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The Time Value of
Money
Introduction to the Time Value of Money
Future Value, Single Amount
Present Value, Single Amount
Annuities
Valuing Multiple Cash Flows
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The Time Value of
Money
(continued)
Additional Detail on Present and Future Values
Yield
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The Time Value of Money > Introduction to the Time Value of Money
Introduction to the Time Value of Money
• Defining the Time Value of Money
• Importance of the Time Value of Money
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The Time Value of Money > Introduction to the Time Value of Money
Defining the Time Value of Money
• Being given $100 today is better than being given $100 in the future because you
don't have to wait for your money.
• Money today has a value (present value, or PV) and money in the future has a
value (future value, or FV).
• The amount that the value of the money changes after one year is called the
interest rate (i). For example, if money today is worth 10% more in one year, the
interest rate is 10%.
Simple Interest Formula
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The Time Value of Money > Introduction to the Time Value of Money
Importance of the Time Value of Money
• Money today is worth more than the same quantity of money in the future. You
can invest a dollar today and receive a return on your investment.
• Loans, investments, and any other deal must be compared at a single point in
time to determine if it's a good deal or not.
• The process of determining how much a future cash flow is worth today is called
discounting. It is done for most major business transactions during investing
decisions in capital budgeting.
Compound Interest
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The Time Value of Money > Future Value, Single Amount
Future Value, Single Amount
• Single-Period Investment
• Multi-Period Investment
• Calculating Future Value
• Approaches to Calculating Future Value
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The Time Value of Money > Future Value, Single Amount
Single-Period Investment
• Single-period investments use a specified way of calculating future and present
value.
• Single-period investments take place over one period (usually one year).
• In a single-period investment, you only need to know two of the three variables
PV, FV, and i. The number of periods is implied as one since it is a single-period.
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The Time Value of Money > Future Value, Single Amount
Multi-Period Investment
• Investments that accrue simple interest have interest paid based on the amount of
the principal, not the balance in the account.
• Investments that accrue compound interest have interest paid on the balance of
the account. This means that interest is paid on interest earned in previous
periods.
• Simple interest increases the balance linearly, while compound interest increases
it exponentially.
Simple Interest Formula
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The Time Value of Money > Future Value, Single Amount
Calculating Future Value
• The future value is the value of a given amount of money at a certain point in the
future if it earns a rate of interest.
• The future value of a present value is calculated by plugging the present value,
interest rate, and number of periods into one of two equations.
• Unless otherwise noted, it is safe to assume that interest compounds and is not
simple interest.
Compound Interest
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The Time Value of Money > Future Value, Single Amount
Approaches to Calculating Future Value
• The "present" can be moved based on whatever makes the problem easiest. Just
remember that moving the date of the present also changes the number of
periods until the future for the FV.
• To find FV, you must first identify PV, the interest rate, and the number of periods
from the present to the future.
• The interest rate and the number of periods must have consistent units. If one
period is one year, the interest rate must be X% per year, and vis versa.
Compound Interest
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The Time Value of Money > Present Value, Single Amount
Present Value, Single Amount
• Single-Period Investment
• Multi-Period Investment
• The Discount Rate
• Number of Periods
• Calculating Present Value
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The Time Value of Money > Present Value, Single Amount
Single-Period Investment
• A single period investment has the number of periods (n or t) equal to one.
• For both simple and compound interest, the PV is FV divided by 1+i.
• The time value of money framework says that money in the future is not worth as
much as money in the present.
FV of a single payment
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The Time Value of Money > Present Value, Single Amount
Multi-Period Investment
• Finding the PV for a multi-period investment is the same as for a single-period
investment: plug FV, the interest rate, and the number of periods into the correct
formula.
• PV varies jointly with FV, and inversely with i and n.
• When n>1, simple and compound interest cease to provide the same answer
(unless the interest rate is 0).
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The Time Value of Money > Present Value, Single Amount
The Discount Rate
• The discount rate represents some cost (or group of costs) to the investor or
creditor.
• Some costs to the investor or creditor are opportunity cost, liquidity cost, risk, and
inflation.
• The discount rate is used by both the creditor and debtor to find the present value
of an amount of money.
Borrowing and lending
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The Time Value of Money > Present Value, Single Amount
Number of Periods
• A period is just a general term for a length of time. It can be anything- one month,
one year, one decade- but it must be clearly defined and fixed.
• For both simple and compound interest, the number of periods varies jointly with
FV and inversely with PV.
• The number of periods is also part of the units of the discount rate: if one period is
one year, the discount rate must be defined as X% per year. If one period is one
month, the discount rate must be X% per month.
FV of a single payment
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The Time Value of Money > Present Value, Single Amount
Calculating Present Value
• The first step is to identify if the interest is simple or compound. Most of the time,
it is compound.
• The interest rate and number of periods must have consistent units.
• The PV is what a future sum is worth today given a specific interest rate (often
called a "discount rate").
Simple Interest Formula
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The Time Value of Money > Annuities
Annuities
• Annuities
• Future Value of Annuity
• Present Value of Annuity
• Calculating Annuities
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The Time Value of Money > Annuities
Annuities
• Annuities have payments of a fixed size paid at regular intervals.
• There are three types of annuities: annuities-due, ordinary annuities, and
perpetuities.
• Annuities help both the creditor and debtor have predictable cash flows, and it
spreads payments of the investment out over time.
