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Time Value of money

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Hello, can you please provide the answers and show full calculations and explanations of each of the problems. Thank you.

Assume that you are nearing graduation and that you have applied for a job with a local bank. As part of the Bank's evaluation process, you have been asked to take an examination that covers several financial analysis techniques. The first section of the test addresses discounted cash flow analysis. See how you would do by answering the following questions.

a. Draw a time line for:
(1) A $100 lump sum cash flow at the end of year 2,
(2) An ordinary annuity of $100 per year for 3 years, and
(3) An uneven cash flow stream of -$50, $100, $75, and $50 at the end of Years 0 through 3.

b. (1) what is the future value of an initial $100 after 3 years if it is invested in an account paying 10% annual interest?
(2) What is the present value of $100 to be received in 3 years if the appropriate interest rate is 10%?

c. We sometimes need to find how long it will take a sum of money (or anything else) to grow to some specified amount. For example, if a company's sales are growing at a rate of 20% per year, how long will it take sales to double?

d. If you want an investment to double in 3 years, what interest rate must it earn?

e. What is the difference between an ordinary annuity and an annuity due?
What type of annuity is shown below? How would you change it to the other type of annuity?

0 1 2 3 years

100 100 100

f. (1) what is the future value of a 3-year ordinary annuity of $100 if the appropriate interest rate is 10%?
(2) What is the present value of the annuity?
(3) What would the future and present values be if the annuity were an annuity due?

g. What is the present value of the following uneven cash flow stream? The appropriate interest rate is 10%, compounded annually.

0 1 2 3 4
0 100 300 300 -50

h. (1) define:
(a) the stated, or quoted, or nominal rate (INOM) and
(b) the periodic rate (IPER).

(2) Will the future value be larger or smaller if we compound an initial amount more often than annually, for example, every 6 months, or semi-annually, holding the stated interest rate constant? Why?

(3) What is the Future value of $100 after 5 years under 12% annual compounding? Semiannual compounding? Quarterly compounding? Monthly compounding? Daily compounding?
(4) What is the effective annual rate (EFF%)? What is the EFF% for a nominal rate of 12%, compounded semiannually? Compounded quarterly? Compounded monthly? Compounded daily?

i. Will the effective annual rate ever be equal to the nominal (quoted) rate?

j. (1) construct an amortization schedule for a $1,000, 10% annual rate loan with 3 equal installments.
(2) What is the annual interest expense for the borrower, and the annual interest income for the lender, during year 2?

k. Suppose on January 1 you deposit $100 in an account that pays a nominal, or quoted, interest rate of 11.33463%, with interest added (compounded) daily. How much will you have in your account October 1, or after 9 months?

l. (1) What is the value at the end of year 3 of the following cash flow stream if the quoted interest rate is 10%, compounded semiannually?

0 1 2 3 years

100 100 100
(2) What is the PV of the same stream?

(3) Is the stream an annuity?

(4) An important rule is that you should never show a nominal rate on a time line or use it in calculations unless what condition holds? (Hint: think of annual compounding, when INOM = EFF% = IPER.) What would be wrong with your answer to question 1-(1) and 1-(2) if you used the nominal rate (10%) rather than the periodic rate (I NOM/2 = 10% /2 = 5%)?

m. Suppose someone offered to sell you a note calling for the payment of $1,000 fifteen months from today. They offer to sell it to you for $850.00. You have $850.00 in a bank time deposit that pays a 6.76649% nominal rate with daily compounding , which is a 7% effective annual interest rate, and you plan to leave the money in the bank unless you buy the note. The note is not risky- you are sure it will be paid on schedule. Should you buy the note? Check the decision in three ways: (1) by comparing your future value if you buy the note versus leaving your money in the bank, (2) by comparing the PV of the note with your current bank account, and (3) by comparing the EFF% on the note versus that of the bank account.

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The solution explains various questions relating to time value of money

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Calculations of the Time Value of Money

1) You invest $20,000 today, at a rate of 10% compound quarterly. What will the investment be worth at the end of year twenty?
2) You are offered an annuity that will pay you $9,000 at the end of each of the next 10 years. What is the maximum amount you would be willing to pay today for this annuity? (Assume you require a 15% rate of return on an investment of this nature.)
3) You have $15,000 to put down on a new house that cost $200,000, and you have been quoted the following finance terms by your local banker: 6% Annual Percentage Rate, for 30 years. If you decide to purchase this home, what will your monthly payment be? Additionally, over the life of the loan what would your total interest expense be?
4) You want to start saving for your child's education. You project that your child will need $170,000 to attend school 15 years from now. If you can earn a rate of return of 10% compounded semi-annually on a given investment, what dollar amount will you need to invest today to ensure your child can attend college?
5) Steaks Galore has $190,000 in excess cash that it wishes to invest. Bank One offers a certificate of deposit that is paying 10%, compounded monthly. Bank Two offers a certificate of deposit paying 9.5%, compounded daily. In which Bank should the firm opt to invest its' surplus cash? (You must use the EAR formula to solve this problem. In addition, you must show all of your work.) Additionally, what is the nominal and period rate of interest offered by Bank One?
6) You plan on depositing $3,000 in an account at the end of each of the next 5 years. If the account is paying interest at an annual rate of 10% per year, what will the total value of your investment be at the end of the 10th year?
7) Your Life Insurance Agent is trying to sell you an investment that will pay you $5,000 a year forever. If your required rate of return is 11% on an investment of this nature, what would you be willing to pay your agent today for this investment opportunity?
8) It is forecasted that you will receive the following cash inflows at the end of the next four years: Year 1 $1,000, Year 2 $2,000, Year 3 $4,000 and Year 4 $1,000. If upon receipt of these cash inflows, you can re-invest the amounts received at a rate of 10%, what will the total future value of this investment be?
9) While Bob Jones was a student at Tiffin University, he borrowed $43,063 in student loans at an annual rate of 7 percent. If Bob repays $500 per month, how long, to the nearest year, will it take him to repay the loan?
10) Company XYZ plans to invest $5 million to clear a tract of land and to set out some young trees. These trees will mature in 12 years, at which time XYZ plans to sell all the trees at an expected price of $10 million. What is XYZ's expected rate of return? In addition, given this rate of return, would you recommend that XYZ proceed with the plan? Why or why not.

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