Chapter 5

What is the present value of a perpetuity paying $150 at the end of each year at 8%?

$1,875 150/.08=1875

Suppose you paid off a $1,200 loan by paying $400 in principal each year plus 10% annual interest over a 3 year period. What is the total payment (interest plus principal) in Year 3?

$440 400+(1200-800)x.10=440

The difference between the present value of an ordinary annuity with payments of $100 per year at 10% compounded annually for 10 years and an annuity due with payments of $100 per year at 10% compounded annually for 10 years is:

$61.45 PV(annuity due)-PV(ordinary annuity) 675.90-614.46=61.45

Suppose you paid a $1,200 loan off by paying $400 in principal each year plus 10% yearly interest. How much is the second interest payment?

$80 (1200-400)x.10=80 -The first year would be paid on the full 1200, so it would be $1,200x0.10=$120

Which of the following is the general formula for the EAR when m is the number of times interest is compounded in a year?

(1 + quoted rate/m)^m - 1

The formula for the present value of annuity is . . .

(1+r)x(PV of an ordinary annuity)

Common Types of Loans

1) Discount Loans 2) Interest-Only Loans 3) Amortized Loans

Annuity Example You just won the lottery and you will receive $100,000 every year for 25 years. 1)How big a loan can you take out today at 9%/year such that the loan payments exactly equal the lottery payments? 2)If you invested the payments in an account that paid 9%/year, how much would you have after receiving the last payment?

1) Find PV PV = 100,000 x ((1-(1/((1.09)^25)) / 0.09 PV = $982,257.96 --> you can take out loan for a little less than a mill 2) Find FV FV= 100,000 x (((1.09)^25)-1)/0.09 FV= $8,470,089.62

You agree to pay back $1100 in four weeks for a $1000 payday loan. Your annual percentage rate (APR) to two decimal places is __%. (assume 52 weeks in a year)

130.00 [(1100/1000)-1] x (52/4)(100)

Compound Interest Example: Consider a bank that pays interest of 1% compounded monthly. How much would you earn in a year? Is it 12% per year?

FV = PV x (1+i)^t FV=100 x (1.01)^12 FV=112.68 i=12.68% 12.68% doesn't equal 12%

You have a bank account that pays 8% interest compounded annually. If you deposit $200 today and $100 a year from now, how much will you have in six years? What if you deposited the money in 2 separate accounts? How much would you have in each account?

FV of $200=200x (1.08)^6 =317.37 FV of $100=100x(1.08)^5 =146.93 FV (total) = 317.37 + 146.93 =464.30

If the interest rate is greater than zero, the value of an annuity due is always ___ an ordinary annuity.

Greater Than

More frequent compounding leads to:

Higher EARs

If the bank quotes an APR of 12% compounded quarterly, what is the APY (EAR)?

If compounding quarterly, quarterly rate is 3% EAR=((1+(0.12/4))^4 -1 EAR= 12.55% per year

Distinguish between Ordinary Annuity and Annuity Due

If you get a complicated problem, draw a timeline. Your answer from one part (a PV or FV) will fit at the beginning or end of the periods for another part of the problem. End: Ordinary Annuity Beginning: Annuity Due

A perpetuity is a constant stream of cash flows for an ____ period of time.

Infinite

When finding the present or future value of an annuity using a spreadsheet (Excel), the ___ should be entered as a decimal.

Interest Rate

An ordinary annuity consists of a ___ stream of cash flows for a fixed period of time.

Level

An insurance company offers you a contract in which they will pay you $10,000 in 10 years and $15,000 in 20 years. If you require a 10% return per year, how much are you willing to pay today?

Looking for: PV Know:FV, interest rate, # of period PV= (10,000/((1.10)^10)) + (15,000/((1.10)^20)) PV = $6,085.08

Because of __ and ___, interest rates are often quoted in many different ways

Tradition and legislation

The first cash flow at the end of week 1 is $100, the second cash flow at the end of month 2 is $100, and the third cash flow at the end of year 3 is $100. This cash flow pattern is an ___ type of a cash flow.

Uneven

The formula for the future value of annuity is . . .

[(1+r)^2 -1]/r

Compound Interest

more frequent the compounding --> the higher the effective rate --> lower PV and higher FV

The general formula for ____ is (1+quoted rate/m)^m -1

the EAR

Which of the following are ways to amortize a loan?

- Pay principal and interest every period in a fixed payment -Pay the interest each period plus some fixed amount of the principal

Notations

- i : Interest Rate per period - n : # of periods - PMT : Cash payments made each period - PV : Present Value (start of stream) -FV : Future Value (end of stream) - t : Time - k, r : Required Rate of Return (i)

Which of the following is perpetuity?

