Finance Questions PS2

Suppose you are able to borrow $10,000 and pay back $23,674 in 10 years. What is the interest rate you are charged compounded yearly?

10,000 * FVIF (i,10) = 23,674 FVIF(i,10) = 2.3674 OR 10,000 = 23,674 * PVIF (i, 10) PVIF (i, 10) = 0.4224 i=9%

You just borrowed $10,000, which you will repay in monthly installments in 2 years. What is the annual interest rate if the monthly payment is $528.71?

10,000 = 528.71 * PVIFA (i%, 24) PVIFA (i%, 24) = 18.9139 i = 2% Annual interest rate = 24%

You have just received an offer from one of the top investment banks. You plan to quit work and go to business school after working for 2 years. You estimate that in two years' time you'll need $100,000 to cover your graduate school expenses. How much money do you need to set aside each month to meet your expenses in two years if the annual interest rate is 12%?

100,000 = monthly payment * FVIFA (1%, 24) 100,000 = x * 26.973 monthly payment = $3,707.41

You're trying to save to buy $129,736 Ferrari. You have $40,000 today that can be invested at your bank. The bank pays 4% annual interest on its accounts. How long will it be before you have enough to buy the car?

12,9736 = 40,000 * FVIF(4%, n) FVIF(4%,n) = 3.2434 n = 30 (years)

If you borrow $1577.38 at 6% interest compounded annually and agree to pay back $200 per year at the end of each year how long will it take you to pay back the loan?

1577.38 = 200 × PV IFA (6%, n years) 7.8869 = ×PV IFA(6%,n years) Look at the 6% column in table A.3 Go down until you find 7.8869 Follow the row to the left to see the number of years. 11 years.

15. You just deposit $20,000 into a savings account that pays interest of 12% per year, compounded semiannually (every six months). What equal dollar amount can you withdraw from the account semiannually, starting today, for a total of twelve withdrawals, such that you have nothing left in the account after the twelfth withdrawal?

20000 = A + A[PVIFA (11n, 6 %)] 20000 = A + A[PVIFA (7.887) A = 20000 / 8.887 = 2250

Lyons Bank offers you a $25,000, seven-year term loan at 11% annual interest. What will you annual loan payment be?

25000 = A * PVIFA(11%, 7) = A * 4.7122A = 5,305.38

You want to lease a new RV from Muscle Motors for $42,000. The lease contract is in the form of 31-month annuity due, i.e., each level payment occurs in the beginning of the month, at a 12% annual interest rate. What will your monthly lease payment be?

42,000 = A[1+PVIF(0.12/12,31-1)] = A * (1+25.8077)A = 42,000/26.8077 = 1,566.71

A 1949 Vincent Black Shadow Series B vintage motorcycle sold for about $46,846 in 1999. If your father was fortunate enough to have purchased one new for $630 in 1949, what return did he earn on his investment?

46846 = 630 * FVIF(i, 50)FVIF(i, 50) = 74.3575i = 9%

15. Imagine the same scenario in the previous problem. Now, how much should you be saving every year to be able to make the down payment?

50,000 = 1,000*FVIF (6%, 10) + A*FVIFA (6%, 8) 50,000 = 1,000*1.7908 + A*9.8975 A = 4870.8

You have $77,217 in an account that earns 5% per year, compounded annually. You plan to withdraw $10,000 per year at the end of each year. How long can you do so?

77, 217 = 10, 000 × PV IFA (5%, n years) 7.7217 = PV IFA (5%, n years) n = 10 years

15. On December 31, XYZ Company buys a building for $60,000, payable 15% down and the balance in 20 equal annual installments that are to include principal plus 10% compound interest on the declining balance. What are the approximate equal installments?

Amount Needed = $ 60000*0.85 = $51000 51000 = A [PVIFA (10%, 20)] A = 51000/8.5136 = $5990

How much are you willing to pay for a 10 year, $1,000 bond with coupon payments of $100 per year if the rate of return required by investors is 10%?

Answer: This bond is selling at par so it is worth $1,000.

15. You have $1000 in your savings account today and plan to graduate college in 2 years. After that you plan to take up a job and save $4000 every year (deposits are to be made at the end of each year) towards buying a house in 10 years. Will your savings be enough for a down payment of 10% on a $500,000 house? Assume that the interest rate is 6% compounded annually.

FV = 1,000*FVIF (6%, 10) + 4,000*FVIFA (6%, 8) FV = 1,000*1.7908 + 4,000*9.8975 FV = 41,380.8 The required down payment = 0.1*500,000 = 50,000. Thus you will not have enough cash to get the mortgage.

