Finance Problems

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%?

$1084.71 (if stuck page 9 part 2) `

You have just received notification that you have won the $58 million first prize in the Centennial Lottery. However, the prize will be awarded on your 100th birthday (assuming you're around to collect), 73 years from now. What is the present value of your windfall if the appropriate discount rate is 2.2 percent?

$11,844,387

Suppose you are looking at the following possible cash flows: Year 1= CF= $100, Year 2 and 3: $200, Year 4 and 5: $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?

$1226.07 $874.17

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? How much will you have in 5 years if you make no more deposits.

$1248.05 $1616.26

Compute the future value of the following: PV: $85,421 Years: 7 Interest Rate: 7.3%

$139,882

You want to receive $5,000per 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 How much could you receive every month for 5 yrs?

$240,867 1.439 48 $4,151.67

First City Bank pays 4.4 percent simple interest on its savings account balances, whereas Second City Bank pays 9.4 percent interest compounded annually. If you made a $300 deposit in each bank, how much more would the Second City Bank account be worth than the First City Bank account at the end of 2 years?

$33

Compute the present value of the following: Years: 6 Interest rate: 7% FV: $518,181

$345,286

You want to begin saving for your daughter's college education and you estimate that she will need 150,000 in 17 years. If you feel confident that you can earn 8% per year, how much do you need to invest today?

$40540.34

Suppose you have a relative who deposited $10 at 5.5% interest 200 years ago. How much would the investment be worth today? Do this problem with simple interest to see the effect of compounding:

$447,189.85 Simple interest: $120 compounding added $447,069.84 to the value of the investment

You are considering an investment that will pay you $1000 in one year, $2000 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?

$4815.93

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.

$50

Suppose your company expects to increase unit sales of widgets by 15% per year for the next 5 years. If you currently sell 3 million widgets in one year, how many widgets do you expect to sell in 5 years?

$6,034,072

Suppose you need $10,000 in one year for the down payment on a new car. If you can earn 7% annually, how much do you need to invest today?

$9,353.79

Your parents set up a trust fund for you 10 years ago that is now worth 19,671.51. If the fund earned 7% per year, how much did your parents invest?

$9999.99

Suppose you have a 1 year old son and you want to provide $75,000 in 17 years towards his college education. You currently have $5,000 to invest. What interest rate must you earn to have the $75,000 when you need it?

%17.27

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-A because, dealing with annual payments for a finite time period and you want to know value TODAY. and PV is what it is worth TODAY $5,124,150.29

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

Although appealing to more refined tastes, art as a collectible has not always performed so profitably. During 2,007, an auction house sold a sculpture at auction for a price of $10,251. Unfortunately for the previous owner, he had purchased it in 1,998 at a price of $14,985. What was his annual rate of return on this sculpture? (Negative amount should be indicated by a minus sign.

-4.13

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?

.1138655152

Suppose you invest the $1,000 from the previous example for 5 years. How much would you have?

1,276.28 the effect of compound interest is small for a small number of periods, but increases as the number of periods increase

Rust Bucket Motor Credit Corporation (RBMCC), a subsidiary of Rust Bucket Motor, offered some securities for sale to the public on March 28, 2,007. Under the terms of the deal, RBMCC promised to repay the owner of one of these securities $79,277 on March 28, 2,049, but investors would receive nothing until then. Investors paid RBMCC $27,693 for each of these securities; so they gave up $27,693 on March 28, 2,007, for the promise of a $79,277 payment in 2,049. Suppose that, on March 28, 2,024, this security's price is $49,302. If an investor had purchased the security at market on March 28, 2,024, and held it until it matured, what annual rate of return would she have earned?

1.92

What is the present value of $500 received in 5 years if the interest rate is at 10%? 15%? What relationship does this define?

10%: 310.46 15%: $248.59 This defines important relationship II: The higher the interest rate, the lower the present value

Suppose you are committed to owning a $188,229 Ferrari. If you believe your mutual fund can achieve a 11.1 percent annual rate of return and you want to buy the car in 6 years, how much must you invest today?

100,093

Assume that in 2,003, a gold dollar minted in 1,890 sold for $194,505. For this to have been true, what annual rate of return did this coin return for the lucky numismatist?

11.38

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?

12,718.56

Suppose you are offered an investment that will allow you to double your money in 6 years. You have $10,000 to invest. What is the implied rate of interest? What if you have only $1 to invest, what would the implied rate of interest be?

12.25% If you only have $1 to invest and it would be doubling, it would not change anything because it is all relative. The rate of interest required to double your money will be the same regardless of how much you invest. This is the same with tripling your money and quadrupling your money.

