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**Discounted Cash Flow Valuation**

Chapter 6

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**Key Concepts and Skills**

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 interest rates are quoted Understand how loans are amortized or paid off

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**Chapter Outline Future and Present Values of Multiple Cash Flows**

Valuing Level Cash Flows: Annuities and Perpetuities Comparing Rates: The Effect of Compounding Loan Types and Loan Amortization

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**Future value of multiple Cash Flows**

Suppose you deposit $100 today in an account paying 8 percent. In one year, you will deposit another $100. how much will you have in two years?

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Ex 6.1 You think you will be able to deposit $4,000 at the end of each year of the next three years in a bank account paying 8 percent interest. You currently have $7,000 in the account. How much will have in three years? Four years?

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**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) (1.09) = 1,248.05

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**Multiple Cash Flows – Example 2 Continued**

How much will you have in 5 years if you make no further deposits? First way: FV = 500(1.09) (1.09)4 = 1,616.26 Second way – use value at year 2: FV = 1,248.05(1.09)3 = 1,616.26

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Example Suppose you deposit $2,000 at the end of each of the next five years. How much will you have at the end of the fifth year if the rate was 10 percent?

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**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) (1.08)2 = =

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Ex 3 Page179 Paradise, Inc., has identified an investment project with the following cash flows. If the discount rate is 8 percent, what is the future value of these cash flows in 4 years? Cash Flow Year 940 1 1090 2 1340 3 1405 4

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Ex 16 Page 180 What is the future value of $ 2,100 in 17 years assuming an interest rate of 8.4 percent compounded semiannually?

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**Present value of multiple Cash Flows**

Suppose you need $1,000 in one year and $2,000 more in two years. If you can earn 9 percent on your money, how much do you have to put up today to exactly cover these amounts in the future?

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Example Suppose that we had an investment that was going to pay $1,000 at the end of every year for the next five years. If this investment owns a 6 percent rate what is the present value of these cash flows?

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**Multiple Cash Flows – Present Value Example 6.3**

You 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 fourth year. You can earn 12 percent on very similar investments . What is the most you should pay for this one?

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Example 6.3 Timeline 1 2 3 4 200 400 600 800 178.57 318.88 427.07 508.41 1,432.93

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**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 $3000 in three years. If you want to earn 10% on your money, how much would you be willing to pay?

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Quick Quiz – Part I 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?

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Ex1 Page 179 Seaborn Co. has identified an investment project with the following cash flows. If the discount rate is 10 percent, what is the present value of these cash flows? Cash Flow Year 950 1 1040 2 1130 3 1075 4

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Ex 2 Page 179 Investment X offers to pay you $6,000 per year for nine years, whereas investment Y offers to pay you $8,000 per year for six years. Which of these cash flow streams has the higher present value if the discount rate is 5 percent?

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Ex 44 Page 183 You just won the TVM lottery. You will receive $1 million today plus another 10 annual payments that increase by $500,000 per year. Thus , in one year, you receive $1,5 million. In two years you get $2 million, and so on. If the appropriate interest rate is 9 percent, what is the present value of your winnings?

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Annuities 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

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**Annuities and Perpetuities – Basic Formulas**

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Example Suppose that we are examining an asset that promised to pay $500 at the end of each of the next three years. If we want to earn 10 percent on our money, how much would we offer for this annuity?

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Annuity – Example 6.5 After carefully going over your budget, you have determined you can afford to pay $632 per month toward a new sport car. You call up your local bank and find out that the going rate is 1 percent per month for 48 months. How much can you borrow?

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**Annuity – Sweepstakes Example**

Suppose you win the Publishers Clearinghouse $10 million sweepstakes. The money is paid in equal annual installments of $333, 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

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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?

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**Buying a House - Continued**

Bank loan Monthly income = 36,000 / 12 = 3,000 Maximum payment = .28(3,000) = 840 PV = 840[1 – 1/ ] / .005 = 140,105 Total Price Closing costs = .04(140,105) = 5,604 Down payment = 20,000 – 5604 = 14,396 Total Price = 140, ,396 = 154,501

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Quick Quiz – Part II 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?

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Finding the Payment Suppose you want to borrow $20,000 for a new car. You can borrow at 8% per year, compounded monthly (8/12 = % per month). If you take a 4 year loan, what is your monthly payment?

