Chapter Discounted Cash Flow Valuation Compounding Periods Other Than Annual Let s examine monthly compounding problems. Future Value Suppose you invest $9,000 today and get an interest rate of 9 percent compounded monthly. How much will you have in 3 years? = Cpt $11,777.81 You can see have introduced another variable. This variable, m, is the number of compounding periods per year. f it is monthly compounding, them m is equal to 12. f it is quarterly compounding, them m is equal to 4. The number of years is multiplied by m and the annual percentage rate (APR) is divided by m. Present Value Assume you need $00,000 in 30 years and can earn an interest rate of 6 percent compounded monthly. How much will you have to invest today? = Cpt $83,020.96 Fin 311 Chapter Handout Page 1
nterest Rate You have $10,000 and need $12,00 in 2. years. What interest rate must you earn to accomplish your goal? Assume monthly compounding. Cpt 0.7466% = r/12 r = (0.7466)(12) = 8.99% umber of Periods You have $100,000 and need $12,000. Your investment will earn a rate of 1.7 % compounded monthly. How many years before you accomplish your goal? Cpt 17.11 = t(12) t=17.11/12 = 1.43 years This method for calculating the different variables in the lump sum equations can be applied to uneven, perpetuity, and annuity cash flows. Page 2 Fin 311 Chapter Handout
Uneven Cash Flows Suppose you are going to receive $10,000 in one year, $1,000 in two years and three years, and $20,000 in five years. f the appropriate interest rate is 10 percent, what is the value of the cash flows today? Draw the time line. r = 10% 0 1 2 3 4 10,000 1,000 1,000 20,000 = 10,000(F 1,10% ) + 1,000(F 2,10% ) + 1,000(F 3,10% ) + 20,000(F,10% ) = = = = = = = = Cpt $9,090.91 Cpt $12,396.69 Cpt $11,269.72 Cpt $12,418.43 = = = = = = = = 0 = $9,090.91 + $12,396.69 + $11,269.72 + $12,418.43 = $4,17.7 What if want to know the value at year? = 10,000(F 4,10% ) + 1,000(F 3,10% ) + 1,000(F 2,10% ) + 20,000 = = = = = = = = = = = = = = = = Cpt = 14,641.00 Cpt = 19,96.00 Cpt = 18,10.00 Cpt = 20,000 V = $14,641.00 + $19,96.00 + $18,10.00 + $20,000.00 = $72,76.00 Fin 311 Chapter Handout Page 3
Another approach is: = (F r,t ) = $4,17.7(F 10%, ) = = = = Cpt = 72,76.00 Cash Flow Worksheet The cash flow worksheet is another alternative to solving this problem. The cash flow worksheet is initiated with the CF key. Push this key and you see CF 0. n this example, the cash flows are: CF 0 0 CF 1 10,000 F 1 1 CF 2 1,000 F 2 2 CF 3 0 F 3 1 CF 4 20,000 F 4 1 10 Cpt 4,17.7 Perpetuities A perpetuity is an even cash flow that occurs at even time intervals forever. You will receive $7,000 per year forever with the first payment occurring one year from today. f the interest rate is 6 percent, what is the value of the perpetuity today? Draw the time line. Perpetuity Perpetuity rate Page 4 Fin 311 Chapter Handout
Suppose the perpetuity payments in the previous problem start 9 years from today. What is the value of the cash flows now? Draw the time line. = = Cpt = 784,26.46 = Cpt = Annuities An ordinary annuity is constant cash flow that occurs at constant and even time intervals for a finite time. The cash flows are at the end of the time period. of an Annuity Suppose you are offered $10,000 per year for three years. f the interest rate is 10 percent, what is the value of the cash flows today? We know several ways of calculating the present value. Here is one: r = 10% 0 1 2 3 10,000 10,000 10,000 9,091 8,264 7,13 0.9091 0.8264 0.713 $24,868 2.4868 Let s look at Table A3 in the 10 percent column and the 3 period row. You find 2.4868. You can see that another way to calculate the of an annuity is to multiply the factor from Table A3 by the annuity. Fin 311 Chapter Handout Page