You are 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% on very similar investments what is the most you should pay for this one?

1432.93

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 yrs?

147,089.22

At 9.4 percent interest, how many years does it take to quadruple your money?

15.43

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 was made today?

157.08 155.50

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 paymnet

172.88

You expect to receive $9,450 at graduation in two years. You plan on investing it at 12.3 percent until you have $71,017. How long must you wait from now to earn that amount?

19.39

Assume that in January 2,010, the average house price in a particular area was $253,564. In January 2,001, the average price was $200,525. What was the annual percentage increase in selling price?

2.64

You have just made your first $5,830 contribution to your retirement account. Assume you earn an 8.5 percent rate of return and make no additional contributions. What will your account be worth when you retire in 44 years?

211,143

You are scheduled to receive $15,116 in two years. When you receive it, you will invest it for 5 more years at 8.1 percent per year. How much will you have at the end of that investment period?

22,313

You have just made your first $5,700 contribution to your retirement account. Assuming you earn a 13 percent rate of return and make no additional contributions, your retirement account will be worth $592,728 when you retire in 38 years. What will your account be worth at retirement if you had waited 7 years before making the $5,700 contribution?

251,945

You think you will be able to deposit $4,000 at the end of each of the next three years in a bank account paying 8 percent interest. You currently have $7,000 in the account. How much will you have in 3 years? How much in 4 years?

3 years: $21,803.58 4 years: $23,547.87 Use it in excel through FV equation with many records and add them up

Suppose you borrow $2,000 at 5% and you are going to make an annual payments of $734.41. How long before you pay off the loan?

3 yrs part 3 pg 5

Your coin collection contains 54 silver dollars minted in 1,951. If your grandparents purchased them for their face value when the year they were minted, how much will your collection be worth when you retire in 2,074, assuming they appreciate at a 3.4 percent annual rate?

3,299

Suppose you want to buy some new furniture for your family room. You currently have $500 and the furniture you want costs $600. If you can earn 6%, how long will you have to wait if you don't add any additional money?

3.13 years

Rust Bucket Motor Credit Corporation (RBMCC), a subsidiary of Rust Bucket Motor, offered some securities for sale to the public on March 28, 2,001. Under the terms of the deal, RBMCC promised to repay the owner of one of these securities $76,869 on March 28, 2,031, but investors would receive nothing until then. Investors paid RBMCC $29,526 for each of these securities; so they gave up $29,526 on March 28, 2,001, for the promise of a $76,869 payment in 2,031. Based on the $29,526 price, what rate was RBMCC paying to borrow money?

3.24

Rust Bucket Motor Credit Corporation (RBMCC), a subsidiary of Rust Bucket Motor, offered some securities for sale to the public on March 28, 2,008. Under the terms of the deal, RBMCC promised to repay the owner of one of these securities $76,305 on March 28, 2,034, but investors would receive nothing until then. Investors paid RBMCC $24,336 for each of these securities; so they gave up $24,336 on March 28, 2,008, for the promise of a $76,305 payment in 2,034. Suppose that, on March 28, 2,023, this security's price is $46,992. If an investor had purchased it for $24,336 at the offering and sold it on this day, what annual rate of return would she have earned?

4.48

Assume the total cost of a college education will be $119,221 when your child enters college in 17 years. You presently have $55,682 to invest. What annual rate of interest must you earn on your investment to cover the cost of your child's college education?

4.58

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%?

485.97

In 1893, the first Putting Green Championship was held. The winner's prize money was $993. If the winner's prize increases at annual rate of 4 percent, what will it be in 2,051?

487,772

Suppose you want to borrow $20,000 for a new car. You can borrow at 8% per year, compounded monthly. If you take a 4 year loan what is your monthly payment?

488.26 You can use the PMT function in excel to solve this

Given a discount rate of 10%, what is the present value of $500 to be received in 5 years? In 10 years? What relationship does this define?

5 yrs: 310.46 10 yrs: 192.77 Here it is defining important relationship I: The longer the time period, the lower the present 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?

595808.111

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?

6% 1% 1% *You would not be able to divide the last APR value by 2 to get the semiannual rate even though they are both 1%. This would not take into consideration the effect of monthly compounding. Your need an APR based on semiannual compounding to find the semiannual rate.

Sometime investments or loans are based on continuous compounding What is the EAR of 7% compounded continuously?

7.25%

At 9.7 percent interest, how many years does it take to double your money?

7.49

You ran a little short on your spring break vacation, so you put $1,000 on your credit card. You can afford 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?