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The Time Value of Money > Annuities
Future Value of Annuity
• To find the FV, you need to know the payment amount, the interest rate of the
account the payments are deposited in, the number of periods per year, and the
time frame in years.
• The first and last payments of an annuity due both occur one period before they
would in an ordinary annuity, so they have different values in the future.
• There are different formulas for annuities due and ordinary annuities because of
when the first and last payments occur.
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The Time Value of Money > Annuities
Present Value of Annuity
• The PV for both annuities-due and ordinary annuities can be calculated using the
size of the payments, the interest rate, and number of periods.
• The PV of a perpetuity can be found by dividing the size of the payments by the
interest rate.
• Payment size is represented as p, pmt, or A; interest rate by i or r; and number of
periods by n or t.
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The Time Value of Money > Annuities
Calculating Annuities
• There are five total variables that go into annuity calculations: PV, FV, interest
rate (i or r), payment amount (A, m, pmt, or p), and the number of periods (n).
• The calculations for ordinary annuities and annuities-due differ due to the different
times when the first and last payments occur.
• Perpetuities don't have a FV formula because they continue forever. To find the
FV at a point, treat it as an ordinary annuity or annuity-due up to that point.
PV of a Perpetuity
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The Time Value of Money > Valuing Multiple Cash Flows
Valuing Multiple Cash Flows
• Future Value, Multiple Flows
• Present Value, Multiple Flows
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The Time Value of Money > Valuing Multiple Cash Flows
Future Value, Multiple Flows
• The FV of multiple cash flows is the sum of the FV of each cash flow.
• To sum the FV of each cash flow, each must be calculated to the same point in
the future.
• If the multiple cash flows are a fixed size, occur at regular intervals, and earn a
constant interest rate, it is an annuity. There are formulas for calculating the FV of
an annuity.
FV of a single payment
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The Time Value of Money > Valuing Multiple Cash Flows
Present Value, Multiple Flows
• To find the PV of multiple cash flows, each cash flow much be discounted to a
specific point in time and then added to the others.
• To discount annuities to a time prior to their start date, they must be discounted to
the start date, and then discounted to the present as a single cash flow.
• Multiple cash flow investments that are not annuities unfortunately cannot be
discounted by any other method but by discounting each cash flow and summing
them together.
Sum FV
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The Time Value of Money > Additional Detail on Present and Future Values
Additional Detail on Present and Future Values
• The Relationship Between Present and Future Value
• Calculating Perpetuities
• Calculating Values for Different Durations of Compounding Periods
• Comparing Interest Rates
• Calculating Values for Fractional Time Periods
• Loans and Loan Amortization
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The Time Value of Money > Additional Detail on Present and Future Values
The Relationship Between Present and Future Value
• The future value (FV) measures the nominal future sum of money that a given
sum of money is "worth" at a specified time in the future assuming a certain
interest rate, or more generally, rate of return. The FV is calculated by multiplying
the present value by the accumulation function.
• PV and FV vary jointly: when one increases, the other increases, assuming that
the interest rate and number of periods remain constant.
• As the interest rate (discount rate) and number of periods increase, FV increases
or PV decreases.
FV of a single payment
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The Time Value of Money > Additional Detail on Present and Future Values
Calculating Perpetuities
• Perpetuities are a special type of annuity; a perpetuity is an annuity that has no
end, or a stream of cash payments that continues forever.
• To find the future value of a perpetuity requires having a future date, which
effectively converts the perpetuity to an ordinary annuity until that point.
• Perpetuities with growing payments are called Growing Perpetuities; the growth
rate is subtracted from the interest rate in the present value equation.
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The Time Value of Money > Additional Detail on Present and Future Values
Calculating Values for Different Durations of Compounding
Periods
• The units of the period (e.g. one year) must be the same as the units in the
interest rate (e.g. 7% per year).
• When interest compounds more than once a year, the effective interest rate
(EAR) is different from the nominal interest rate.
• The equation in skips the step of solving for EAR, and is directly usable to find the
present or future value of a sum.
EAR with Continuous Compounding
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The Time Value of Money > Additional Detail on Present and Future Values
Comparing Interest Rates
• A nominal interest rate that compounds has a different effective rate (EAR),
because interest is accrued on interest.
• The Fisher Equation approximates the amount of interest accrued after
accounting for inflation.
• A company will theoretically only invest if the expected return is higher than their
cost of capital, even if the return has a high nominal value.
Fisher Equation
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The Time Value of Money > Additional Detail on Present and Future Values
Calculating Values for Fractional Time Periods
• The balance of an account only changes when interest is paid. To find the
balance, round the fractional time period down to the period when interest was
last accrued.
• To find the PV or FV, ignore when interest was last paid an use the fractional time
period as the time period in the equation.
• The discount rate is really the cost of not having the money over time, so for
PV/FV calculations, it doesn't matter if the interest hasn't been added to the
account yet.
Compounding Interest
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The Time Value of Money > Additional Detail on Present and Future Values
Loans and Loan Amortization
• Each amortization payment should be equal in size and pays off a portion of the
principal as well as a portion of the interest.
• The percentage of interest versus principal in each payment is determined in an
amortization schedule.
• If the repayment model for a loan is "fully amortized," then the very last payment
pays off all remaining principal and interest on the loan.
Amortization Schedule
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The Time Value of Money > Yield
Yield
• Calculating the Yield of a Single-Period Investment
• Calculating the Yield of an Annuity
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The Time Value of Money > Yield
Calculating the Yield of a Single-Period Investment
• There are a number of ways to calculate yield, but the most common ones are to
calculate the percent change from the initial investment, APR, and APY (or EAR).