The Equation Approach The equation for the of an annuity is: A A 1 1 1 r r t 10,000 1 1 1.10 0.10 3 $24,868.2 The Table Approach A = A(FA r,t ) A = 10,000(FA 10%,3 ) A = 10,000(2.4868) A = 24,868 The Calculator Approach = = Cpt = $24,868.2 = = if the interest rate is 12% = $24,018.31 at 8% = $2,770.97 We can use the cash flow worksheet CF 0 0 CF 1 10,000 F 1 3 10 Cpt 24,868.2 Page 6 Fin 311 Chapter Handout
Future Value of Annuities You are planning to save $,000 per year for the next 30 years toward retirement with the first payment occurring one year from today. f you can earn an 11 percent interest rate, what will the value of your retirement portfolio be? r = 11% 0 30 A =,000 The Equation Approach A A 1 r r t 1,000 30 1.11 0.11 1 99,104.39 The Table Approach A = A(FA r,t ) A =,000(FA 11%,30 ) A =,000(199.02) A = 99,100 The Calculator Approach = = = = Cpt = $99,104.39 We cannot directly use the cash flow worksheet. This calculator does not have a function. Fin 311 Chapter Handout Page 7
You will receive $0,000 per year for 10 years with the first payment occurring 7 years from today. f the appropriate interest rate is 10. percent, what is the value of the cash flows today? 0 7 r = 10.% 16 A = 0,000 The Equation V 0 = 0,000(FA 10.%,10 ) (F 10.%,6 ) = = = = Cpt $300,738.64 Cpt $16,202.10 = = = = We can use the cash flow worksheet CF 0 CF 1 F 1 6 CF 2 F 2 10 Cpt 16,202.10 Page 8 Fin 311 Chapter Handout
Annuity Payments You want to retire in 40 years with $1. million. You plan to save an equal amount each year and feel you can earn an interest rate of 11 percent. How much do you have to save each year? The Equation 1,00,000 = A(FA 40, 11% ) Cpt $2,78.09 You want to buy a car that costs $29,000. You plan to put $2,000 down and finance the remainder for years. The dealer offers you a loan with a 4. percent annual interest rate with monthly compounding. How much are your payments? The Equation 27,000 = A(FA ()(12), 4.%/12 ) Cpt $03.36 Fin 311 Chapter Handout Page 9
Annuity nterest Rates You want to have $2 million when you retire in 3 years. You plan to invest $6,000 each year to fund your retirement. What interest rate do you have to earn? The Equation 2,000,000 = A(FA 3, r ) Cpt 10.89 You plan to put $20,000 down on your new house. The house sells for $200,000 and you want to get a 30 year fixed rate mortgage. f the highest payment you can afford is $1,12 per month, what is the highest interest rate you could afford? The Equation 180,000 = A(FA 30(12), r/12 ) Cpt 0.327% = r/12 r = (0.327)(12) = 6.39% Page 10 Fin 311 Chapter Handout
Comparing Cash Flows Suppose a quarterback and wide receiver have both signed new contracts. The quarterback s contract calls for an immediate signing bonus of $6 million and an annual salary of $4 million for three years. The wide receiver s contract calls for a signing bonus of $3. million and an annual salary of $ million for three years. Assuming all salary payments are at the end of the year and the interest rate is 18 percent, which contract is worth more? QB: The Equation = Cpt 8,697,091.72 = $8,697,091.72 + 6,000,000 = $14,697,091.72 = $24,147,792.00 Wide receiver: The Equation = Cpt $10,871,364.6 = $10,871,364.6 + 3,00,000 = $14,371,364.6 = $23,612,612.00 Difference: Time 0 = $32,727.07 Time 3 = $3,180.00 Fin 311 Chapter Handout Page 11
