FIN3403- Chapter 5: Q.1, 2, 3, 4, 7 and Key Terms
5-6
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Step by Step Approach
- Multiple the amount at the beginning of each period by (1 + I ). - Shows exactly what is happening - Time consuming, especially when a number of years are involved; so streamlined procedures have been developed.
Interest rates on Excel you use
- Percentages or decimals (e.g., .05 or 5%)
Spreadsheets
- Used in business when problems involve the time value of money (TVM). - Show in detail what is happening, and they help reduce both conceptual and data- entry errors.
When using the time value of money function on most financial calculators you generally enter the interest rate as a
- Whole number (e.g., 5)
Amortized loan
A loan that is repaid in equal payments over its life.
Annuity Due
An annuity whose payments occur at the beginning of each period.
Ordinary (deferred) annuity
An annuity whose payments occur at the end of each period.
Time Line
An important tool used in time value analysis; it is a graphical representation used to show the timing of cash flows.
Interest earned on interest is called
Compound Interest
Most financial contracts are based on
Compound interest but in legal proceedings, the law often specifies that simple interest much be used.
We find the future value of uneven cash flow streams by
Compounding
INT
Dollars of interest earned during the year = Beginning amount x I
In an Annuity Due
Each payment occurs one period earlier, and all of the payments earn interest for one additional period. Therefore, the FV of an annuity due will be greater than that of a similar ordinary annuity.
The formula for FV with simple interest is
FV = PV + PV * (I) * (N)
Annuity Due Formula
FVA due= FVA ordinary (1+I)
Formula Approach (Future Value)
FVn = PV (1+I) ^N
The most you should pay for a bond is its
Fair price
On the calculator: there is a key that enables us to switch between ordinary annuities and annuities due.
For ordinary annuities the designation is "End Mode" or something similar, while for annuities due the designation is "Begin" or "Begin Mode" or "Due" or something similar.
5-4. Would you rather have a savings account that pays 5% interest compounded semiannually or one that pays 5% interest compounded daily?
For the same stated rate, daily compounding is best. You would earn more "interest on interest."
If the beginning balance of interest is higher each successive year, the interest earned each year
Increases
Finding the Number of Years, N
Input I/YR; PV; PMT; and FV. Then press N to calculate.
I
Interest rate earned per year
A dollar in hand today is worth
More than a dollar to be received in the future because if you had it now, you could invest it, earn interest, and own more than a dollar in the future.
Financial Calculators
N; I/YR; PV; PMT; FV
N
Number of periods involved in the analysis
Compound Interest
Occurs when interest is earned on prior periods' interest.
Simple Interest
Occurs when interest is not earned on interest.
With Excel, all inputs are shown in
One place, which makes checking data entries relatively easy.
Step by step formula for PV of an ordinary annuity
PVAn=PMT[ 1 - ( 1 / ( 1 + I ) ^ n ) / I ] When using a calculator for ordinary annuity make sure to set to end mode. (Note that the PV of the annuity due is larger because each payment is discounted back one less year.)
PMT
Payment. PMT = 0
Time Lines
Show the timing of cash flows. The tick marks occur at the end of periods, so time 0 is today; time 1 is the end of the first period (year, month, etc) or the beginning of the second period.
Future Value (FVn)
The amount to which a cash flow or series of cash flows will grow over a given period of time when compounded at a given interest rate.
Effective (equivalent) annual rate (EAR or EFF%)
The annual rate of interest actually being earned, as opposed to the quoted rate.
Nominal (quoted or stated) interest rate, Indm
The contracted interest rate.
5-1. What is an opportunity cost? How is this concept used in TVM analysis and where is it shown on a time line? Is a single number used in all situations?
The opportunity cost is the rate of interest one could earn on an alternative investment with a risk equal to the risk of the investment in question. This is the value of I in the TVM equations, and it is shown on the top of a time line, between the first and second tick marks. It is not a single rate-- the opportunity cost rate varies depending on the riskiness and maturity of an investment, and it also varies from year to year depending on inflationary expectations.
Annual percentage rate (APR)
The periodic rate times the number of periods per year.
PVAn
The present value of an annuity of N periods.
Discounting
The process of finding the present value of a cash flow or a series of cash flows; discounting is the reverse of compounding.
Opportunity Cost
The rate of return you could earn on an alternative investment of similar risk.
Present Value (PV)
The value today of a future cash flow or series of cash flows.
Cash Flow (CFt)
This term designates a cash flow that's not part of an annuity.
Payment (PMT)
This term designates equal cash flows coming at regular intervals.
t
Time period
5-3. If a firm's earnings per share grew from $1 to $2 over a 10- year period, the total growth would be 100%, but the annual growth rate would be less than 10%. True or false? (Hint: If you aren't sure, plug in some numbers and check it out.)