• APR (annual percentage rate) is a commonly used calculation that figures out the
nominal amount of interest accrued per year. It does not account for compounding
interest.
• APY (annual percentage yield) is a way of using the nominal interest rate to
calculate the effective interest rate per year. It accounts for compounding interest.
• EAR (effective annual rate) is a special type of APY that uses APR as the nominal
interest rate.
Percent Change
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The Time Value of Money > Yield
Calculating the Yield of an Annuity
• The yield of an annuity may be found by discounting to find the PV, and then
finding the percentage change from the PV to the FV.
• The Internal Rate of Return (IRR) is the discount rate at which the NPV of an
investment equals 0.
• The IRR calculates an annualized yield of an annuity.
IRR Example
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Appendix
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The Time Value of Money
Key terms
• accrue To add, or grow.
• amortization The distribution of the cost of an intangible asset, such as an intellectual property right, over the projected useful
life of the asset.
• amortization schedule a table detailing each periodic payment over the life of the loan
• amortized loan a form of debt where the principal is paid down over the life of the debt according to some amortization
schedule, typically through equal payments
• annuity A specified income payable at stated intervals for a fixed or a contingent period, often for the recipient's life, in
consideration of a stipulated premium paid either in prior installment payments or in a single payment. For example, a
retirement annuity paid to a public officer following his or her retirement.
• annuity-due An investment with fixed-payments that occur at regular intervals, paid at the beginning of each period.
• annuity-due An annuity where the payments occur at the beginning of each period.
• annuity-due a stream of fixed payments where payments are made at the beginning of each period
• capitalization The process of finding the future value of a sum by evaluating the present value.
• cash flow The sum of cash revenues and expenditures over a period of time.
• compound interest Interest, as on a loan or a bank account, that is calculated on the total on the principal plus accumulated
unpaid interest.
• compound interest Interest, as on a loan or a bank account, that is calculated on the total on the principal plus accumulated
unpaid interest.
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The Time Value of Money
• compound interest Interest, as on a loan or a bank account, that is calculated on the total on the principal plus accumulated
unpaid interest.
• compounding period The length of time between the points at which interest is paid.
• discount To find the value of a sum of money at some earlier point in time. To find the present value.
• discount to account for the time value of money
• discount rate The interest rate used to discount future cash flows of a financial instrument; the annual interest rate used to
decrease the amounts of future cash flow to yield their present value.
• discounting The process of finding the present value using the discount rate.
• discounting The process of determining how much money paid/received in the future is worth today. You discount future values
of cash back to the present using the discount rate.
• discounting The process of finding the present value using the discount rate.
• Effective Interest The amount of interest accrued per year after accounting for compounding.
• effective-interest method amortizing a debt according to the effective interest rate paid
• Future Value The value of an asset at a specific date. It measures the nominal future sum of money that a given sum of money
is "worth" at a specified time in the future, assuming a certain interest rate, or more generally, rate of return, it is the present
value multiplied by the accumulation function.
• Future Value (FV) The value of the money in the future.
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The Time Value of Money
• growth rate The percentage by which the payments grow each period.
• incremental cash flows the additional money flowing in or out of a business due to a project
• inflation An increase in the general level of prices or in the cost of living.
• interest rate The percentage of an amount of money charged for its use per some period of time. It can also be thought of as
the cost of not having money for one period, or the amount paid on an investment per year.
• interest rate The percentage of an amount of money charged for its use per some period of time. It can also be thought of as
the cost of not having money for one period, or the amount paid on an investment per year.
• Interest Rate (i or r) The cost of not having money for one period, or the amount paid on an investment per year.
• Internal Rate of Return (IRR) The discount rate that will cause the NPV of an investment to equal 0.
• multi-period More than one unit of time.
• Multi-period investment An investment that takes place over more than one periods.
• net present value the present value of a project or an investment decision determined by summing the discounted incoming and
outgoing future cash flows resulting from the decision
• Net Present Value (NPV) The present value of a project or an investment decision determined by summing the discounted
incoming and outgoing future cash flows resulting from the decision.
• Nominal Interest The amount of interest accrued per year without accounting for compounding.
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The Time Value of Money
• ordinary annuity An annuity where the payments occur at the end of each period.
• ordinary annuity An investment with fixed-payments that occur at regular intervals, paid at the end of each period.
• ordinary repair expense accrued in normal maintenance of an asset
• period The length of time during which interest accrues.
• period The length of time during which interest accrues.
• period The length of time during which interest accrues.
• Periods (t or n) Units of time. Usually one year.
• perpetuity An annuity in which the periodic payments begin on a fixed date and continue indefinitely.
• perpetuity An annuity in which the periodic payments begin on a fixed date and continue indefinitely.
• present value Also known as present discounted value, is the value on a given date of a payment or series of payments made
at other times. If the payments are in the future, they are discounted to reflect the time value of money and other factors such
as investment risk. If they are in the past, their value is correspondingly enhanced to reflect that those payments have been (or
could have been) earning interest in the intervening time. Present value calculations are widely used in business and
economics to provide a means to compare cash flows at different times on a meaningful "like to like" basis.
• present value a future amount of money that has been discounted to reflect its current value, as if it existed today
• Present Value (PV) The value of the money today.