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**Finding number of payments**

How many $100 payments will pay off a $5,000 loan at 1% per period?

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**Finding the Number of Payments – Example 6.6**

You ran little short on your spring money vacation, so you put $1,000 on your credit card. You can offered only the minimum payment of $20 per month. The interest rate on the credit card is 1.5 percent per month. How long will you need to pay off the $1,000?

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**Finding the Number of Payments – Another Example**

Suppose you borrow $2,000 at 5% and you are going to make annual payments of $ How long before you pay off the loan? 2,000 = (1 – 1/1.05t) / .05 = 1 – 1/1.05t 1/1.05t = = 1.05t t = ln( ) / ln(1.05) = 3 years

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Finding the Rate An insurance company offers to pay you $1,000 per year for 10 years if you will pay $6,710 up front. What rate is implicit in this 10-year annuity?

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Finding the Rate Suppose a relative of yours wants to borrow $3,000. she offers to repay you $1,000 every year for four years. What interest rate you bring offered?

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Finding the Rate Suppose you borrow $10,000 from your parents to buy a car. You agree to pay $ per month for 60 months. What is the monthly interest rate? Sign convention matters!!! 60 N 10,000 PV PMT CPT I/Y = .75%

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Ex 4 Page 179 An investment offers $5,300 per year for 15 years , with the first payment occurring one year from now. If the required return is 7 percent what is the value of the investment?

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Ex 26 Page 181 Beginning three months from now, you want to be able to withdraw $2,300 each quarter from your bank account to cover college expenses over the next four years, if the account pays .65 percent interest per quarter, how much do you need to have in you bank account today to meet you expense needs over the next four years?

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Quick Quiz – Part III You want to receive $5,000 per month for the next 5 years. How much would you need to deposit today if you can earn .75% per month? What monthly rate would you need to earn if you only have $200,000 to deposit? Suppose you have $200,000 to deposit and can earn .75% per month. How many months could you receive the $5,000 payment? How much could you receive every month for 5 years?

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**Future Values for Annuities**

Suppose you plan to contribute $2,000 every year to a retirement account paying 8 percent. If you retire in 30 years, how much will you have?

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**Future Values for Annuities**

Suppose you begin saving for your retirement by depositing $2,000 per year in an IRA. If the interest rate is 7.5%, how much will you have in 40 years?

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Ex 8 Page 180 You want to have $ 90,000 in your savings account 10 years from now, and you are prepared to make equal annual deposits into the account at the end of each year. If the account pays 6.8 percent interest, what amount must you deposit each year?

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Annuity Due If the first payment occurs at the beginning of the period, it is called an annuity due

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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?

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Annuity Due Timeline 32,464

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**Perpetuity Perpetuity – infinite series of equal payments**

Perpetuity formula: PV = C / r An investment offers a perpetual cash flow of $500 every year. The return you require on such an investment is 8 percent. What is the value of this investment?

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Example 6.7 Suppose the fellini Co. wants to sell preferred stock at $100 per share. A similar issue of preferred stock already outstanding has a price of $40 per share and offers a dividend of $1 every quarter. What dividend will Fellini have to offer if the preferred stock is going to sell?

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**Growing Annuities and perpetuities**

Suppose that we are looking at a lottery payout over 20-year period. The first payment, made one year from now, will be $200,000 every year thereafter, the payment will grow by 5 percent, what’s the present value if the appropriate discount rate is 11 percent?

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Quick Quiz – Part IV You want to have $1 million to use for retirement in 35 years. If you can earn 1% per month, how much do you need to deposit on a monthly basis if the first payment is made in one month? What if the first payment is made today? You are considering preferred stock that pays a quarterly dividend of $1.50. If your desired return is 3% per quarter, how much would you be willing to pay?

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Table 6.2

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**Effective Annual Rate (EAR)**

This is the actual rate paid (or received) after accounting for compounding that occurs during the year If you want to compare two alternative investments with different compounding periods you need to compute the EAR and use that for comparison.

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**Example Suppose that you have three options to borrow**

Bank A: 15 percent compounded daily Bank B : 15.5 percent compounded quarterly Bank C: 16 percent compounded annually Which of these is the bets if you are thinking of opining a savings account?

