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Level 109

Time Value of Money


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Time Value
the value of a particular sum of money changes as time passes
Risk
the possibility that the realized or actual return will differ from our expected return
Present Value
How much spending power money has today
Discount Rate
risk Free Rate + risk premium
Compounding
(calculating the future value of a sum)
Discounting
(calculating the present value of a sum)
Future Value Formula
Future value = present value × ( 1 + i ) n
2 Cautions when doing TVM Calculations on Financial Calculator
First, ensure that your calculator is set to the correct number of payments per year for the problem you are solving.
PV Sign
The sign of the PV (positive or negative) indicates the direction of the cash flows, money in and out of your wallet
Present Value Formula
Present value = PV = FV / ( 1 + i ) n
Annuity
an equally spaced sequence of equal cash flows
Annuity-FV
where PMT= annuity payment; n=number of payments; i=discount rate; Bracketed value sometimes called: future value interest factor for an annuity, or FVIFA
Annuity-PV
where PMT= annuity payment; n=number of payments; i=discount rate; Bracketed value sometimes called: present value interest factor for an annuity (PVIFA)
Examples of ways to interpret a PV of an annuity
This value is the value today of the three future $1,000 cash flows. Assuming that we really have an 8% opportunity cost, we should be indifferent between receiving three annual end-of-year future cash f…
Compounding Annuities
The key thing to remember in these compounding problems is that all the variables need to be stated for the same time period.
Effective Yield
Also known as the APY or Annual Percentage Yield
Perpetuities
A perpetuity is an infinite stream of equally spaced, equal cash flows.
Annuities Due
An annuity due, in contrast, is an annuity whose payments occur at the beginning of the period.
Solving for Annuities Due
Method 1: All we need to do is find the present value of the other payments using the ordinary annuity method and then simply add in the first payment.
Distinction between Annuities Due and Annuities
Annuities Due will have greater value than ordinary annuity: With an annuity due we receive our payments earlier than we would with an ordinary annuity. As we learned at the beginning of this to…
Uneven Cash Flows
When all of the cash flows are different
Solving Uneven Cash Flows
When all of the cash flows are different, we have to discount or compound each individual flow separately using the present/future value approach that we used for single sums and then add them together.
Deferred Annuity
As the name implies, this is a standard annuity whose first payment is deferred to some point in the future.
Solving Deferred Annuity
Alternatively, we can find the present value of the annuity at its beginning (which, if we consider it to be an ordinary annuity, will be one period before the first payment) then discount…
The Personal Application of the Time Value of Money
○ Understanding this concept and honing the ability to use it will greatly enhance your ability to plan for and achieve financial goals, but it will also profoundly impact your sense of well-being.
LOS 5.a. Interpret interest rates as required rates of return, discount rates or opportunity costs
An interest rate can be interpreted as the RATE OF RETURN required in equilibrium for a particular investment, DISCOUNT RATE for calculating the present value of future cash flows, or OPPORTUNITY COST of consuming …
Real risk-free rate of interest
Theoretical rate on a single-period loan that has no expectation of inflation in it.
Nominal risk-free rate
Since we know future inflation rates are not zero, nominal risk-free rates contain an inflation premium. Ex: T-Bills
Effective Annual Rate
Represents the annual rate of return actually being earned after adjustments have been made for different compounding periods.
PV= FV / (1+I/Y)^n
Future Value of a Single Cash Flow
Annuity
Stream of equal cash flows that occurs at equal intervals over a given period.
Solving TVM when Compounding periods are other than annual
Divide I/Y by the # of periods being compounded in the year, than multiply N by the same number
Perpetual Annuities
Perpetuities are annuities with infinite lives
net present value (NPV)
Capital investment decision model in which the PV of a project's cash inflows is compared to the PV of the projects outflows; the diff between these values determines whether or not the project is an acceptable invstmt.
Internal Rate of Return (IRR)
IRR is defined as the rate of return that equates the PV of an investment's expected benefits (inflows) with the PV of its costs (outflows)
NPV Decision Rule
Accept projects with a positive NPV. Positive NPV increases shareholders' wealth.
IRR Decision Rule
Accept projects with an IRR that is greater than the firm's (investor's) required rate of return
Holding Period Return and Total Return
Holding Period Return: percentage change in the value of an investment over the period it is held.
Money-weighted Return*
The internal rate of return (IRR) on a portfolio, taking into account all cash inflows and outflows.
Time-Weighted Rate of Return*
Measures the compound growth of an investment. Rate at which $1 compounds over a specified performance horizon.
Bank Discount Yield (BDY)
T-Bills are quoted on a bank discount basis, which is based on the face value of the instrument instead of the purchase price. See pg 147 for formula.
Holding Period Yield (HPY)
Also known as holding period return, is the total return an investor earned between the purchase date and the sale or maturity date. Formula on Pg 148
Effective Annual Yield (EAY)
Annualized value, based on a 365-day year, accounts for compound interest.
Money Market Yield (or CD Equivalent Yield)
Using a money market yield makes the quoted yield on a T-bill comparable to a yield quotes for interest-bearing money market instruments that pay interest on a 360-day basis (but uses simple interest). Formula pg 149
HPY- is the actual return an investor will receive if the money market instrument is held until maturity
Once we have HPY, EAY or MMY we can use one as a basis for calculating the other two. What is the relationship between these three functions?
Bond-Equivalent Yield
Refers to 2x the semiannual discount rate.
Distinguish between Descriptive Statistics and Inferential Statistics:
Descriptive Statistics - summarize the characteristics of a data set. (Focus on large data sets)
Distinguish between a Population and a Sample
Population - Includes all members of a specified group
Different types of Measurement Scales
French word for black, noir, to remember the types of scales in order of precision:
Parameter
A measure used to describe a characteristic of a population.