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The Time Value of Money
• principal The money originally invested or loaned, on which basis interest and returns are calculated.
• quarter A period of three consecutive months (1/4 of a year).
• simple interest interest paid only on the principal.
• Single-period investment An investment that takes place over one period, usually one year.
• time period assumption business profit or loses are measured on timely basis
• time value of money the value of an asset accounting for a given amount of interest earned or inflation accrued over a given
period
• yield In finance, the term yield describes the amount in cash that returns to the owners of a security. Normally it does not
include the price variations, at the difference of the total return. Yield applies to various stated rates of return on stocks
(common and preferred, and convertible), fixed income instruments (bonds, notes, bills, strips, zero coupon), and some other
investment type insurance products
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The Time Value of Money
Simple Interest Formula
Simple interest is when interest is only paid on the amount you originally invested (the principal). You don't earn interest on interest you previously
earned.
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The Time Value of Money
Compound Interest
Interest is paid at the total amount in the account, which may include interest earned in previous periods.
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The Time Value of Money
Compounding Interest
The effect of earning 20% annual interest on an initial $1,000 investment at various compounding frequencies.
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The Time Value of Money
Simple Interest Formula
Simple interest is when interest is only paid on the amount you originally invested (the principal). You don't earn interest on interest you previously
earned.
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The Time Value of Money
Simple Interest Formula
Simple interest is when interest is only paid on the amount you originally invested (the principal). You don't earn interest on interest you previously
earned.
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The Time Value of Money
Amortization Schedule
An example of an amortization schedule of a $100,000 loan over the first two years.
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The Time Value of Money
Borrowing and lending
Banks like HSBC take such costs into account when determining the terms of a loan for borrowers.
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The Time Value of Money
IRR Example
The setup to find the IRR of the investment with cash flows of -4000, 1200, 1410, 1875, and 1050. By setting NPV = 0 and solving for r, you can find the
IRR of this investment.
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The Time Value of Money
Fisher Equation
The nominal interest rate is approximately the sum of the real interest rate and inflation.
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The Time Value of Money
Compound Interest
In this formula, your deposit ($100) is PV, i is the interest rate (5% for Bank 1, 6% for Bank 2), t is time (5 years), and FV is the future value.
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The Time Value of Money
Compound Interest
Interest is paid at the total amount in the account, which may include interest earned in previous periods.
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The Time Value of Money
Annual Percentage Yield
The Annual Percentage Yield is a way or normalizing the nominal interest rate. Basically, it is a way to account for the time factor in order to get a more
accurate number for the actual interest rate.inom is the nominal interest rate.N is the number of compounding periods per year.
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The Time Value of Money
PV Annuity-due
The PV of an annuity with the payments at the beginning of each period
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The Time Value of Money
FV Periodic Compounding
Finding the FV (A(t)) given the PV (Ao), nominal interest rate (r), number of compounding periods per year (n), and number of years (t).
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The Time Value of Money
Present Value Single Payment
Finding the PV is a matter of plugging in for the three other variables.
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The Time Value of Money
Simple Interest Formula
Simple interest is when interest is only paid on the amount you originally invested (the principal). You don't earn interest on interest you previously
earned.
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The Time Value of Money
Percent Change
The percent change in value is the change in value from PV to FV (V2 to V1) divided by PV (V1) times 100%.
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The Time Value of Money
FV of a single payment
The PV and FV are directly related.
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The Time Value of Money
Solving for n
This formula allows you to figure out how many periods are needed to achieve a certain future value, given a present value and an interest rate.
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The Time Value of Money
Calculating the effective annual rate
The effective annual rate for interest that compounds more than once per year.
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The Time Value of Money
Simple Interest Formula
Simple interest is when interest is only paid on the amount you originally invested (the principal). You don't earn interest on interest you previously
earned.
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The Time Value of Money
Sum FV
The PV of an investment is the sum of the present values of all its payments.
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The Time Value of Money
Car
Car loans, mortgages, and student loans all generally have compound interest.
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The Time Value of Money
Compound Interest
Interest is paid at the total amount in the account, which may include interest earned in previous periods.
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The Time Value of Money
EAR
The Effective Annual Rate is the amount of interest actually accrued per year based on the APR. n is the number of compounding periods of APR per
year.
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The Time Value of Money
FV Ordinary Annuity
The FV of an annuity with the payments at the end of each period
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The Time Value of Money
FV Annuity-Due
The FV of an annuity with payments at the beginning of each period: m is the amount amount, r is the interest, n is the number of periods per year, and t
is the number of years.
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The Time Value of Money
FV of a single payment
The FV of multiple cash flows is the sum of the future values of each cash flow.
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The Time Value of Money
PV Ordinary Annuity
The PV of an annuity with the payments at the end of each period
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The Time Value of Money
FV of a single payment
The PV and FV are directly related.
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The Time Value of Money
PV of a Perpetuity
The PV of a perpetuity is the payment size divided by the interest rate.
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The Time Value of Money
EAR with Continuous Compounding
The effective rate when interest compounds continuously.
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The Time Value of Money
FV of a single payment
The FV is related to the PV by being i% more each period.
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The Time Value of Money
Which of these is a variable used to calculate the time value of
money?
A) The interest rate.
B) The amount of time that has passed.
C) The present value of the sum you have.
D) All of these answers.
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The Time Value of Money
Which of these is a variable used to calculate the time value of
money?
A) The interest rate.
B) The amount of time that has passed.
C) The present value of the sum you have.
