FINC 331-WEEK 3: Time Value of Money
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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
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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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Defining the Time Value of Money
Importance of the Time Value of Money
Introduction to the Time Value of Money
The Time Value of Money > Introduction to the Time Value of Money
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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%.
Defining the Time Value of Money
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Simple Interest Formula
The Time Value of Money > Introduction to 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.
Importance of the Time Value of Money
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Compound Interest
The Time Value of Money > Introduction to the Time Value of Money
Single-Period Investment
Multi-Period Investment
Calculating Future Value
Approaches to Calculating Future Value
Future Value, Single Amount
The Time Value of Money > Future Value, Single Amount
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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.
Single-Period Investment
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The Time Value of Money > Future Value, Single Amount
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.
Multi-Period Investment
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Simple Interest Formula
The Time Value of Money > Future Value, Single Amount
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.
Calculating Future Value
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Compound Interest
The Time Value of Money > Future Value, Single Amount
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.
Approaches to Calculating Future Value
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Compound Interest
The Time Value of Money > Future Value, Single Amount
Single-Period Investment
Multi-Period Investment
The Discount Rate
Number of Periods
Calculating Present Value
Present Value, Single Amount
The Time Value of Money > Present Value, Single Amount
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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.
Single-Period Investment
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FV of a single payment
The Time Value of Money > Present Value, Single Amount
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).
Multi-Period Investment
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The Time Value of Money > Present Value, Single Amount
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.
The Discount Rate
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Borrowing and lending
The Time Value of Money > Present Value, Single Amount
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.
Number of Periods
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FV of a single payment
The Time Value of Money > Present Value, Single Amount
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").
Calculating Present Value
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Simple Interest Formula
The Time Value of Money > Present Value, Single Amount
Annuities
Future Value of Annuity
Present Value of Annuity
Calculating Annuities
Annuities
The Time Value of Money > Annuities
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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.
Annuities
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The Time Value of Money > Annuities
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.
Future Value of Annuity
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The Time Value of Money > Annuities
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.
Present Value of Annuity
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The Time Value of Money > 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.
Calculating Annuities
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PV of a Perpetuity
The Time Value of Money > Annuities
Future Value, Multiple Flows
Present Value, Multiple Flows
Valuing Multiple Cash Flows
The Time Value of Money > Valuing Multiple Cash Flows
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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.
Future Value, Multiple Flows
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FV of a single payment
The Time Value of Money > Valuing Multiple Cash 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.
Present Value, Multiple Flows
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Sum FV
The Time Value of Money > Valuing Multiple Cash Flows
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
Additional Detail on Present and Future Values
The Time Value of Money > Additional Detail on Present and Future Values
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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.
The Relationship Between Present and Future Value
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FV of a single payment
The Time Value of Money > Additional Detail on Present and Future Values
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.
Calculating Perpetuities
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The Time Value of Money > Additional Detail on Present and Future Values
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.
Calculating Values for Different Durations of Compounding Periods
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EAR with Continuous Compounding
The Time Value of Money > Additional Detail on Present and Future Values
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.
Comparing Interest Rates
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Fisher Equation
The Time Value of Money > Additional Detail on Present and Future Values
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.
Calculating Values for Fractional Time Periods
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Compounding Interest
The Time Value of Money > Additional Detail on Present and Future Values
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.
Loans and Loan Amortization
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Amortization Schedule
The Time Value of Money > Additional Detail on Present and Future Values
Calculating the Yield of a Single-Period Investment
Calculating the Yield of an Annuity
Yield
The Time Value of Money > Yield
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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.
Calculating the Yield of a Single-Period Investment
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Percent Change
The Time Value of Money > Yield
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.
Calculating the Yield of an Annuity
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IRR Example
The Time Value of Money > Yield
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Appendix
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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Wikipedia. "Future value." GNU FDL http://en.wikipedia.org/wiki/Future_value View on Boundless.com
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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Wikipedia. "Future value." GNU FDL http://en.wikipedia.org/wiki/Future_value View on Boundless.com
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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Wikimedia. CC BY-SA http://upload.wikimedia.org/wikipedia/commons/thumb/7/7f/Compound_Interest_with_Varying_Frequencies.svg/800px-Compound_Interest_with_Varying_Frequencies.svg.png View on Boundless.com
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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Wikipedia. "Future value." GNU FDL http://en.wikipedia.org/wiki/Future_value View on Boundless.com
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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Wikipedia. "Future value." GNU FDL http://en.wikipedia.org/wiki/Future_value View on Boundless.com
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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Wikipedia. "Future value." GNU FDL http://en.wikipedia.org/wiki/Future_value View on Boundless.com
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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Wikipedia. "Future value." GNU FDL http://en.wikipedia.org/wiki/Future_value View on Boundless.com
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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Wikipedia. "Annual percentage yield." GNU FDL http://en.wikipedia.org/wiki/Annual_percentage_yield View on Boundless.com
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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Math Major. CC BY-SA http://mathmajor.org/accounting-finance-and-investing/annuities-and-perpetuities/ View on Boundless.com
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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Wikipedia. "Compound interest." GNU FDL http://en.wikipedia.org/wiki/Compound_interest View on Boundless.com
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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Wikipedia. "Future value." GNU FDL http://en.wikipedia.org/wiki/Future_value View on Boundless.com
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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Wikipedia. "Compound interest." GNU FDL http://en.wikipedia.org/wiki/Compound_interest View on Boundless.com
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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