Sample Statistic
Used to measure a characteristic of a sample
Frequency Distribution
Frequency distribution groups observations into classes (also known as intervals). An interval is a range of values.
Relative Frequency
Relative recency is the percentage of total observations falling within each interval
Cumulative Absolute Frequency or Cumulative Relative Frequency
Summing the absolute or relative frequencies starting at the lowest interval and progressing through the highest.
Histogram
Bar chart showing intervals on the horizontal axis and frequency on the vertical. Allows us to quickly identify where the most observations were concentrated
Frequency Polygon
Midpoint of each interval is plotted on the horizontal axis, and the absolute frequencies for that interval is plotted on the vertical axis. Each point is connected with a straight line.
Weighted Mean
Recognizes that different observations may have a disproportionate influence on the mean.
Geometric Mean
[(1+R1) X (1+RT)]^(1/t)-1
Measures of Location
Quartiles and measures of central tendencies are known collectively as
Harmonic Mean
Used for certain computations, like average cost of shares purchased overtime. Pg174
Range =
Maximum Value-Minimum Value
Population Variance and Population Standard Deviation
Population variance is defined as the average of the squared deviations from the mean.
Chebyshev Inequality
States that for any set of observations, regardless of shape of distribution, the percentage of the observations that lie within k standard deviations of the mean is AT LEAST 1-(1/k)^2 for all k >1
Coefficient of Variation (CV)
standard deviation of X / average value of X
Sharpe Ratio (Sharpe Measure or Reward-to Variability ratio)
Measures excess return per unit of risk. Pg 181 study formula. Excess return per unit of risk
skewness
lack of symmetry
Skewness on....
Symetrical Distribution: Mean=Median=Mode
kurtosis
the amount of dispersion in a distribution
A Random Variable: is an uncertain value determined by chance.
LOS 8.a.: Define a random variable, an outcome, an event, mutually exclusive events, and exhaustive events
The two properties of a probability are:
(1) The sum of the probabilities of all possible mutually exclusive events is 1.
Distinguish between empirical, subjective and a priori probabilities
Priori probability: measures predetermined probabilities based on well-defined inputs.
Unconditional Probability (Marginal Probability)
Refers to the probability of an event regardless of the past or future occurrence of other events
Conditional Probability
is where the occurrence of one event affects the probability of the occurrence of another event. Key word is "given".
Multiplication Rule of Probability
Used to determine the joint probability of two events:
Addition Rule of Probability
Used to determine the probability that at least one of two events will occur:
Total Probability Rule
Used to determine the unconditional probability of an event, given conditional probabilities. Pg203
Bayes' Formula
Used to update a given set of prior probabilities for a given event in response to the arrival of new information.
Time Value of Money Purpose
time value of money concerns equivalence relationships between cash flows occurring on different dates
Three Ways to Consider Interest Rates
rates of return: the minimum amount an investor must receive in order to Accept an investment
real risk-Free Interest Rate
The Composition of the Interest Rate (r)
Inflation premium
reflection of change in general price level
Default Risk Premium
Compensates investors for the possibility that the borrower Will fail to make a promised payment at the contracted time and in the contracted amount
liquidity premium
-Compensates investors for the risk of loss relative to an investment's fair value if the investment needs to be converted to cash quickly
Maturity premium
-Compensates investors for the increased sensitivity of the market value of debt to a change in market interest rates as maturity is extended, in general
Nominal Risk-free Interest Rate
the sum of the real risk-Free Interest Rate and the inflation premium
FV = PV(1 + r)
Single Cash Flow Investment Equation for One Period
Simple Interest
Interest Rate times the Principal
Principal
the amount of money borrowed in a loan of the amount of your money in savings account that is earning interest
compounding
the calculation of interest on a principal amount, plus interest on the interest accrued during a previous period
FV = PV(1 + r)^N
Single Cash Flow Investment Equation for Multiple Periods (FV)
Important Note about N and r
both variables must be defined in the same time Units
Parentheses First
PEMDAS Order of Operations
FV = PV(1 + r/m)^(m*N)
Single Cash Flow Investment Equation for Multiple Compounding Periods per Year (FV)
FV = PVe^(r*N)
Single Cash Flow Investment Equation with Continuous Compounding (FV)
Effective Annual Rate (EAR) Equation with Multiple Compounding Periods
EAR = (1 + Periodic interest rate)^m - 1
EAR = e^r - 1
Effective Annual Rate (EAR) Equation with Continuous Compounding
Ordinary Annuity
-Has a first cash flow that occurs one period from now (t = 1)
Annuity Due
-Has a first cash flow that occurs immediately (t = 0)
Perpetuity
an annuity with infinite life
Simple Annuity Equation (FV)
FV = A*[((1 + r)^N -1)/r]
How to Solve for Unequal Cash Flows
Solve for each cash flow at a time using the simple FV/PV equation
PV = FV[1/(1 + r)^N]
Single Cash Flow Investment Equation for Multiple Periods (PV)
How Present Value Relates to the Discount Rate
-For a given discount rate, the farther in the future the amount to be received, the smaller that amount's present value
PV = FV/[(1 + (r/m))^(m*N)]
Single Cash Flow Investment Equation for Multiple Compounding Periods per Year (PV)
PV = A[(1 - (1/((1 + r)^N))/r]
Present Value of a Series of Equal Cash Flows Equation (PV)
PV = A/r
The Present Value of an Infinite Series of Equal Cash Flows - Perpetuity
Growth Rate Equation
g = (FV/PV)^(1/N) - 1
Solving for N when Compounding Annually
N ln(1 + r) = ln(FV/PV)
Cash Flow Additivity Principle
-When dealing with uneven cash flows, we take maximum advantage of the principle that dollar amounts indexed at the same point in time are additive