D) All of these answers.
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The Time Value of Money
Which of the following is an explanation for why the concept of the
time value of money is important to a business?
A) The present value of a dollar you get in a year is less than the value of
a dollar you get today.
B) By figuring out the present value of future income, a company can
better compare possible projects.
C) A project's future unadjusted revenues can be misleading when trying
to determine its profitability.
D) All of these answers.
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The Time Value of Money
Which of the following is an explanation for why the concept of the
time value of money is important to a business?
A) The present value of a dollar you get in a year is less than the value of
a dollar you get today.
B) By figuring out the present value of future income, a company can
better compare possible projects.
C) A project's future unadjusted revenues can be misleading when trying
to determine its profitability.
D) All of these answers.
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The Time Value of Money
You have $300,000 that you want to invest in a one year
Certificate of Deposit (CD) with a 4% annual interest rate. What
will be the value of that CD in a year?
A) $420,000
B) $301,200
C) $312,000
D) $315,000
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The Time Value of Money
You have $300,000 that you want to invest in a one year
Certificate of Deposit (CD) with a 4% annual interest rate. What
will be the value of that CD in a year?
A) $420,000
B) $301,200
C) $312,000
D) $315,000
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The Time Value of Money
What is the future value in 30 years of $100,000 invested today in
a savings account earning a 1% compound interest rate every
year (rounded up to the nearest dollar)?
A) 30000
B) 130000
C) More than $134785
D) 134785
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The Time Value of Money
What is the future value in 30 years of $100,000 invested today in
a savings account earning a 1% compound interest rate every
year (rounded up to the nearest dollar)?
A) 30000
B) 130000
C) More than $134785
D) 134785
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The Time Value of Money
You plan to invest $100,000 in a 3 year Certificate of Deposit that
has a 5% compound interest rate. What is its future value?
A) $115,763
B) $115,000
C) $105,000
D) $115,927
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The Time Value of Money
You plan to invest $100,000 in a 3 year Certificate of Deposit that
has a 5% compound interest rate. What is its future value?
A) $115,763
B) $115,000
C) $105,000
D) $115,927
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The Time Value of Money
You plan to invest $100,000 in a 3 year Certificate of Deposit that
has a simple interest rate of 5%. What is its future value?
A) $115,763
B) $105,000
C) $115,000
D) $115,927
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The Time Value of Money
You plan to invest $100,000 in a 3 year Certificate of Deposit that
has a simple interest rate of 5%. What is its future value?
A) $115,763
B) $105,000
C) $115,000
D) $115,927
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The Time Value of Money
The future value concept answers which of the following
questions?
A) How much money will you have in 10 years if you invest $10,000
today earning 10% as interest every year?
B) How much money will you have in 10 years if you invest $10,000
every year at 10% interest every year?
C) All of these answers
D) How much money will you have in 10 years if you invest $10,000
every year at 10% net return?
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The Time Value of Money
The future value concept answers which of the following
questions?
A) How much money will you have in 10 years if you invest $10,000
today earning 10% as interest every year?
B) How much money will you have in 10 years if you invest $10,000
every year at 10% interest every year?
C) All of these answers
D) How much money will you have in 10 years if you invest $10,000
every year at 10% net return?
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The Time Value of Money
Which of the following is the correct formula for calculating future
value with simple interest?
A) FV = PV * (1+i)t
B) FV = PV * i
C) All of these answers
D) FV = PV * (1+i*t)
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The Time Value of Money
Which of the following is the correct formula for calculating future
value with simple interest?
A) FV = PV * (1+i)t
B) FV = PV * i
C) All of these answers
D) FV = PV * (1+i*t)
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The Time Value of Money
What is the future value in 30 years of $100,000 invested today in
a savings account earning a 1% simple interest rate every year
(rounded up to the nearest dollar)?
A) 130,000
B) 30,000
C) 134,785
D) More than $134,785
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The Time Value of Money
What is the future value in 30 years of $100,000 invested today in
a savings account earning a 1% simple interest rate every year
(rounded up to the nearest dollar)?
A) 130,000
B) 30,000
C) 134,785
D) More than $134,785
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The Time Value of Money
A period of three consecutive months (1/4 of a year).
A) quarter
B) Discounting
C) perpetuity
D) annuity
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The Time Value of Money
A period of three consecutive months (1/4 of a year).
A) quarter
B) Discounting
C) perpetuity
D) annuity
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The Time Value of Money
You decide to make an investment on a bond that pays 5%
interest on the principal only. Which of the following describes
how interest accrues on this investment?
A) Compound interest.
B) Incepted interest.
C) Simple interest.
D) All of these answers.
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The Time Value of Money
You decide to make an investment on a bond that pays 5%
interest on the principal only. Which of the following describes
how interest accrues on this investment?
A) Compound interest.
B) Incepted interest.
C) Simple interest.
D) All of these answers.
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The Time Value of Money
In a year, you expect to receive a payment of $1 million in a year.
That annual interest rate is 5%. What is the present value of the
future payment?
A) $666,667
B) $952,381
C) $995,025
D) $1,050,000
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The Time Value of Money
In a year, you expect to receive a payment of $1 million in a year.
That annual interest rate is 5%. What is the present value of the
future payment?
A) $666,667
B) $952,381
C) $995,025
D) $1,050,000
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The Time Value of Money
What is the present value of $100,000 that will be received 5
years from today if you face a 10% compound interest rate every
year (rounded up to the nearest dollar)?