7.75 years (this is the really long equation) you can use the NPER equation this is only if you do not add anything else to the card! part 3 page 4

In 1,908, the first Putting Green Championship was held. The winner's prize money was $226. In 2,013, the winner's check was $1,277,660. What was the percentage increase per year in the winner's check over this period?

8.58

You're trying to save to buy a new $133,797 Ferrari. You have $61,508 today that can be invested at your bank. The bank pays 8.7 percent annual interest on its accounts. How long will it be before you have enough to buy the car?

9.32

Solve for the unknown interest rate in the following: PV: 975 FV 4770 Years 17

9.79

Imprudential, Inc. has an unfunded pension liability of $252 million that must be paid in 25 years. To assess the value of the firm's stock, financial analysts want to discount this liability back to the present. If the relevant discount rate is 4 percent, what is the present value of this liability?

94,529,434

A) Suppose you need $15,000 in 3 years. If you can earn 6% annually, how much do you need to invest today? B) If you could invest the money at 8%, would you have to invest more or less at 6%? How much?

A) $12594.29 B) At 8% you would be earning $11,907.48. So you would have to invest less of a PV at a higher interest rate. You would need to invest $686.81 less. This goes along with the relationship that the higher the interest rate, the lower the present value.

A) Suppose you have $500 to invest and you believe that you can earn 8% per year over the next 15 years. How much would you have at the end of 15 years using compound interest? B) How much would you have at the end of the 15 years using simple interest?

A) $1586.08 B) $1400

A)Suppose you invest $1,000 for one year at 5% per year. What is the future value in one year? B) Suppose you invest $1,000 for one year at 5% per year. What is the future value in one year? C) Do Part B with simple and compound interest to see the difference

A)$1,050 B) $1,102.50 C) Simple: 1,100 Compound: 1,102.5

Annual Percentage Rate (APR)

AKA nominal, stated, APR, or just rate This is the annual rate that is quoted by law APR= period rate times the number of periods per year To get the period rate we are earing we use APR but doing period rate=APR/number of periods per year You should NEVER divide the effect rate by the number of periods per year because this will not give you the period rate

Suppose you can earn 1% per month on $1 invested today. What is the APR? How much are you effectively earning? What is the ACTUAL rate of interest you are earning, taking into consideration the effect of compounding? Suppose if you put it in another account, you can earn 3% per quarter. What is APR? How much are you effectively earing?

APR=12% you are effectively earning? 1.1268 (this is just the FV) *With simple interest you would have earned only .12 the additional .0068 comes from the compounding of the interest. (using months) Using time value of money equation, we can solve for the rate. 12.68%. because we are taking the value we got from compounding monthly as the FV. If I put money in the bank and only pay once a year 12% but since they are doing it monthly it is 12.68% APR= 12% EAR=12.55%

Annuity Due (for PV-A)

Because payments occur at the beginning of the period of the period when calculating present value, all cash flows of an annuity due are discounted for one less period. Therefore, the PV-A due is greater than the PV-A of a similar ordinary annuity by (1 +r) PV-A (annuity due)= PV of ordinary annuity * (1+r)

What rate should you use to compare alternative investments or loans?

EAR Used to compare two alternative investments with different compounding periods

Annuity:

FINITE series of EQUAL payments that occur at REGULAR intervals For annuity C=PMT in excel

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? (EAR) Let's verify this choice. Suppose you invest $100 in each account. How much will you have in each account after one year? (APR)

How much they are actually earning taking compounding into account_ First account: 5.39% Second account: 5.37% remember keep in mind monthly and semiannually for calc rate and time. first account: 105.39 second account: 105.37 first account wins

Perpetuity

INFINITE series of EQUAL payments

Ordinary Annuity

If the first payment occurs at the end of the period

Annuity Due

If the first payments occurs at the beginning of the period

When might you want to compute the number of periods?

If we know the present value (PV), the future value (FV), and the interest rate per period of compounding (i), the future value factors allow us to calculate the unknown number of time periods of compound interest (n).

You are looking at an investment that will pay $1,200 in 5 years if you invest $1,000 today. What is the implied rate of interest?

In excel you use the Rate function (Pv is neg because what you are giving up and FV is positive because what you are getting) 3.714%

You want to purchase a new car and you are willing to pay $20,000. If you can invest at 10% per year and you currently have $15,000, how long will it be before you have enough money to pay cash for the car?

Number of periods is NPER function in excel: 3.02 years

The frequency of the _______ drives _______

PAYMENT, EVERYTHING

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?

PV because what you borrow is what you take out today

Suppose you borrow $10,000 from your parents to buy a car. You agree to pay $207.58 per month for 60 months. What is the monthly interest rate?

