Assume that you are nearing graduation and that you have applied for a job with a local bank, First National 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 time value of money analysis. See how you would do by answering the following questions.
a. Draw time lines 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 percent, annual compounding?(2) What is the present value of $\$ 100$ to be received in 3 years if the appropriate interest rate is 10 percent, annual compounding?
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 percent per year, how long will it take sales to double?
d. 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?
e. (1) What is the future value of a 3-year ordinary annuity of $\$ 100$ if the appropriate interest rate is 10 percent, annual compounding?
(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?
f. What is the present value of the following uneven cash flow stream? The appropriate interest rate is 10 percent, compounded annually.
g. What annual interest rate will cause $\$ 100$ to grow to $\$ 125.97$ in 3 years?
h. A 20 -year-old student wants to begin saving for her retirement. Her plan is to save $\$ 3$ a day. Every day she places $\$ 3$ in a drawer. At the end of each year, she invests the accumulated savings $(\$ 1,095)$ in an online stock account that has an expected annual return of 12 percent.
(1) If she keeps saving in this manner, how much will she have accumulated by age $65 ?$
(2) If a 40 -year-old investor began saving in this manner, how much would he have by age 65 ?
(3) How much would the 40 -year-old investor have to save each year to accumulate the same amount at age 65 as the 20 -year-old investor described above?
i. (1) Will the future value be larger or smaller if we compound an initial amount more often than annually, for example, every 6 months, or semiannually, holding the stated interest rate constant? Why?
(2) Define (a) the stated, or quoted, or nominal, rate, (b) the periodic rate, and (c) the effective annual rate $(\mathrm{EAR})$
(3) What is the effective annual rate corresponding to a nominal rate of 10 percent, compounded semiannually? Compounded quarterly? Compounded daily?
(4) What is the future value of $\$ 100$ after 3 years under 10 percent semiannual compounding? Quarterly compounding?
j. When will the effective annual rate be equal to the nominal (quoted) rate?
k. (1) What is the value at the end of Year 3 of the following cash flow stream if the quoted interest rate is 10 percent, compounded semiannually?
(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? (I Iint: Think of annual compounding, when $\left.i_{\mathrm{Nom}}=\mathrm{EAR}=\mathrm{i}_{\mathrm{PER}} .\right) \mathrm{What}$
would be wrong with your answer to parts $\mathrm{k}(1)$ and $\mathrm{k}(2)$ if you used the nominal rate, 10 percent, rather than the periodic rate, $\mathrm{i}_{\mathrm{Nom}} / 2=10 \% / 2=5 \% ?$
1. (1) Construct an amortization schedule for a $\$ 1,000,10$ percent, annual compounding 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 ?