Chapter 6
EAR =
(1 + [quoted rate/m])^m - 1 m = # of compoundings
Present Value Interest Factor for Annuities =
(1 - PV factor) / r
It is important to work only with effective rates because ...
... (1) the highest quoted rate is not necessarily the best, and (2) because compounding during the year can lead to a significant difference between the quoted rate and the effective rate
It is implicitly assumed that the cash flows occur at the ...
... end of each period
When money is continuously compounded ...
... interest is being credited the instant it is earned, so the amount of interest grows continuously
Most corporate bonds have the general form of an ...
... interest-only loan
There can be a HUGE difference between the APR and EAR when interest rates are ...
... large Ex: Payday loans - Write a post date check and receive cash upfront
The PV of a series of future cash flows is ...
... simply the amount you would need today to exactly duplicate those future cash flows (for a given discount rate)
When calculating PV for annuities ...
... you use the PMT function instead of the FV function key *Leave FV blank*
1. Describe how to calculate the future value of a series of cash flows 2. Describe how to calculate the present value of a series of cash flows 3. Unless we are explicitly told otherwise, what do we always assume about the timing of cash flows in present and future value problems?
1. $(1+R)^t 2. $/(1+R)^t 3. They occur at the end of each period
Annuity Due Value Steps:
1. Calculate the PV or FV as though it were an ordinary annuity 2. Multiply your answer by (1 + r)
Two ways to calculate FV for multiple cash flows:
1. Compound the accumulated balance forward one year at a time 2. Calculate the future value of each cash flow first and then add them up
Two ways to calculate PV for multiple cash flows:
1. Discount back one period at a time 2. Calculate the present values individually and add them up
Two ways to pay an amortized loan:
1. Have the borrower pay the interest each period plus some fixed amount 2. Have the borrower make a single, fixed, payment every period *Total interest is greater for the equal total payment case (because the loan is repaid more slowly early on, so the interest is somewhat higher)*
Three Types of Loans
1. Pure Discount Loans 2 Interest-Only Loans 3. Amortized Loans
1. In general, what is the present value of an annuity of C dollars per period at a discount rate of r per period? The future value? 2. In general, what is the present value of a perpetuity?
1. The present value of $C per period at a discount rate of "r" is C X{[1-(1+r)^t]/r}; The future value is C X {[(1+r)^t]-1/r} 2. The present value of a perpetuity is C/r
1. If an interest rate is given as 12% compounded daily, what do we call this rate? 2. What is an APR? What is an EAR? Are they the same thing? 3. In general, what is the relationship between a stated interest rate and an effective interest rate? Which is more relevant for financial decisions? 4. What does continuous compounding mean?
1. When ever a quoted rate is accompanied by the compounding frequency that really is an annual percentage rate, or APR, provided that rate is for an entire year - A standard quoted interest rate 2. An APR is that interest rate per period multiplied by the number of periods per year. An EAR is the interest rate expressed as if it were compounded once per year. The APR and EAR are the same only when compounding occurs annually - NOT THE SAME THING - The EAR is higher than the APR except when compounding occurs on an annual basis; in that instance they are the same. That EAR is more relevant for financial decisions because it indicates what is actually being earned or charged 3. A stated interest rate is expressed in terms of the interest payment made each period (Ex. 10%); whereas, the EAR is expressed as if it were compounded once per year (Ex. 10.25%). EAR are most relevant for financial decisions 4. Continuous compounding means that interest is being credited the instant it is earned
Annuity
A series of constant (or level) cash flows that occur at the end of each period for some fixed number of periods *Cash flows are said to be in "ordinary annuity form"*
An APR is in fact a quoted (stated) rate, in the sense we've been discussing
An APR of 12% on a loan calling for monthly payments is really 1% per month EAR = [1 + (APR/12)]^12 -1 EAR = 1.01^12 - 1 EAR = 12.6825%
Perpetuity
An annuity in which the cash flows continue forever - Has an infinite number of cash flows, so we obviously can't compute its value by discounting each one
PV for a Perpetuity Ex:
An investment offers a perpetual cash flow of $500 every year and the return you require on such an investment is 8%. What is the value of this investment? PV = C / r PV = $500 / .08 PV = $6250
Growing Perpetuity Present Value
C / (r - g)
PV for a Perpetuity =
C / r
Growing Annuity Present Value =
C X [(1-((1+g)/(1+r))^t)/(r-g)] g = growth rate *lottery example* pg. 167
Annuity Present Value =
C x (1 - PV factor / r) [C x (1 - (1 / ( 1 + r )^t) / r]
Future Value for Annuities =
C x [(FV factor - 1) / r] C x ([(1 + r)^t -1] / r)
Future Value of C per Period for t Periods at r Percent per Period:
FV = C x {[(1 + r)^t - 1] / r} A series of identical cash flows is called an annuity, and the term [(1 + r)^t - 1] / r is called the annuity future value factor
Is an APR an effective annual rate?
