Finance Chapter 6
Annuity
A level stream of cash flows for a fixed period of time A series of constant or level cash flows that occur at the end of each period for some fixed number of periods In ordinary annuity form
A Lease is an Example of What?
An annuity due
Annuity Due
An annuity for which the cash flows occur at the beginning of each period Almost any type of arrangement in which we have to repay the same amount each period
Perpetuity
An annuity in which the cash flows continue forever Also called consols
Relationship between Annuity Due and Ordinary Annuity
Annuity Due value = Ordinary Annuity value x (1+i)
There can be a huge difference between the APR and EAR when interest rates are what?
Are large
Payday Loans
Are short-term loans made to consumers, often for less than two weeks
As you increase the length of time involved, what happens to the present value of annuity? What happens to the future value?
Assuming positive cash flows, both the present and future values will rise
What happens to the future value of an annuity if you increase the rate r? What happens to the present value?
Assuming positive cash flows, the present value will fall and the future value will rise
It is implicitly assumed that the cash flows occur?
At the end of each period
Steps for Calculating Annuity Due
First calculate the present or future value as though it were an ordinary annuity then multiply your answer by (1+i)
IF you were an athlete negotiating a contract, would you want a big singing bonus payable immediately and smaller payments in the future, or vice versa? How about looking at it from the team's perspective?
If the total money is fixed, you want as much as possible as soon as possible. The team (or, more accurately, the team owner) wants just the opposite
Amortized Schedule
Interest paid is given by the beginning balance x by the interest rate
Consol
Is a type of perpetuity
Annual Percentage Yield
Is an effective annual rate (EAR) Used on things like savings accounts
What is the most common way of amortizing a loan?
Is to have the borrower make a single, fixed payment every period Almost all consumer loans (such as car loans) and mortgages work this way
Growing Perpetuities
Play a key role in our analysis of stock prices
What are the 4 Pieces to an Annuity Present Value?
Present value (PV) Periodic cash flow (C) The discount rate (r) The number of payments or the life of the annuity t
Growing Perpetuity
Principal / (i - g)
The Present Value of a Perpetuity is
Principal / i
Types of Loans
Pure discount loans Interest-only loans Amortized loans
Borrowers (APRs) and Savers (APYs)
Rates quoted to borrowers and to savers are not computed the same way
Suppose two athletes sign 10-year contracts for $80 million. In one case, we're told that the $80 million will be paid in 10 equal installments. In the other case, we're told that the $80 million will be paid in 10 installments, but the installments will increase by 5 percent per year. Who got the better deal?
The better deal is the one with equal installments
Interest-Only Loans
The borrower pays interest each period and to repay the entire principal at some point in the future Example: corporate bonds Principal is repaid all at once
Pure Discount Loans
The borrower receives money today and repays a single lump sum at some time in the future Is the simplest form of loan Are common when the loan term is short (a year or less) Principal is repaid all at once
Stated Rate and Effective Annual Rate
The highest stated rate is not necessarily the best Compounding during the year can lead to a significant difference between the stated rate and the effective rate
Annual Percentage Rate (APR)
The interest rate charged per period multiplied by the number of periods per year Is equal to the interest rate per period multiplied by the number of periods in a year Is a stated rate
Effective Annual Rate (EAR)
The interest rate expressed as if it were compounded once per year Is what you actually get or what you pay
Stated Interest Rate
The interest rate expressed in terms of the interest payment made each period Also known as the quoted interest rate
Amortized Loans
The lender may require the borrower to repay parts of the loan amount over time Example: medium-term business loans
Principal
The original loan amount
Growing Perpetuities and Growing Annuities
The principal is the cash flow that is going to occur exactly one period from today
Amortizing the Loan
The process of providing for a loan to be paid off by making regular principal reductions A simple way of amortizing a loan is to have the borrower pay the interest each period plus some fixed amount
Reason for why total payment is fixed and the principal paid is rising each period
The reason for this is that the loan is repaid more slowly early on, so the interest is somewhat higher
What happens to the future value of a perpetuity if interest rates increase? What if interest rates decrease?
This is a trick question. The future value of a perpetuity is undefined since the payments are perpetual Finding the future value at any particular point automatically ignores all cash flows beyond that point
EAR Continuous Compounding
Used for the the highest possible EAR EAR = e^q - 1
Should lending laws be changed to require lenders to report EARs instead of APRs? Why or why not?
Yes, they should. APRs generally don't provide the relevant rate The only advantage is that they are easier to compute, but with modern computing equipment, that advantage is not very important
Present Value Interest Factor of Annuities
[1-{1/(1+i)^n}/i]
Future Value Interest Factor of Annuities
{(1+i)^n - 1}/i