If you borrow $100,000 for 10 years at 5% interest compounded annually how much will you have to pay back at the end?

FV = 100, 000 × FV IF (5%, 10y) = 100, 000 × 1.6289 = 162, 890

You are scheduled to receive $22,000 from Charles Foundation in two years. When you receive it, you will invest it for six more years at 6% per year. How much will you have in eight years?

FV = 22000 * FVIF(6%, 6) = 31,207.42

Commonwealth Inc. has identified an investment project with the following cash flows: Year 1: $500; Year 2: $600; Year 3: $700, Year 4: $800. What is the future value of these cash flows in Year 4, if the discount rate is 8%?

FV = 500 FVIF(8%, 3) + 600 FVIF(8%, 2) + 700 FVIF(8%, 1) + 800 = 2,885.70

If you put $500 per year at the end of each year into a bank account paying 3% interest compounded annually how much will you have in 6% years?

FV = 500 × FV IFA (3%, 6y) = 500 × 6.4684 = 3234.20

What is the future value of $750 in 6 years assuming an interest rate of 10% compounded semiannually?

FV = 750 * FVIF(10%/2, 6*2) = 750 * FVIF(5%, 12) = 750*1.7959 = 1,346.89

You decide to take up on Bank of America's offer to open up a High Yield CD account with a deposit $5000. The interest rate offered is 5% and it is compounded annually. How much will you have in your account after 1 year? 5 years? 10 years? Why is the interest earned ($ amount) in the last 5 years greater than the first 5 years?

FV=5000*FVIF (5%, 1) =5000*1.0500 =5250 FV=5000*FVIF (5%, 5) =5000*1.2763 =6381.5 FV=5000*FVIF (5%, 10) =5000*1.6289 =8144.5 The interest earned in the last 5 years is higher because you already have a higher balance invested by the end of the first 5 years - $6381 as opposed to $5000. This problem illustrates the fact that compound interest grows faster as the time horizon increases.

After you graduate, you will be getting phone calls from life insurance companies looking to sell you a wide variety of financial services products. Those of you who work for such companies will probably be the ones making the calls. One of the products you hear about promises that if you pay $500 per year for the next 10 years, you will receive $500 per year forever, beginning in year 11. If the interest rate is 9%, is this a wise investment?

Figure out how much you will end up paying (FV) if you accept the insurance, then calculate what you will receive in the future from the insurance (calculate this as a perpetuity). Compare the two: Pay: = 500*FVIFA (9%, 10) = 500*15.193 = $7,596.50 Receive Perpetuity of $500 per year at 9% interest = c/r = 500/0.09 = $5,555.56 So at 9% interest, you pay more than you receive! Not a wise investment.

8 years later, when there is only 1 coupon payment left, the bond sells for $1047.62. What is the interest rate at that time?

Next year the bondholder will receive 1 coupon payment and the principle so: 1047.62 = 100 × PV IF (I%, 1y) + 1000 × PV IF (I%, 1y) = 1, 100 × PV IF (I%, 1y) 0.9524 = PV IF (I%1y) I = 5%

Suppose you are considering purchasing a new car and you have choice between two cars that are identical in every way except that car "A" uses gasoline at the rate of 20 miles per gallon and car "B" uses gasoline at the rate of 40 miles per gallon. Suppose that you drive 10,000 miles each year, pay all your gasoline bills at the end of each year, throw the car away at the end of 10 years, and gasoline sells for $1.30 per gallon for ten years. How much extra will you be willing to pay for car "B" over car "A" if your opportunity cost is 8%?

Number of gallons for Car A per year = 10000 / 20 = 500g Number of gallons for Car B per year = 10000 / 40 = 250g Saving of gallons = 500 - 250 = 250g Savings of cash = 250 * 1.3 = $325 yearly P = 325 [PVIFA (10y, 8%)] = 325 (6.71) = $2180

You decide to buy a 2009 Porsche Boxster and go to the dealer. He offers you two financing options. You can either take out a 30 month loan for the full purchase price of the car and pay $1800 every month, or alternatively, you can put in a down payment of $5000 and take a 50 month loan with payments of $1,057.60 per month. Which one do you think is a better deal? (Assume an annualized interest rate of 12%)

Option 1: PV = $1800*PVIFA [1%, 30] = $46453 Option 2: PV = $5000 + $ $1,057.60*PVIFA [1%, 50] = $46453 Since the present values are equal, neither financing option is superior.