PV-A because we are borrowing money Remember these relationships: If computed PV is greater than loan amt, then the interest rate is too low If computed PV is lower than the loan amt then the interest rate is too high

Preferred stock (or preference stock) is an important example of a perpetuity. When a corporation sells preferred stock, the buyer is promised a fixed cash dividend every period (usually every quarter) forever. This dividend must be paid before any dividend can be paid to regular stockholders, hence the term preferred.

Preferred stock is a promised fixed dividend that can go on forever

What is the relationship between future value and present value?

Present Value is the sum of money that must be invested in order to achieve a specific future goal. Future value is the dollar amount that will accrue over time when that sum of money is invested. The present value is the amount you must invest to realize the future value. The mathematical relationship is Fv = PV(1 + r)^t. The present value is always less than the future value when we have positive rates of interest. The present value is the current value of an amount to be received in the future

Effective Annual Rate (EAR)

Rate you are actually paying/receiving but takes into consideration compounding This is the actual rate paid or received after accounting for compounding that occurs during the year If you want to compute two alternative investments with different compounding periods you need to compute the EAR and use that for comparison

What is the difference between simple interest and compound interest?

Simple Interest: You only earn on the original amount deposited (the principal) Compound Interest: You earn interest on the interest

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?

Since first payment is made TODAY it is an annuity due. FV-A Due In excel place a 1 in type field for annuity due Since payment is due today you can start on 0 instead of 1 or you can do normal equation * (1+r)

Suppose you want to buy a new house. You currently have $15,000 and you figure you need to have a 10% down payment plus an additional 5% in closing costs. (Closing costs =5% of the amount borrowed). If the type of house you want costs about $150,000 and you can earn 7.5% per year, how long will it be before you have you enough money for the down payment and closing costs?

Step 1: How much do you need in the future? $21,750 Step 2: How long? 5.14 years

Interest Rate:

The "exchange rate" between earlier money and later money AKA: discount rate, cost of capital, opportunity cost of capital, required return

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?

The PV of the future cash flows is 91.49. You should not accept this because the broker is charging you more than you would be willing to pay.

Suppose you are offered the following investment choices: *You can invest $500 today and receive $600 in 5 years. The investment in considered low risk. *You can invest the $500 in a bank account paying 4%. *What is the implied interest rate for the first choice and what investment should you choose?

The implied rate for the first choice is: 3.71% You should pick the one with 4% because you will receive $608.33 instead of $600.

What is the definition of EAR?

The rate expressed as though it were compounded once per year

Annuity Due (for FV-A)

These are payments that come at the beginning instead of the end Since the payments occur at the beginning of the period when calculating future value, all the cash flows of an annuity due are compounded for one additional period. Therefore, the FV-A due is great than the FV of a similar ordinary annuity by (1+r) FV-A (annuity due) = FV of ordinary annuity * (1 + r)

What are some situations in which you might want to compute the implied interest rate?

When there are more than one opportunities to invest, we might want to compute for their risks and Future Values. higher interest and low risk is more preferable. When there are more than one opportunities to invest, we might want to compute for their risks and Future Values. higher interest and low risk is more preferable.

After carefully going over your budget, you have determined you can afford to pay a $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?

Why is this a PV-A? *the payment is 632 PER month (equal payment) for 48 months (finite series) *It is PV because you want to know how much you can borrow. So you want that TODAY *PV-A is for LOANS because you want to know how much you can borrow today. Since rate and payment is in months no need to change anything $23,999.54

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?

current required return: 40= 1/r r=.025 or .25% per quarter Dividend for new preferred: 100=C/.025 C=2.50 per quarter

Present Value:

earlier money on a time line

for calc APR use NOMINAL function

effective rate= EAR npery= m

Future value:

later money on a time line

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. 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?

loan? Monthly income: 36,000/12= 3,000 Max. Payment= .28(3,000)=840 Max. Loan amt= PV=840[1-1/1.005^360]/.005=140105 Offer? Loan+Downpayment Closing costs= .04(140,150)=5,604 Downpayment= 20,000-5604= 14396 Total Price: 140,105+14396= 154501

Discount rate:

often, we will want to know the implied interest rate on an investment, so we will want to solve for r. A saving bond does not specifically say the interest rate, so you can figure it out through the values you do have

APR is the..?

quoted rate

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?

this is a FV-A because set payment per year and want to know how much we will have in 40 yrs so FV $454,513.04

You solve for EAR in excel by..

using the EFFECT function nominal rate= APR Npery= # of periods per yr.

When we talk about the "value" of something...

we are talking about the present value unless we specifically indicate that we want the future value

What do we mean by "discounting"?

when we talk about discounting, we mean finding the present value of some future amount, If I have a dollar today, I can be investing and earing something