-A constant stream of cash flows forever

Amortized Loans

-Equal payments during the life of loan -Payments include both principle and interest -No extra payment at the end

EAR

Actual return earned in a year (APY or effective annual rate) 1%/month --> 12.68% /year EAR -True Interest Rate -a.k.a. APY EAR= (1+(APR/m))^m-1 -Effective Rate on calculator

APR

Compounding period return x Number of periods in a year (Nominal Rate) 1%/month --> 12%/year APR -monthly rate times number of months -periodic rate times the number of periods -Doesn't tell us the true interest rate -Nominal Rate on calculator

What is the appropriate excel function to convert a quoted rate of 12% compounded quarterly to an EAR?

EFFECT (0.12.4)

For a stated positive interest rate, the EAR is always ___ the APR.

Equal to or greater than

The entire principal of an interest-only loan is the :

Original Loan Amount

The present value form for a ____ is PV = C/r, where C is the constant and regularly timed cash flow to infinity, and r is the interest rate.

Perpetuity

The original amount of a loan is termed the loan __.

Principal

Which of the following processes can be used to calculate the future value of multiple cash flows?

-Calculate the future value of each cash flow first and then add them up -Compound the accumulated balance forward one year at a time

Interest-Only Loans

-Equal payments during the life of loan PLUS all principal repaid at maturity

Annuity Due

-Has the payments at the beginning of the periods -"You will make the first payment today" -You earn (pay) an extra period's interest on every payment so multiply an ordinary annuity by (1+i)

Ordinary Annuity

-Has the payments at the ends of the period -"You will make the first payment one month from now"

Discount Loans

-Only one payment at maturity -Repays all of the principal and accrued interest

Perpetuity

-Pay the same amount every period forever -Doesn't make sense to talk about future values of these (no end to investment period) -PV=Payment/interest rate

Which of the following are real-world examples of annuities?

-Pensions -Mortgages -Leases

Special Cases / Shortcuts

-Perpetuity: Infinite equal payments -Annuity: Finite equal payments ~Ordinary Annuity ~Annuity Due

When entering variables in an Excel function (or in a financial calculator) the "sign convention" can be critical to achieving a correct answer. The sign convention says that outflows are negative values and inflows are positive values. For which variables is this a consideration?

-Present Value -Future Value -Payment

Suppose you expect to receive $5,000 in one year, $4,300 more in two years, and an additional $5,000 in three years. What is the present value amount with the corresponding cash flow assuming a discount rate of 17%?

-Present Value of the Year 1 Cash Flow: $4,273.50 (5000/1.17) -Present Value of the Year 2 Cash Flow: $3,141.21 (4300/(1.17)^2) -Present Value of the Year 3 Cash Flow: $3,121.85 (5000/(1.17)^3)

An interest rate expressed in terms of the interest payment made each period is called a _____

-Quoted interest rate -Stated interest rate

Annuities

-Same cash flow every period but for a limited amount of time -2 Types: Ordinary Annuity and Annuity Due

In the Excel setup of a loan amortization problem, which of the following occurs?

-The payment is found with =PMT (rate, nper, -pv, fv) -To find the principal payment each month, you subtract the dollar interest payment from the fixed payment

Which of the following could not be evaluated as annuities or annuities due?

-Tips to a waiter -Monthly electric bills

A credit card charges 18 percent interest per year (APR) (1.5 percent each month). What is the EAR?

19.56% (1.015)^12 - 1= 19.56%

You borrow $100 and agree to pay back your payday loan in 2 weeks for 10% interest over that 2 week period. What is your stated annual interest rate?

260.0% 26 x (10%) = 260

The present value interest factor for a 30 year annuity with an interest rate of 10% per year is ____

9.4269 [1-(1/1.10^30) /.10]=9.4269

Which of the following is the simplest form of loan?

A pure discount loan

The most common way to repay a loan is to pay ____

A single fixed payment every period

If the bank quotes an APY of 10.47% and compounds monthly, what is the APR?

APR =[(1+EAR)^(1/m)-1] x m APR=[(1.1047)^(1/12) -1]x12 APR=0.10000 or 10%

Commonly Used Interest Rates

APR and EAR

An annuity with payments beginning immediately rather than at the end of the period is called an ____.

Annuity Due

An annuity due is a series of payments that begin __

At the beginning of each period

The effective annual rate (EAR) takes into account the ____ of interest that occurs within a year.

Compounding

Spreadsheet functions used to calculate the present value of multiple cash flows assume, by default, that all cash flows occur at the ___ of the period

End

Which of the following is NOT a way to amortize a loan?

Fixed interest payments only -Ways to: ~fixed payments ~interest plus fixed amounts ~Pay the interest each period plus some fixed amount of the principal ~Pay principal and interest every period in a fixed payment

The present value of an annuity due is equal to the present value of an __ annuity multiplied by (1+r)

Ordinary

If you borrow $15,000 today at 5% annual interest to be repaid in one year as a lump sum, this is termed a ____

Pure Discount Loan