Reading 5 LOS's Flashcards
LOS 5a: Interpret interest rates as require rates of return, discount rates, or opportunity costs
- Minimum rate of return what you require to accept a payment at a later date
- Discount Rate is applied to a future cash flow in order to determine its present value
- Opportunity cost is the cost of spending the money today as opposed to saving it for a certain period and earning a return on it
LOS 5b: Explain an interest rate as the sume of a real risk-free rate, and premiums that compensate investors for bearing distinct types of risk
Interest rates are determine by the demand and supply of funds. They are composed of the real risk-free rate plus compensation for bearing different types of risks:
- The real risk-free rate is the single period return on a risk-free security assuming zero inflation. This rate reflects individuals preferences for current versus future consumption
- An inflation premium is added to the real risk-free rate to reflect the expected loss in purchasing power over the term of a loan. The real risk free rate plus the inflation premium equals the nominal risk-free rate
- The default risk premium compensates investors for the risk that the borrower might fail to make promised payments in full in a timely manner
- the liquidity premium compensates investors for any difficulty they might face in converting their holdings into cash at fair value
- The maturity premium compensates investors for the higher sensitiviy of the market values of longer term debt instruments to changes in interest rates
LOS 5c: Calculate and interpret the effective annual rate, given the stated annual interest rate and the frequency of compounding
LOS 5d: Solve time value of money problems for different frequencies of compounding
The Frequency of Compounding
Compounding and Future Values
If interest payments are made monthly, the interest rate is not quoted on a monthly basis but is the stated annual interest rate. We need to unannualize this rate to work with our problem. To do this, we will simply divide the stated annual interest rate by the amount of payments throughout the year. So for example with monthly, if the state annual interest rate is 8%, then the periodic interest rate is 8%/12 = .667%
Continuous Compounding and Future values
In continous compounding, the number of compounding periods is infinite, and the expression for FV of an amount after N years is given as:
- FV = PV x er x N
Effective Annual rates
Once we break down the stated annual interest rate into the periodic interest rate, we would like to find our Effective Annual Rates, to see how much interest we are earning annualy. To do this we simply
EAR = ( 1 + peroidic rate) number of periods yearly - 1
So for our previous monthly example, the effective annual rate would be:
(1 + .00667) 12 - 1 = 8.3%
Notice that as the number of compounding periods increases, the effective annual rate (EAR) increases as well.
Continuous Compounding and PV
Pretty much the opposite of FV. To find the PV compounded use the calculator
N= number of periods
I/Y = periodic interest rate
FV = amount to be received.
For continous the formula will be:
- PV = FV / e r x N
AS the number of compounding periods increases, the present value of the investment decreases
LOS 5e: calculate and interprety the future value and present value of a single sum of money, an ordinary annuity, an annuity due, a perpetuity, and a series of unequal cash flows
Future Value of a Single Cash Flow
FV = PV (1+ r ) N
When we hold the cash flow for multiple periods, we start to compound interest (that is interest is earned on interest).
The future value will increase as the number of periods increases and as the interest rate increases
The Present value of a Single Cash Flow
PV = FV / ( 1 + r) N
We must discount the future cash flow to equal todays dollars.
For a given discount rate, the longer the time period the lower the PV
For a given time period, the higher the discount rate the lower the PV
FV and PV of a Series of Cash Flows
Ordinary Annuties
An annuity is a finite set of level squential cash flows
An ordinary annuity is an annuity where the cash flows occure at the end of each compounding period
This is different than calculating single sums because we have to find the value of a stream of periodic payments
FV of an Ordinary Annuity
We will use a calculator.
PMT = - annual payment
N= number of payments
/Y= interest rate over payments
PV of an Ordinary Annuity
N =number of periods
I/Y = interest rate
PMT = - payment so that way we get PV in positive
FV= 0
Annuties due
an annuity due is an annuity where the periodic cash flows occur at the beginning of every period. There are two ways to calculate the present and future values of annuties due:
- You can set your calculator in BGN mode and then insert all the variables as you normally would. Your calculators are usually in END mode, but nothing shows up on your calculator screens to indicate this. When in BGN mode, you will see a symbol indicating you are in this mode.
- You can treat the cash flow stream as an ordinary annuity over N compounding periods and simply multiply the resulting PV or FV by (1 + periodic interest rate). Given an ordinary annuity and an annuity due, the annuity due must be greater because each cash flow is received one period earlier
Present Value of a Perpetuity
A perpetuity is a never ending series of level payments, where the first cash flow occurs in the next period
PV = PMT / I/Y
LOS 5f: Demonstrate the use of a timeline in modeling and solving time value of money problems
Loan Payments and Amortization
For Loan payments and amortizations, questions will be asked that want you to figure out how large payments need to be to get the loan paid off in time. In this case we would put into our calculator
N= number of periods to pay off loan
I/Y = Interest rate on loan
PV = face value of loan
FV = 0, since we will pay it off by then
Then we simply compute the payment.
Once we have the payment, a problem may ask you how much of the payment is to principal and to interest. To do this we would have to figure out how much interest is being charged on the principle and subtract that from the payment.
Pretty much for all of these problems we will be given all but one of FV, PV, PMT, I/Y, and N, with the one missing the value we are solving for. It is important to make sure that we keep our periods and our interest rates in proper order.
