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# Time Value of Money Concepts

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1. You deposit \$5,000 in an account that earns 12% compounded annually. Compute the account balance at the end of:

a. 1 year
b. 20 years
c. 40 years

2. You deposit \$100,000 in an account that earns a 10% annual rate of return. You have decided that at the end of each year you will withdraw the interest that is credited to your account. If you plan on closing the account at the end of the tenth year, how much are you able to withdraw at that point in time?

3. You have the opportunity to deposit \$10,000 into one of two accounts. The Optimum account offers to pay 12% compounded annually while the prime account offers to pay 12% compounded monthly. If you plan on leaving the money in the account for 10 years, what will be the difference between the Optimum and Prime account balances at the end of the tenth year?

4. If you expect to inherit \$50,000 at the end of the 20 years, compute the present value of the inheritance at the following annual interest rates.
a. 6%
b. 12%
c. 24%

5. If you require a 10% rate of return on the following investment opportunities, which would you prefer--\$100,000 today, \$250,000 ten years from today, or \$12,000 per year at the end of each of the next 20 years?

6. Mirage Investments is offering a investment opportunity that pays a 10% annual rate of return, compounded quarterly. How much would you have to pay for this investment if Mirage has agreed to pay you \$32,000 at the beginning of the sixth year?

7. How long will it take to double your initial deposit into an Escalator Account, if the bank offering the account pays the following interest rates:

a. 9% compounded annually
b. 14% compounded annually
c. 9% compounded monthly

8. Insolvent Savings and Loan has advertised an account that pays 8% compounded annually. In the advertisement, Insolvent provides a example of a hypothetical account. The advertisement shows that a customer depositing \$10,000 with the bank can achieve an account balance of \$100,627. Due to a misprint, it is not clear from the advertisement how long a customer would have to leave the money in the account. It is up to you to correct that misprint--how long does this hypothetical customer have to leave the money in the account?

9. Compute the effective annual interest rate paid by your bank if your initial deposit with the bank doubles in:

a. 5 years
b. 10 years
c. 20 years

10. Over a 13 year time period, your \$3,000 investment in gold coins has increased in value to \$15,000. What effective annual rate of return have you earned on your investment in gold? What is your holding period return on this investment?

11. Mutual Benefit is offering a "Ultimate Opportunity" annuity contract that offers investors a 14% annual rate of return. The contract specifies that Mutual Benefit will pay the investor an equal amount at the end of each contract year. You have decided to buy a contract that provides an annual payment of \$10,000 per year at the end of each of the next 20 years. How much are you going to have to pay Mutual Benefit for this annuity contract?

12. You just lost a lawsuit which requires you to pay the plaintiff \$500,000. The judge, not being well versed in finance, has agreed to your request that you be allowed to pay \$25,000 per year at the end of each of the next 20 years.

a. If you can invest in government securities that compound interest at an annual rate of 7%, how much do you have to set aside today in order to insure that the payments are made to the plaintiff?

b. If your broker tells you that by investing in securities issued by corporations you can earn a compound annual rate of 12%, how much do you have to set aside today in order to insure that the payments are made to the plaintiff?

c. Which investment alternative would you select? Should the plaintiff care about your decision?

13. You have decided to deposit \$1,000 per year at the end of each of the next 15 years in an account that earns 5% compounded annually. What is the account balance at the end of the fifteenth year?

14. Since the big promotion and the raise that goes with it finally came through, you have decided to get serious about saving some money. You are going to save \$5,000 per year at the end of each of the next 25 years, but are unsure where to invest the money. If you invest the money in certificates of deposit, you forecast that you will earn a 7.5% effective annual rate of return. An alternative is to invest the money in a High Yield bond fund. The fund claims that last year it generated a 14% effective annual rate of return, but notes that past performance may not reflect future returns.

a. Compute the account balance at the end of the twenty-fifth year if you select the certificate of deposit investment alternative.

b. Compute the account balance at the end of the twenty-fifth year if you select the High Yield investment alternative. Assume that the fund earns a 14% annual return.

c. Given the difference in the account balances, why would anyone purchase certificates of deposit?