Put another way, if a bank quotes a car loan at 12% APR, is the consumer actually paying 12% interest?
FV for Annuities Example:
n = 30 i = 8 PMT = $2000 FV = 2000 x [((1.08)^30 -1) / .08] FV = 2000 x [(10.0627 -1) / .08] FV = 2000 x 113. 2834 FV = $226,556.42
Effective Annual Rates and Compounding
If a rate is quoted as 10% compounded semiannually, this means the investment actually pays 5% every six month -- Which is NOT the same as 10% per year
Effect Annual Rates and Compounding Example:
If you invest $1 at 10% per year, you will have $1.10 at the end of the year. If you invest at 5% every six months, then you'll have the future value of $1 at 5% for two periods: $1 x 1.05^2 = $1.1025 This is $.0025 more. Your account was credited with $1 x .05 = 5 vents in interest after six months. In the following six months, you earned 5% on that nickel, for an extra 5 x .05 = .25 cents
Time Line
Illustrates the process of calculating the future value
Annuity Due Value =
Ordinary annuity value x (1 + r) *Works for both PV and FV annuities*
Present Value of a Perpetuity of C per Period
PV = C / r A perpetuity has the same cash flow every year forever
Present Value of C per Period for t Periods at r Percent per Period:
PV = C x {1 - [1/91 + r)^t]} / r The term {1 - [1/91 + r)^t]} / r is called the annuity present value factor
EAR Example:
Suppose you are offered 12% compounded monthly, what is the EAR? EAR = [1 + (.12/12)]^12 - 1 EAR = 1.01^12 -1 EAR = 1.126825 -1 EAR = 12.6825% Where: Quoted Rate = .12 m = 12
Interest-Only Loan
The borrower pays interest each period and repays the entire principal (of the original loan) at some point in the future *If there is just one period, a pure discount loan and an interest-only loan are the same thing* Ex: With a 3 year, 10%, interest-only loan, the borrower would pay $1000 x .10 = $100 in interest at the end of the first and second years. At the end of the third year, the borrower would return the $1000 along with another $100 in interest for that year
Pure Discount Loans
The borrower receives money today and repays a single lump sum at some time in the future - Usually for short-term loans Ex: Suppose a borrower was able to repay $25,000 in five years. If we, acting as the lender, wanted a 12% interest rate, how much would we be willing to lend? n = 5 i = 12 FV = $25,000 PV = ? PV = 25,000/1.12^5 PV = $14,186
Amortized Loans
The borrower repays parts of the loan amount over time
As the number of times a deposit is compounded grows, the higher the EAR is
The differences get very small though
Annual Percentage Rate (APR)
The interest rate charged per period multiplied by the number of periods per year
Effective Annual Rate (EAR)
The interest rate expressed as if it were compounded once per year - In our example, the 10.25% is the EAR
Stated (quoted) Interest Rate
The interest rate expressed in terms of the interest payment made each period - In our example, 10% is the stated interest rate
Amortizing the Loan
The process of providing for a loan to be paid off by making regular principal reductions
Comparing Effective Annual Rates Example:
Which is best? Bank A: 15% compounded daily Bank B: 15.5% compounded quarterly Bank c: 16% compounded yearly Bank A : .15/365 = .000411^365 = $1.1618 or 16.18% Bank B: .155/4 = .03875^4 = $1.1642 or 16.42% Bank C: .16/1 = .16^1 = $1.16 or 16%
EAR =
e^q -1 Where: q = quoted rate e = 2.71828 (number on calculator)
PV for Annuities Example:
i = 10 n = 3 FV = $500 PV = 500 x ([(1/1.1)^3] / .10) PV = 500 x 2.48685 PV = $1,243.43