A) 62092
B) 52092
C) 72092
D) 82092
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The Time Value of Money
What is the present value of $100,000 that will be received 5
years from today if you face a 10% compound interest rate every
year (rounded up to the nearest dollar)?
A) 62092
B) 52092
C) 72092
D) 82092
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The Time Value of Money
Assume you invest money in a bond that will pay you $250,000 in
four years. The bond has an annual interest rate of 5%. You do
not receive interest payments while you own the bond; it is zerocoupon. What is the bond's present value?
A) $205,676
B) $205,482
C) $238,095
D) $240,385
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The Time Value of Money
Assume you invest money in a bond that will pay you $250,000 in
four years. The bond has an annual interest rate of 5%. You do
not receive interest payments while you own the bond; it is zerocoupon. What is the bond's present value?
A) $205,676
B) $205,482
C) $238,095
D) $240,385
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The Time Value of Money
Which of the following is a cost to the investor that is included in
the calculation of an investment's interest rate?
A) Opportunity Cost.
B) Inflation.
C) Risk of a bad investment.
D) All of these answers.
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The Time Value of Money
Which of the following is a cost to the investor that is included in
the calculation of an investment's interest rate?
A) Opportunity Cost.
B) Inflation.
C) Risk of a bad investment.
D) All of these answers.
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The Time Value of Money
Which of the following could be an appropriate period used in a
present value calculation?
A) A year.
B) A month.
C) Three months.
D) All of these answers.
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The Time Value of Money
Which of the following could be an appropriate period used in a
present value calculation?
A) A year.
B) A month.
C) Three months.
D) All of these answers.
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The Time Value of Money
Which of the following the correct formula for calculating present
value using compound interest?
A) PV = FV/(1+(i*t))
B) PV = FV/(i)
C) All of these answers.
D) PV = FV / (1+i)n
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The Time Value of Money
Which of the following the correct formula for calculating present
value using compound interest?
A) PV = FV/(1+(i*t))
B) PV = FV/(i)
C) All of these answers.
D) PV = FV / (1+i)n
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The Time Value of Money
A security offers to pay the holder $1000 at the end of every
month for five years. What type of annuity is this?
A) Annuity-due.
B) Ordinary annuity.
C) Perpetuity.
D) Regular annuity.
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The Time Value of Money
A security offers to pay the holder $1000 at the end of every
month for five years. What type of annuity is this?
A) Annuity-due.
B) Ordinary annuity.
C) Perpetuity.
D) Regular annuity.
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The Time Value of Money
An annuity pays $1500 at the beginning of every month for five
years. The interest rate of the annuity is 4%. What is this annuity's
future value?
A) $97,948
B) $99,448
C) $101,280
D) $99,780
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The Time Value of Money
An annuity pays $1500 at the beginning of every month for five
years. The interest rate of the annuity is 4%. What is this annuity's
future value?
A) $97,948
B) $99,448
C) $101,280
D) $99,780
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The Time Value of Money
A five year annuity pays $1000 at the end of every month for four
years. It has an interest rate of 3%. What is its present value?
A) $3,717
B) $26,024
C) $3,828
D) $25,266
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The Time Value of Money
A five year annuity pays $1000 at the end of every month for four
years. It has an interest rate of 3%. What is its present value?
A) $3,717
B) $26,024
C) $3,828
D) $25,266
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The Time Value of Money
A person purchases a security that pays $1000 a year in
perpetuity. The interest rate of that security is 4%. What is the
present value of that security?
A) $25,000
B) $1040
C) $250,000
D) $4160
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The Time Value of Money
A person purchases a security that pays $1000 a year in
perpetuity. The interest rate of that security is 4%. What is the
present value of that security?
A) $25,000
B) $1040
C) $250,000
D) $4160
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The Time Value of Money
The sum of cash revenues and expenditures over a period of
time.
A) Cash
B) Cash Flow
C) Assets
D) Account Receivables
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The Time Value of Money
The sum of cash revenues and expenditures over a period of
time.
A) Cash
B) Cash Flow
C) Assets
D) Account Receivables
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The Time Value of Money
You purchase two annuities. The first is for three years and pays
$1500 annually. The second is for four years and pays $2000
annually. The interest rate for both is 4%. What is the Future
Value of this portfolio?
A) $12,612.90
B) $13,175.33
C) $506.74
D) $485.11
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The Time Value of Money
You purchase two annuities. The first is for three years and pays
$1500 annually. The second is for four years and pays $2000
annually. The interest rate for both is 4%. What is the Future
Value of this portfolio?
A) $12,612.90
B) $13,175.33
C) $506.74
D) $485.11
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The Time Value of Money
You can purchase two three-year annuities today. One is valued
at $2000, the other at $4000. The 1st annuity begins paying
$1000 in a year. The 2nd annuity begins paying $1500 in two
years. The interest rate is 5%. What is the PV of the portfolio?
A) $613.60
B) $6613.60
C) $808.12
D) $6808.12
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The Time Value of Money
You can purchase two three-year annuities today. One is valued
at $2000, the other at $4000. The 1st annuity begins paying
$1000 in a year. The 2nd annuity begins paying $1500 in two
years. The interest rate is 5%. What is the PV of the portfolio?
A) $613.60
B) $6613.60
C) $808.12
D) $6808.12
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The Time Value of Money
Of the following car financing options, which one would you prefer
while assuming that you prefer paying the least amount of dollars
and that you face a 10% annual compound interest rate on all
your financial decisions?