15. Three years ago you borrowed \$30,000 in order to purchase a boat. The lender agreed to a 7 year loan at a 14% annual interest rate. A unique feature of the loan is the requirement that the annual payment of \$6,136.64 be made at the beginning of each year. It is the end of the third year and you are about to write a check for the fourth payment. Since this last year was fairly prosperous you have decided to pay off the entire loan balance at this time. How much do you have to pay in order to repay the entire remaining balance on the loan?

16. You have decided to deposit \$1,500 per year at the beginning of each of the next 10 years in an account that earns 8% compounded annually. What is the account balance at the end of the tenth year?

#### Solution Preview

Solution is attached in MS Word file.

1. You deposit \$5,000 in an account that earns 12% compounded annually. Compute the account balance at the end of:

a. 1 year
Present value of deposit=PV=\$5000
Rate of interest=i=12% per period
Number of periods=n=1 year
Future Value of deposit=FV=?
FV=PV*(1+i)^n=5000*(1+12%)^1=\$5600

b. 20 years
Present value of deposit=PV=\$5000
Rate of interest=i=12% per period
Number of periods=n=20 years
Future Value of deposit=FV=?
FV=PV*(1+i)^n=5000*(1+12%)^20=\$48,231.47

c. 40 years
Present value of deposit=PV=\$5000
Rate of interest=i=12% per period
Number of periods=n=40 years
Future Value of deposit=FV=?
FV=PV*(1+i)^n=5000*(1+12%)^40=\$465,254.85

2. You deposit \$100,000 in an account that earns a 10% annual rate of return. You have decided that at the end of each year you will withdraw the interest that is credited to your account. If you plan on closing the account at the end of the tenth year, how much are you able to withdraw at that point in time?

Since, interest earned during each year is withdrawn, principal amount at the beginning of 10th year will be equal to the initial deposit.
Principal amount at the beginning of 10th year=P=\$100000
Rate of interest=i=10%
Number of period=n=1 (10th year)
Interest earned=P*i*n=100000*10%*1=\$10,000

Amount available at the end of 10th year=100000+10000=\$110000

3. You have the opportunity to deposit \$10,000 into one of two accounts. The Optimum account offers to pay 12% compounded annually while the prime account offers to pay 12% compounded monthly. If you plan on leaving the money in the account for 10 years, what will be the difference between the Optimum and Prime account balances at the end of the tenth year?

Optimum Account
Present Value of deposit=PV=\$10000
Rate of interest=i=12% per period
Number of periods=n=10
Future Value of deposit=FV=?
FV=PV*(1+i)^n=10000*(1+12%)^10=\$31058.48

Prime Account
Present Value of deposit=PV=\$10000
Nominal Rate of interest=i=12% per year with monthly compounding
Number of compounding in a year=c=12
Effective Rate of interest=r=(1+(i/c))^c-1=(1+(12%/12))^12-1=12.6825%
Number of periods=n=10
Future Value of deposit=FV=?
FV=PV*(1+r)^n=10000*(1+12.6825%)^10=\$33003.86

Difference in balances=33003.86-31058.48=\$1945.38

4. If you expect to inherit \$50,000 at the end of the 20 years, compute the present value of the inheritance at the following annual interest rates.

a. 6%
Future value of inheritance=FV=\$50000
Rate of interest=i=6% per period
Number of periods=n=20
Present Value of inheritance=FV/(1+i)^n=50000/(1+6%)^20=\$15,590.24

b. 12%
Future value of inheritance=FV=\$50000
Rate of interest=i=12% per period
Number of periods=n=20
Present Value of inheritance=FV/(1+i)^n=50000/(1+12%)^20=\$5,183.34

c. 24%
Future value of inheritance=FV=\$50000
Rate of interest=i=24% per period
Number of periods=n=20
Present Value of inheritance=FV/(1+i)^n=50000/(1+24%)^20=\$676.92

5. If you require a 10% rate of return on the following investment opportunities, which would you prefer--\$100,000 today, \$250,000 ten years from today, or \$12,000 per year at the end of each of the next 20 years?

PV of option 1 i.e. \$100,000 ...

#### Solution Summary

There are 16 problems. Solution to each problem provides step by step methodology to find out the asked parameter/s.

\$2.19