True, because of compounding effects-- growth on growth. The following example demonstrates the point. The annual growth rate is I in the following equation: $1(1+I)^10=$2 We can find I in the equation above as follows: Using a financial calculator input N= 10, PV = -1, PMT = 0, FV = 2, and I/YR = ? Solving for I/YR you obtain 7.18% Viewed another way, if earnings had grown at the rate of 10% per year for 10 years, then EPS would have increased from $1.00 to $2.59, found as follows: Using a financial calculator, input N = 10, I/YR = 10, PV = -1, PMT = 0, and FV = ?. Solving for FV you obtain $2.59. This formulation recognizes the "interest on interest" phenomenon.
Finding the interest Rate, I
Use calculator and input the N, PV, PMT, and FV. and hit I/YR to calculate.
The definition of an annuity includes the words constant payment. In other words,
annuities involve payments that are equal in every period.
Time value concepts can be applied to
anything that grows -- sales, population, earnings per share, or future salary.
If a bonds price is more than the fair price, you should
buy the CD
Discounting is the reverse of
compounding
Finding a present value is the reverse of
finding a future value
We can find an annuity's
future and present values, the interest rate built into annuity contracts, and the length of time it takes to reach a financial goal using an annuity.
If you could buy the bond for less than the fair price, you should buy it rather than
invest in the CD
With annuity due, each payment is shifted to the left by
one year When a deposit is made each year, we show the payments with minus signs.
There are two important classes of uneven cash flows:
1) A stream that consists of a series of annuity payments plus an additional final lump sum. Ex. Bonds 2) All other uneven streams. Ex. Stocks and capital investments
Uneven (non-constant) cash flow
A series of cash flows where the amount varies from one period to the next.
Annuity
A series of equal payments at fixed intervals for a specified number of periods.
Perpetuity
A stream of equal payments at fixed intervals expected to continue forever
Amortization Schedule
A table showing precisely how a loan will be repaid. It gives the required payment on each payment date and a breakdown of the payment, showing how much is interest and how much is repayment of principal.
The process of going to future value (FV) from present value (PV) is called
Compounding.
We can use _____ different procedures to solve time value problems.
Four - Step by step approach - Formula Approach - Financial Calculators - Spreadsheets
_____________ annuities are more common in finance
Ordinary
Present Value =
PV = FVn/(1+I)^N
Formula for Perpetuities
PV of a perpetuity = PMT/I
5-7. Banks and other lenders are required to disclose a rate called the APR. What is this rate? Why did Congress require that it be disclosed? Is it the same as the effective annual rate? If you were comparing the costs of loans from different lenders, could you use their APRs to determine the loan with the lowest effective interest rate?.
The annual percentage rate (APR) is the periodic rate times the number of periods per year. It is also called the nominal, or stated, rate. With the "Truth in Lending" law, Congress required that financial institutions disclose the APR so the rate charged would be more "transparent" to consumers. The APR is equal to the effective annual rate only when compounding occurs annually. If more frequent compounding occurs, the effective rate is always greater than the annual percentage rate. Nominal rates can be compared with one another, but only if the instruments being compared use the same number of compounding periods per year. If this is not the case, then the instruments being compared should be put on an effective annual rate basis for comparisons.
Annual compounding
The arithmetic process of determining the final value of a cash flow or series of cash flows when interest is added once a year.
Semiannual compounding
The arithmetic process of determining the final value of a cash flow or series of cash flows when interest is added twice a year.
Compounding
The arithmetic process of determining the final value of cash flow or series of cash flows when compound interest is applied.
Ordinary Annuity Formula FVAn= PMT (1 + I ) ^N-1 +PMT (1 + I) ^N-2 + PMT (1 + I) ^N-3 With Time 3:
The first payment earns interest for two periods, the second payment earns interest for one period, and the third payment earns no interest at all because it is made at the end of the annuity's life.
FVAn
The future value of an annuity over N periods
5-2. Explain whether the following statement is true or false: $100 a year for 10 years is an annuity; but $100 in Year 1, $200 in Year 2, and $400 in Years 3 through 10 does not constitute an annuity. However, the second series contains an annuity.
True. The second series is an uneven cash flow stream, but it contains an annuity of $400 for 8 years. The series could also be thought of as a $100 annuity for 10 years plus an additional payment of $100 in Year 2, plus additional payments of $300 in Years 3 through 10.
If you know the PV, you can________________ to find the FV, while if you know the FV, you can __________________ to find the PV
compound; discount
Annuities must have
constant payments at fixed intervals for a specified number of periods. If these conditions don't hold, then the payments do not constitute an annuity.
The present value of a cash flow due N years in the future is the amount which,
if it were on hand today, would grow to equal the given future amount.
For perpetuities
the payments go on forever, so you can't apply the step-by-step approach.