You are trying to plan for retirement on 10 years. You currently have $200,000 in a bond account and $400,000 in a stock account. You plan to add $10,000 per year at the end of each of the next 10 years to your bond account. The stock account earns 12.% and the bond account earns 8.%. When you retire, you plan to withdraw an equal amount for the next 20 years (at the end of each year) and have nothing left. Additionally, when you retire you will transfer your money to an account that earns 7.2%. How much can you withdraw each year? The Bond Account Equation V B 10 = Cpt 42,196.69 + Cpt 148,30.99 = 600,47.68 Or Cpt 600,47.68 The Stock Account Equation V S 10 = Cpt 1,298,928.41 You will have $600,47.68 + 1,298,928.41 = $1,899,476.09 Page 12 Fin 311 Chapter Handout
The Retirement Equation $1,899,476.09 = A(FA 20, 7.2% ) Cpt $182,79.78 You are planning to save for the college education of your children. They are two years apart; one will begin college in 1 years and the other will begin college in 17 years. Assume both will be on the four year plan. You estimate each child s education will cost $23,000 per year, payable at the beginning of each school year. The annual interest rate is 6. percent. How much must you deposit each year in an account to fund your children s education? You will make your last deposit when your oldest child enters college. CF 0 CF 1 F 1 CF 2 F 2 CF 3 F 3 Cpt $17,899.32 Cpt $6,29.8 Growing Perpetuities You will receive perpetuity with a payment of $10,000 next year. The payment will grow by 2. percent per year forever. f the appropriate interest rate is 8 percent, what is the value of the cash flows today? CF r 1 g 10,000 0.08 0.02 $181,181.18 Fin 311 Chapter Handout Page 13
Examples with Multiple Annuities want to withdraw $,000 a year for the next five years and $8,000 a year for the following five years. can earn 10. percent. How much do have to invest today? am asking for the present value of these two annuities. There are several ways to calculate this value. First, draw the time line. 0 r = 10.% 10 A =,000 The next step is to write an equation for the present value: V 0 =,000(FA, 10.% ) + 8,000(FA, 10.% )(F, 10.% ) cpt $18,714.29 cpt $29,942.87 cpt $18,17.32 V 0 = 18,714.29 + 18,17.32 = $36,889.61 A = 8,000 Page 14 Fin 311 Chapter Handout
Here is another way cpt ow my time line looks like this: 0 r = 10.% A =,000 29,942.87 This is the equation: V 0 = cpt $36,889.61 The cash flow worksheet is the last way will show you. CF 0 0 CF 1 F 1 CF 2 F 2 Cpt 36,889.61 0 r = 10.% A =,000 Let s look at this time line again. could ask the following question: want to withdraw $,000 a year for years and have $29,942.87 left in the investment account. How much do have to deposit today if can earn 10. percent? know the answer is $36,889.61 29,942.87 Work this outside of class. Fin 311 Chapter Handout Page 1
Let s look at this problem in a different way. will invest $36,889.61 today in an asset that will earn 10. percent. plan to withdraw $,000 a year for the next five years. How much can withdraw each year for the next years? We know the answer is $8,000 but how would we calculate it? Here is the time line: 0 r = 10.% 10 A =,000 36,889.61 A =? 36,889.61 = cpt $29,942.87 ow have the following timeline: r = 10.% 10 And the equation is: solve the equation like this: A =? 29,942.87 cpt $8,000 Page 16 Fin 311 Chapter Handout
will invest $36,889.61 today in an asset that will earn 10. percent. plan to withdraw $,000 a year for the next five years and $8,000 a year after the first five year period. How many years can withdraw the $8,000? We know the answer is years but how would we calculate it? Here is the time line: 0 r = 10.% =? A =,000 36,889.61 A = 8,000 36,889.61 = cpt $29,942.87 ow have the following timeline: r = 10.% And the equation is: solve the equation like this: A = 8,000 Cpt 29,942.87 Fin 311 Chapter Handout Page 17
