flow is $22 base case + $0.40 financing benefits= $22.40 in this example. If you use a cash flow of $22.40, DO NOT discount with an after tax cost of debt (like you would when you use WACC.) Alternatively, you could consider the cash flow as $22 and instead discount using the aftertax cost of debt (like you would when you use WACC).