Determine how much you must deposit today, January 1, to be able to withdraw $100 on July1, August 1, September 1, and October 1. Assume that the interest rate is 24% per year compounded monthly.

P = 100 [PVIFA (9, 2%) - PVIFA (5, 2%)] = 100 (8.1622 - 4.7135) = 100 (3.45) = $345 OR P = 100 [PVIFA (4, 2%)] = 100 (3.808) = 380.8 P = 380.8 [PVIF (5, 2%)] = 380.8 (.906) = 345

How much is it worth 1 year later if the interest rate falls to 8%?

P = 100 × PV IFA (8%, 9y) + 1000 × PV IF (8%, 9y) = 100 × 6.2469 + 1000 × 0.5002) = 1124.89

What is the present value of receiving $300 for 6 times every quarter starting from today, nothing for three quarters and $250 for 6 more quarters. Discount rate is 12% compounded quarterly.

P = 300 + 300 * PVIFA (3%, 5) + 250 * PVIFA (3%, 6) * PVIF (3%,8) P = 300 + 300 * 4.580 + 250 * 5.417 * 0.789 = $ 2742.503 OR: P= 300 + 300 * PVIFA (3%, 5) + 250 [PVIFA (3%, 14) - PVIFA (3%, 8)] P = 300 + 300 * 4.580 + 250 [11.30 - 7.020] = $2744 Differences in results are due to rounding errors...!

You have just received notification that you have won the $1 million first prize in the Monstah Lottery. However, the prize will be awarded on your 50th birthday. Assuming you're 20 years old now. What is the present value of your windfall if the discount rate is 16%?

PV = 1,000,000 * PVIF(16%, 50-20) = 1,000,000 * 0.0116 = 1,164.82

Suppose you are still committed to owning a $129,736 Ferrari. If your mutual fund can achieve a 9% annual return and you want to buy the car in 10 years, how much must you invest today?

PV = 129,736 * PVIF(9%, 10) = 129,736 * 0.4224 = 54,802

An investment offers $2,250 per year for 15 years, with the first payment occurring one year from now. If the required return is 10%, what is the value of the investment?

PV = 2250 * PVIFA(10%, 15) = 2250 * 7.6061 = 17,114

How much do you need to put into a bank account today if it earns 3% per year compounded annually if you want to buy a $30,000 boat 4 years from now?

PV = 30, 000 × PV IF (3%, 4y) = 30, 000 × 0.8885 = 26, 655

What is the present value of $300 paid at the end of each year for the next 6 years at 7% interest?

PV = 300 × PV IFA (7%, 6y) = 300 × 4.7665 = 1429.95

Here's a retirement problem. You start your first job at 22 and want to make payments towards your pension. The question is how much of a cut should you be taking from each of your paychecks. You expect to live until you're 90 years old, but want to retire at 72 and draw $60,000 income from your pension plan until you die. At a constant interest rate of 8%, find the amount you should be depositing every year.

PV = 60,000*PVIFA (8%, 18) = 60,000*9.3719 = 562,314 FV = A* FVIFA (8%, 50) 562,314 = A* 573.7702 A = 980

You learn that you've won the $206 million jackpot in the Mega Millions lottery! But now you have to decide on the payment option. You can choose either an annuity payment of $7.9 million over 25 years or an immediate cash payment of $120 million. If the interest rate today is 4% (and assumed to stay at that level) which one would you choose? If you expect the interest rates to rise would that affect your choice?

PV = 7.9m*PVIFA (4%, 25) PV = 7.9m*15.6221 PV = 123.4m Since 123.4>120, it's better to wait and collect the annuity payments. However, if the interest rates were to increase the present value of the annuity would decrease and the $120m in cash might turn out to be a better deal. Another way to look at it is that if the rates increased I could invest the cash at a higher rate.

Fidelity has just signed a 15-year lease to rent out office space for $60,000 per year. Payments are made at the beginning of each year. If the firm's interest rate is 8%, find the present value of the lease payments.

PV= 60,000 + 60,000*PVIFA (8%, 14) = 60,000 + 60,000 *8.244 = 60,000 + 494,640 = 554,640

You have decided to buy a laptop. The store offers you two financing plans. You can either buy the computer for $1600 in cash or pay $100 each month for 24 months. You have $1600 in cash right now. The interest rate your bank offers you on money invested is 12% annually, compounded monthly. What will you do, i.e. buy the computer using your cash on hand or buy the computer in 100$ installments?