will invest $6,000 a year for the next five years and $10,000 a year for the following five years in an asset that earns 11.2 percent. How much do have at the end of 10 years? am asking for the future value of these two annuities. There are several ways to calculate this value. First, draw the time line. 0 r = 11.2% 10 The next step is to write an equation for the future value: V 10 = cpt $37,3.0 cpt $63,994.89 cpt $62,88.42 V 10 = 63,994.89 + 62,88.42 = $126,83.31 Here is another way A = 6,000 cpt $37,3.0 ow my time line looks like this: A = 10,000 r = 11.2% 10 This is the equation: V 10 = A = 10,000 37,3.0 Page 18 Fin 311 Chapter Handout
cpt $126,83.31 The cash flow worksheet is the last way will show you. CF 0 CF 1 F 1 CF 2 F 2 Cpt 126,83.31 Cpt 43,88.94 Let s look at this problem in a different way. want $126,83.31 in 10 years. plan to invest $6,000 a year for the next years in an asset that will earn 11.2 percent. How much must invest each year for the last years in order to meet my goal? We know the answer is $10,000 but how would we calculate it? Here is the time line: 0 r = 11.2% 10 A = 6,000 A =? 126,83.31 126,83.31 = cpt $37,3.0 Fin 311 Chapter Handout Page 19
ow have the following timeline: r = 11.2% 10 And the equation is: solve the equation like this: A =? 37,3.0 126,83.31 cpt -$10,000 want $126,83.31 in 10 years. plan to invest $6,000 a year for the next years and then $10,000 a year in an asset that will earn 11.2 percent. How many years must invest the $10,000? We know the answer is years but how would we calculate it? Here is the time line: 0 r = 11.2% =? A = 6,000 A = 10,000 126,83.31 126,83.31 = 6,000(FA, 11.2% )(F, 11.2% )+ 10,000(FA, 11.2% ) Cpt $37,3.0 Page 20 Fin 311 Chapter Handout
ow have the following timeline: r = 11.2% And the equation is: solve the equation like this: 37,3.0 A = 10,000 126,83.31 Cpt Suppose you borrow $2000 and you are going to make annual payments of $700 each year for 3 years. What is the annual rate of this loan? You want to receive $000 per year for the next years. How much would you need to deposit today if you can earn 9 percent? What rate would you need to earn if you only have $1,000 to deposit? Suppose you have $1,000 to deposit and can earn 9 percent. How many years could you receive the $000 payment? How much could you receive each year for years? Fin 311 Chapter Handout Page 21
Effective Annual Rates (EAR) APR EAR = 1 1 m m Use the nterest Rate Conversion Worksheet 12% monthly compounding = 12.68% 12% semiannual compounding = 12.36% 12% daily compounding = 12.7% EAR versus APR Suppose you go Vito, a local loan shark, to inquire about the interest rate on loans. You are told the interest will be a 20 percent per month. f you are brave enough to ask, what APR will Vito say you are paying? What is the EAR you are paying? APR = om Cpt Eff C/Y 12 Page 22 Fin 311 Chapter Handout
nflation (or Why nflation Doesn t Matter) Today nterest rate One year $100 10% $110 Hot dogs $1 4% $1.04 You can buy 100 10.7692308 What was your rate of return? The Fisher Effect (1 + R) = (1 + r)(1 + h) 1 1.10.04 1 0.07692309 Approximate Fisher Effect R r + h Fin 311 Chapter Handout Page 23
Question You want to buy a motorcycle one year from now, two years from now, and three years from now. The motorcycle currently costs $20,000. You can earn a 10 percent return, and the price of the motorcycle will increase at 4 percent per year. How much do you have to deposit today in order to be able to pay cash for each motorcycle? ominal cash flows r = 10% 0 1 2 3 20,800 21,632 22,497.28 CF 0 0 CF 1 20,800 F 1 1 CF 2 21,632 F 2 1 CF 3 22,497.28 F 3 1 10 Cpt $3,689.32 Real cash flows CF 0 0 3 CF 1 20,000 F 1 3.7692308% or Cpt $3,689.32.7692308% $20,000 Cpt $3,689.32 0 Page 24 Fin 311 Chapter Handout