What$$s the difference between an ordinary annuity and an annuity due? What type of annuity is shown below? How would you change the time line to show the other type of annuity?

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A timeline shows periods of $0$ to $3$ years with an increment of $1$ year. The cash flows for periods $1,2,$ and $3$ are $100$ dollars each.

$(1)$

What$$s the future value of a $3-$year ordinary annuity of $$100$ if the appropriate interest rate is $10\%?$

$(2)$

What$$s the present value of the annuity?

$(3)$

What would the future and present values be if the annuity were an annuity due?

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

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$(1)$

Define the nominal rate $($

I

NOM

$),$ which also is called the stated rate and the quoted rate. Also define the periodic rate $($

I

PER

$).$ If the nominal rate is $6\%$ and is compounded quarterly, what is the periodic rate $($

I

PER

$)?$ If it is compounded monthly?

$(2)$

If the stated interest rate is constant, will the future value be larger or smaller if we compound an initial amount more often than annually $($for example, semiannually$)?$ 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$\%),$ also called the annual equivalent rate $($AER$)?$ What is the EFF$\%$ for a nominal rate of $12\%,$ compounded semiannually? Compounded quarterly? Compounded monthly? Compounded daily?

$(5)$

Can the effective annual rate ever be equal to the nominal $($quoted$)$ rate?

Define the annual percentage rate $($APR$).$ Suppose you borrow $$3,621\mathrm{.}15$ and take out a loan with an $8\%$ nominal interest rate. You must repay the loan with $48$ monthly payments of $$90$ each. In addition, you must pay an initial loan processing fee of $$100.$ What is the APR for this loan?

Explain when you should use the: nominal rate $($

I

NOM

$),$ periodic rate $($

I

PER

$),$ effective annual rate $($EFF$\%),$ and annual percentage rate $($APR$).$ Can you think of a way to modify the APR so that it would be more helpful? If so$,$ how would you calculate that rate?

$(1)$

Construct an amortization schedule for a $$1,000,10\%$ annual rate loan with three equal installments.

$(2)$

During Year $2,$ what is the annual interest expense for the borrower, and what is the annual interest income for the lender? Hint: Construct an amortization schedule.

Suppose that on January $1$ you deposit $$100$ in an account that pays a nominal $($or quoted$)$ interest rate of $11\mathrm{.}33463\%,$ with interest added $($compounded$)$ daily. How much will you have in your account on October $1,$ or $9$ months $(273$ days$)$ later?

$(1)$

Consider the time line shown here. Is the stream of cash flows an annuity?

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$(2)$

What is the value at the end of Year $3$ of the previous cash flow stream if the quoted interest rate is $10\%,$ compounded semiannually?

$(3)$

What is the PV of the same stream?

$(4)$

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

I

NOM

$=$

EFF

$\%$

$=$

I

PER

$.)$ What would be wrong with your answers to parts $(1)$ and $(2)$ if you used the nominal rate of $10\%$ rather than the periodic rate,

I

NOM

$/$

$2$

$=$

$10$

$\%$

$/$

$2$

$=$

$5$

$\%$

$?$

You have the chance to buy a guaranteed promissory note for $$850.$ The note pays $$1,000$ in $15$ months $($i$.$e$.,$ exactly $456$ days$).$ You have $$850$ in a bank account that pays a $7\%$ nominal rate compounded daily. Which is a better investment, the note or the bank account? Answer this question using three approaches:

$(1)$

compare your future value if you buy the note versus leaving your money in the bank;

$(2)$

compare the PV of the note with your current bank balance; and

$(3)$

compare the effective rate or return on the note with that of the bank account.