Present value of buying the computer with ready cash = $1600 Present value of buying the computer paying 100$ monthly = 100 * PVIFA (1%, 24) = 100 * 21.2434 = $2,124.34 You are better off buying the laptop with immediate cash. Indeed, were you to choose the installment plan and invest your $1600 in the bank, drawing the deposit down by $100 each month, you would be out of money in the 16th month. 160000 * PVIFA (1%, n) PVIFA (1%, n) = 16 n=15

You have won a small sweepstakes. You receive a letter that claims you have won $5,150, which will be divided into three payments. You will receive $750 at the end of year 1, $1,800 at the end of year 2, and $2,600 at the end of year 3. What is the present value of this prize? Assume r = 17%.

Simply discount back the three payments with interest rate = 17%: PV = 750/1.17 + 1800/1.17^2 + 2600+1.17^3= $3,579.31

1. If you deposit $750 in a money market deposit account paying 7%, how much will it be worth in 9 years?

Since this is only one payment, it is calculated as a simple future value problem: FV=750*FVIF (7%, 9) =750*1.838 =1,378.5

Janet purchased a home for $250,000, and made a $30,000 down payment. She took out a 30-year mortgage on the $220,000 balance. If her mortgage rate is 12% per year, compounded yearly, what are her yearly mortgage payments?

This is a simple annuity problem. Remember to subtract the down payment from original sale price. A * PVIFA (12%, 30) = 220,000 A * 8.055 = 220,000 A = $27,312.23

1. Suppose Janet has $2,500 in her bank account today. She would like to have $30,000 saved at the end of 6 years to make a down payment on a house. If she wants to make equal annual deposits at the end of each of the next 6 years, how much should she deposit each year if the interest rate is 6%?

You have $2500 today plus you'll make 6 equal annual deposits at 6%(made at the end of each year). You want $30,000 6 years from now. 2,500*FVIF (6%, 6) + A [FVIFA (6%, 6)] = 30000 2,500 * 1.419 + A * 6.975 = 30000 A * 6.975 = 26452.5 A = $ 3,792.47

Suppose you are 35 years old today and you are beginning to plan for your retirement. You want to invest an equal amount at the end of each year for the next 25 years so you can retire when you turn 60. You expect you can live until just after you turn 80 and you want to be able to withdraw $30,000 per year from your retirement account from your 61st birthday through your 80th birthday. If you can make your investments at a 10% rate of return, how much must you deposit each year?

You will deposit equal amounts for 25 years at 10% (ordinary annuity). You want $30,000 annual income for 20 years, starting 25 years from now at 10%. ($0 at 60 t=35 60 70 80 t=61 $30K for each year from 61-80) P = 30,000*PVIFA (10%, 20)] P = 30,000*(8.514) = 255,420 F = A * FVIFA (10%, 25) 255,420 = A*(98.347) A=$2597

5. Your parents plan to retire in 10 years (on Dec 31st). They would like to have an annual income of $35,000 starting 11 years from now (again, on Dec 31st), continuing forever (unbeknownst to you, they've discovered the Fountain of Youth and won't tell you about it). They've been able to save $40,000 so far. How much should they invest at the end of each of the next 10 years to reach their goal of retirement bliss if their interest rate is 5%?

Your parents have $40,000 today and will make equal annual deposits at 5% (made at the end of each year). They want $35,000 annual income forever beginning 11 years from now. First, calculate the annual $35,000 income as a perpetuity (use 5% interest rate) = c/r = 35,000/0.05 = $700,000 Now, calculate how much your parents must save (including what they already have) to match the $700,000 that they will need 10 years from now to receive their income starting at the end of year 11: 40,000*FVIF (5%, 10) + A*FVIFA (5%, 10) = 700,000 40,000 * 1.629 + A * 12.578 = 700,000 65,160 + A * 12.578 = 700,000 634,840 = A * 12.578 A = $50,472

You take out a variable-rate mortgage of $300,000 with a term of 30 years. The initial interest rate is 4% which is fixed for 5 years. After that the rate will be adjusted annually. a. What is your original yearly payment amount? (Assume one payment is made every year.) b. What will be your remaining balance after 5 years? c. If the interest rates go up to 8% after 5 years, what will be your new payment amount?

a) $300,000 = A* PVIFA[4%, 30] A = $17,349 b) PV = $17349*PVIFA[4%, 25] PV = $271,027 c) $271,027 = A* PVIFA[8%, 25] A = $25,389