A) A lump-sum payment of $20,000 in two years from today.
B) A payment $10,000 today and another of $10,000 in one year from
today.
C) A lump-sum payment of $19,000 today only.
D) A lump-sum payment of $20,000 today only.
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The Time Value of Money
Of the following car financing options, which one would you prefer
while assuming that you prefer paying the least amount of dollars
and that you face a 10% annual compound interest rate on all
your financial decisions?
A) A lump-sum payment of $20,000 in two years from today.
B) A payment $10,000 today and another of $10,000 in one year from
today.
C) A lump-sum payment of $19,000 today only.
D) A lump-sum payment of $20,000 today only.
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The Time Value of Money
Which of the following describes the relationship between present
value and future value?
A) When present value increases, the future value decreases, assuming
all variables are constant.
B) When one increases, the other increases, assuming all variables are
constant.
C) The higher the interest rate, the higher the present value and the
lower the future value.
D) The more time that passes, the higher the present value and the lower
the future value.
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The Time Value of Money
Which of the following describes the relationship between present
value and future value?
A) When present value increases, the future value decreases, assuming
all variables are constant.
B) When one increases, the other increases, assuming all variables are
constant.
C) The higher the interest rate, the higher the present value and the
lower the future value.
D) The more time that passes, the higher the present value and the lower
the future value.
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The Time Value of Money
You own a perpetuity that pays $1000 annually. It has a 5%
annual interest rate and a 2% annual growth rate. What is the
present value of the perpetuity?
A) $20,000
B) $33,333
C) $50,000
D) $14,286
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The Time Value of Money
You own a perpetuity that pays $1000 annually. It has a 5%
annual interest rate and a 2% annual growth rate. What is the
present value of the perpetuity?
A) $20,000
B) $33,333
C) $50,000
D) $14,286
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The Time Value of Money
A bond currently valued at $100,000 has a quarterly interest rate
of 5%. The bond matures in 3 years. What is its future value?
A) $1,157,625
B) $1,219,391
C) $1,160,755
D) $1,050,945
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The Time Value of Money
A bond currently valued at $100,000 has a quarterly interest rate
of 5%. The bond matures in 3 years. What is its future value?
A) $1,157,625
B) $1,219,391
C) $1,160,755
D) $1,050,945
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The Time Value of Money
Which of the following is a definition for the term "real interest
rate"?
A) The rate of return that capital could earn in an alternative investment
of equivalent risk.
B) The amount of interest actually accrued in a given period.
C) All of these answers.
D) The interest rate accrued after accounting for inflation.
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The Time Value of Money
Which of the following is a definition for the term "real interest
rate"?
A) The rate of return that capital could earn in an alternative investment
of equivalent risk.
B) The amount of interest actually accrued in a given period.
C) All of these answers.
D) The interest rate accrued after accounting for inflation.
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The Time Value of Money
Which of the following is a definition for cost of capital?
A) The rate of return that capital could earn in an alternative investment
of equivalent risk.
B) The interest rate accrued after accounting for inflation.
C) The amount of interest actually accrued in a given period.
D) All of these answers.
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The Time Value of Money
Which of the following is a definition for cost of capital?
A) The rate of return that capital could earn in an alternative investment
of equivalent risk.
B) The interest rate accrued after accounting for inflation.
C) The amount of interest actually accrued in a given period.
D) All of these answers.
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The Time Value of Money
You purchase a three year certificate of deposit (CD) for $100,000
on January 1st, 2000. This CD has an annual interest rate of 5%.
The interest compounds continuously. What is the balance for the
CD account on July 1, 2001?
A) $107,593
B) $105,000
C) $107,500
D) $110,250
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The Time Value of Money
You purchase a three year certificate of deposit (CD) for $100,000
on January 1st, 2000. This CD has an annual interest rate of 5%.
The interest compounds continuously. What is the balance for the
CD account on July 1, 2001?
A) $107,593
B) $105,000
C) $107,500
D) $110,250
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The Time Value of Money
Which of the following statements about the amortization of a loan
is true?
A) A loan is fully amortized once the amortization schedule is drafted.
B) Amortization is the process of paying off a debt over time through
regular payments.
C) 50% of each payment is for interest while the rest is applied to the
principal balance.
D) All of these answers.
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The Time Value of Money
Which of the following statements about the amortization of a loan
is true?
A) A loan is fully amortized once the amortization schedule is drafted.
B) Amortization is the process of paying off a debt over time through
regular payments.
C) 50% of each payment is for interest while the rest is applied to the
principal balance.
D) All of these answers.
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The Time Value of Money
Which of the following correctly defines a method of determining a
single period investment's yield?
A) The Effective Annual rate is the interest rate multiplied by the number
of payment periods per year.
B) Annual Percentage Rate = (1+(i/N))^N - 1.
C) Change-in-value equals the investment's FV minus its PV. Divide that
by PV and multiply by 100%.
D) All of these answers.
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The Time Value of Money
Which of the following correctly defines a method of determining a
single period investment's yield?
A) The Effective Annual rate is the interest rate multiplied by the number
of payment periods per year.
B) Annual Percentage Rate = (1+(i/N))^N - 1.
C) Change-in-value equals the investment's FV minus its PV. Divide that
by PV and multiply by 100%.
D) All of these answers.
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The Time Value of Money
You purchase a two year annuity for $2800. The annuity pays
$1500 each year. What is the annuity's approximate IRR?