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Example 6.9 As a lender you want to quote a rate that features monthly compounding. What rate do you quote?

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**Annual Percentage Rate**

This is the annual rate that is quoted by law By definition APR = period rate times the number of periods per year Consequently, to get the period rate we rearrange the APR equation: Period rate = APR / number of periods per year You should NEVER divide the effective rate by the number of periods per year – it will NOT give you the period rate

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**Computing APRs What is the APR if the monthly rate is .5%?**

What is the APR if the semiannual rate is .5%? What is the monthly rate if the APR is 12% with monthly compounding?

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EAR - Formula

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Things to Remember You ALWAYS need to make sure that the interest rate and the time period match. If you are looking at annual periods, you need an annual rate. If you are looking at monthly periods, you need a monthly rate. If you have an APR based on monthly compounding, you have to use monthly periods for lump sums, or adjust the interest rate appropriately if you have payments other than monthly

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**Computing APRs from EARs**

If you have an effective rate, how can you compute the APR? Rearrange the EAR equation and you get:

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Example Suppose you are going to have $50 deducted from your paycheck every two weeks and have it placed in an account that pays 8% compounded daily. How much will you have in 35 years?

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**Continuous Compounding**

Sometimes investments or loans are figured based on continuous compounding EAR = eq – 1 The e is a special function on the calculator normally denoted by ex Example: What is the effective annual rate of 7% compounded continuously? EAR = e.07 – 1 = or 7.25%

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**Computing EARs - Example**

Suppose you can earn 1% per month on $1 invested today. What is the APR? 1(12) = 12% How much are you effectively earning? FV = 1(1.01)12 = Rate = ( – 1) / 1 = = 12.68% Suppose if you put it in another account, you earn 3% per quarter. What is the APR? 3(4) = 12% FV = 1(1.03)4 = Rate = ( – 1) / 1 = = 12.55%

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**Decisions, Decisions II**

You are looking at two savings accounts. One pays 5.25%, with daily compounding. The other pays 5.3% with semiannual compounding. Which account should you use? First account: EAR = ( /365)365 – 1 = 5.39% Second account: EAR = ( /2)2 – 1 = 5.37% Which account should you choose and why?

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**Decisions, Decisions II Continued**

Let’s verify the choice. Suppose you invest $100 in each account. How much will you have in each account in one year? First Account: Daily rate = / 365 = FV = 100( )365 = Second Account: Semiannual rate = / 2 = .0265 FV = 100(1.0265)2 = You have more money in the first account.

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APR - 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?

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**Computing Payments with APRs**

Suppose you want to buy a new computer system and the store is willing to sell it to allow you to make monthly payments. The entire computer system costs $3,500. The loan period is for 2 years and the interest rate is 16.9% with monthly compounding. What is your monthly payment? Monthly rate = .169 / 12 = Number of months = 2(12) = 24 3,500 = C[1 – (1 / )24] / C =

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**Future Values with Monthly Compounding**

Suppose you deposit $50 a month into an account that has an APR of 9%, based on monthly compounding. How much will you have in the account in 35 years? Monthly rate = .09 / 12 = .0075 Number of months = 35(12) = 420 FV = 50[ – 1] / = 147,089.22

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**Present Value with Daily Compounding**

You need $15,000 in 3 years for a new car. If you can deposit money into an account that pays an APR of 5.5% based on daily compounding, how much would you need to deposit? Daily rate = .055 / 365 = Number of days = 3(365) = 1,095 FV = 15,000 / ( )1095 = 12,718.56

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**Quick Quiz – Part V What is the definition of an APR?**

What is the effective annual rate? Which rate should you use to compare alternative investments or loans? Which rate do you need to use in the time value of money calculations?

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**Pure Discount Loans – Example 6.12**

Treasury bills are excellent examples of pure discount loans. The principal amount is repaid at some future date, without any periodic interest payments. If a T-bill promises to repay $10,000 in 12 months and the market interest rate is 7 percent, how much will the bill sell for in the market? PV = 10,000 / 1.07 = 9,345.79

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**Interest-Only Loan - Example**

Consider a 5-year, interest-only loan with a 7% interest rate. The principal amount is $10,000. Interest is paid annually. What would the stream of cash flows be? Years 1 – 4: Interest payments of .07(10,000) = 700 Year 5: Interest + principal = 10,700 This cash flow stream is similar to the cash flows on corporate bonds and we will talk about them in greater detail later.