A) 8.6%
B) 4.5%
C) 2.3%
D) 10%
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The Time Value of Money
You purchase a two year annuity for $2800. The annuity pays
$1500 each year. What is the annuity's approximate IRR?
A) 8.6%
B) 4.5%
C) 2.3%
D) 10%
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The Time Value of Money
Attribution
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• Wikipedia. "Cost of capital." CC BY-SA 3.0 http://en.wikipedia.org/wiki/Cost_of_capital
• Wikipedia. "Discounting." CC BY-SA 3.0 http://en.wikipedia.org/wiki/Discounting
• Wikipedia. "Future value." CC BY-SA 3.0 http://en.wikipedia.org/wiki/Future_value
• Boundless Learning. "Boundless." CC BY-SA 3.0 http://www.boundless.com//finance/definition/multi-period-investment
• Boundless Learning. "Boundless." CC BY-SA 3.0 http://www.boundless.com//finance/definition/single-period-investment
• Boundless Learning. "Boundless." CC BY-SA 3.0 http://www.boundless.com//finance/definition/periods-t-or-n
• Connexions. "Mathematics of Finance." CC BY 3.0 http://cnx.org/content/m18905/latest/
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• Wikipedia. "Future value." CC BY-SA 3.0 http://en.wikipedia.org/wiki/Future_value
• Wikipedia. "Future value." CC BY-SA 3.0 http://en.wikipedia.org/wiki/Future_value
• Wiktionary. "compound interest." CC BY-SA 3.0 http://en.wiktionary.org/wiki/compound+interest
• Wikipedia. "Present value." CC BY-SA 3.0 http://en.wikipedia.org/wiki/Present_value
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The Time Value of Money
• Wiktionary. "simple interest." CC BY-SA 3.0 http://en.wiktionary.org/wiki/simple+interest
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• Wikipedia. "Future value." CC BY-SA 3.0 http://en.wikipedia.org/wiki/Future_value
• Wiktionary. "accrue." CC BY-SA 3.0 http://en.wiktionary.org/wiki/accrue
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The Time Value of Money
• Wiktionary. "principal." CC BY-SA 3.0 http://en.wiktionary.org/wiki/principal
• Connexions. "Mathematics of Finance." CC BY 3.0 http://cnx.org/content/m18905/latest/
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http://en.wikibooks.org/wiki/Principles_of_Finance/Section_1/Chapter_3/Applications_of_Time_Value_of_Money/Annuities
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• Boundless Learning. "Boundless." CC BY-SA 3.0 http://www.boundless.com//finance/definition/nominal-interest
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The Time Value of Money
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• Wikipedia. "Yield (finance)." CC BY-SA 3.0 http://en.wikipedia.org/wiki/Yield_(finance)
• Wikipedia. "Future value." CC BY-SA 3.0 http://en.wikipedia.org/wiki/Future_value
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• Wikipedia. "Cash flow." CC BY-SA 3.0 http://en.wikipedia.org/wiki/Cash_flow
• Wiktionary. "annuity." CC BY-SA 3.0 http://en.wiktionary.org/wiki/annuity
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http://en.wikibooks.org/wiki/Principles_of_Finance/Section_1/Chapter_2/Time_Value_of_Money/Introduction
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• Boundless Learning. "Boundless." CC BY-SA 3.0 http://www.boundless.com//finance/definition/present-value-pv
• Wikipedia. "Annuity (finance theory)." CC BY-SA 3.0 http://en.wikipedia.org/wiki/Annuity_(finance_theory)
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• Wikipedia. "Internal rate of return." CC BY-SA 3.0 http://en.wikipedia.org/wiki/Internal_rate_of_return
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The Time Value of Money
• Wiktionary. "yield." CC BY-SA 3.0 http://en.wiktionary.org/wiki/yield
• Boundless Learning. "Boundless." CC BY-SA 3.0 http://www.boundless.com//finance/definition/net-present-value-npv
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• http://fonetika3.appspot.com/wiki/Time_value_of_money#Present_value_of_a_future_sum. CC BY-SA
http://fonetika3.appspot.com/wiki/Time_value_of_money#Present_value_of_a_future_sum
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• Wikipedia. "Future Value." CC BY-SA 3.0 http://en.wikipedia.org/wiki/Future%20Value
• Wikipedia. "Present value." CC BY-SA 3.0 http://en.wikipedia.org/wiki/Present_value
• Wikipedia. "Time value of money." CC BY-SA 3.0 http://en.wikipedia.org/wiki/Time_value_of_money
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The Time Value of Money
• Boundless Learning. "Boundless." CC BY-SA 3.0 http://www.boundless.com//finance/definition/discounting--2
• Wiktionary. "interest rate." CC BY-SA 3.0 http://en.wiktionary.org/wiki/interest+rate
• Connexions. "Mathematics of Finance." CC BY 3.0 http://cnx.org/content/m18905/latest/
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• Wikipedia. "period." CC BY-SA 3.0 http://en.wikipedia.org/wiki/period
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• Boundless Learning. "Boundless." CC BY-SA 3.0 http://www.boundless.com//finance/definition/ordinary-annuity--2
• Boundless Learning. "Boundless." CC BY-SA 3.0 http://www.boundless.com//finance/definition/annuity-due--2
• http://fonetika3.appspot.com/wiki/Time_value_of_money#Present_value_of_a_future_sum. CC BY-SA
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