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**Amortized Loan with Fixed Principal Payment - Example**

Consider a $50,000, 10 year loan at 8% interest. The loan agreement requires the firm to pay $5,000 in principal each year plus interest for that year.

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**Amortized Loan with Fixed Payment - Example**

Each payment covers the interest expense plus reduces principal Consider a 4 year loan with annual payments. The interest rate is 8% and the principal amount is $5,000.

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Ex 55, 56 Page 184 Prepare an amortization schedule for a five-year loan of $42,000. the interest rate is 8 percent per year, and the loan calls for equal annual payments. How much interest is paid in the third year? How much total interest is paid over the life of the loan? Rework the problem assuming that the loan agreements calls for a principle reduction of $8,400 every year instead of equal annual payment

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Ex 69 Page 187 You have just arranged for a $750,000 mortgage to finance the purchase of a large tract of land. The mortgage has an 8.1 percent APR, and it calls for monthly payments over the next 30 years. However, the loan has an eight-year balloon payment, manning that the loan must be paid off then. How big will the balloon payment be?

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Quick Quiz – Part VI What is a pure discount loan? What is a good example of a pure discount loan? What is an interest-only loan? What is a good example of an interest-only loan? What is an amortized loan? What is a good example of an amortized loan?

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**Comprehensive Problem**

An investment will provide you with $100 at the end of each year for the next 10 years. What is the present value of that annuity if the discount rate is 8% annually? What is the present value of the above if the payments are received at the beginning of each year? If you deposit those payments into an account earning 8%, what will the future value be in 10 years? What will the future value be if you opening the account with $1,000 today, and then make the $100 deposits at the end of each year?

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Ex 10 & 11 Page 180 The Maybe Pay Life Co. is trying to sell you an investment policy that will pay you and your heirs $25,000 per year for ever. If the required return on this investment is 7.2 percent, how much will you pay for the policy? Suppose a sales associate told you the policy costs $375,000. at what rate would this be a fair deal?

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Ex 15 Page 180 Barcian Credit Crop. Wants to earn an effective annual return on its consumer loans of 16 percent per year. The bank uses daily compounding on its loans. What interest rate the bank required by law to report to potential borrowers? Explain why this rate is misleading to an uninformed borrower?

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Ex 22 Page 181 Friendly Quick Loans, Inc., offers you “three for four or 1 knock on your door”. This means you get $3 today and repay $4 when you get your paycheck in one week (or else). What’s the effective annual return Friendly’s earns on this lending business? If you were brave enough to ask, what APR would Friendly’s say you were paying?

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Ex 30 Page 182 You are looking at an investment that has an effective annual rate of 17 percent. What is the effective monthly return?

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Ex 32 Page 182 You are planning to save for your retirement over the next 30 years. To do this, you will invest $700 a month in a stock account and $300 a month in a bond account. The return of the stock account is expected to be 11 percent, and the bond account will pay 6 percent. When you retire you will combine your money into an account with 9 percent return. How much can you withdraw each month from your account assuming a 25-year withdraw period?

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Ex 35 Page 182 Suppose an investment offers to triple your money in 12 months. What rate of return per quarter are you being offered?

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Ex 36 Page 182 You’ve just joined the investment banking firm of Dewey, Cheatum, and Howe. They’ve offered you two different salary arrangements. You can have $95,000 per year for the next two years, or you can have $70,000 per year for the next two years, along with $45,000 signing bonus today. The bonus is paid immediately, and the salary is paid at the end of each year. If the interest rate is 10 percent compounded monthly, which do you prefer?

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Ex 38 Page 183 Your job pays you only once a year for all the work you did over the previous 12 months. Today, December 31, you just received your salary of $50,000 and you plan to spend all of it. However, you want to start saving for retirement beginning next year. You have decided that one year from today you will begin depositing 5 percent of your annual salary in an account that will earn 11 percent per year. Your salary will increase at 4 percent per year throughout your career. How much money will you have on the date of your retirement 40 years from today?

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