The Time Value of Money

Relationship Between PV and FV

PV= FV x [1/(1+ I/Y)^N]= FV/(1+I/Y)^N

PMT

Payment Variable on a financial calculator. The PMT Variable defines the size of periodic cash flows for an annuity and perpetuity.

Principal Component of Loan

Payment amount - Interest Component for that particular period.

Cash Flow Additivity Principle

Refers to the fact that the present value of any stream of cash flows equals the sum of the present values of the cash flows. If we have two series of cash flows, the sum of the present values of the two series is the same as the present values of the two series taken together, adding cash flows that will be paid at the same point in time. You can divide up a series of cash flows any way you like, and the present value of the "pieces" will equal the present value of the original series

Effective Annual Rate (EAR)

Represents the annual rate of return actually being earned after adjustments have been made for different compounding periods. ** A bank will quote a savings rate of 8% compounded quarterly, rather than 2% per quarter.

Required Interest Rate on a Security

Required interest rate on a security= Nominal risk-free rate + default risk premium+ liquidity premium + maturity risk premium

Annuity Due

Second type of annuity where payments or receipts occur at the beginning of each period (the first payment is today at t=0).

Risk

Securities may have one or more "types of risk", and each added risk increases the required rate of return on the security.

Equilibrium Interest rates

The "Required rate of Return" for a particular investment, in the sense that the market rate of return is the return that investors and savers require to get them to willingly lend their funds. Interest rates are also referred to as "discount rates". ** If an individual can borrow funds at an interest rate of 10%, then that individual should "discount" payments to be made in the future at that rate in order to get their equivalent value in current dollars or other currency.

Key Chapter Concept II

The Real Risk-Free Rate is a theoretical rate on a single-period loan when there is no expectation of inflation. Securities may have several risks, and each increases the required rate of return. These include default risk, liquidity risk, and maturity risk. The required rate of return on a security= real risk-free rate + expected inflation + default risk premium + liquidity risk premium + maturity risk premium.

Present Value (PV)

The computation of an investment's PV works in the opposite direction of computing FV--it brings the cash flows from an investment back to the beginning of the investment's life based on an appropriate compound rate of return.

Key Chapter Concept III

The effective annual rate when there are "m" compounding periods = [1 + (stated annual rate/m)]^m-1, Each dollar invested will grow to [1 + (stated annual rate/m)]^m-1, in one year.

Future Vale Factor/ Future Value Interest Factor

The factor (1 + I/Y)^N, which represents the compounding rate on an investment for a single cash flow at I/Y over N compounding periods

PV of a Perpetuity

The fixed periodic cash flow divided by the appropriate periodic rate of return. PV of a Perpetuity= PMT/I/Y

Future Value (FV)

The future value of an investment's cash flows as a result of compound interest. Computing FV involves projecting the the cash flows forward, on the basis of an appropriate compound interest rate, to the end of the investment's life.

Continuous Compounding

The limit of shorter and shorter compounding periods. To convert an annual stated rate to the EAR with continuous compounding, we use the formula: e^r-1= EAR

Ordinary Annuity

The most common type of annuity. It is characterized by cash flows that occur at the END of each compounding period.

Ending Balance in a Given Period, t

The period's beginning balance - the Principal component of the payment, where the beginning balance for period t is the ending balance from period t-1.

Loan Amortization

The process of paying off a loan with a series of periodic loan payments, whereby a portion of the outstanding loan amount is paid off, or amortized, with each payment. When a company or individual enters into a long-term loan, the debt is usually paid off over time with a series of equal, periodic loan payments, and each payment includes the repayment of principal and an interest charge. The payments may be made monthly, quarterly, or even annually. Regardless of the payment frequency, the size of the payment remains fixed over the life of the loan. The amount of the principal and interest component of the loan payment, however, does not remain fixed over the term of the loan.

Liquidity Risk

The risk of receiving less than fair value for an investment if it must be sold for cash quickly.

Default Risk

The risk that a borrower will not make the promised payments in a timely manner.

Present Value Factor/Present Value Interest Factor/Discount Factor

1/(1+I/Y)^N

Time Lines

A diagram of the cash flows associated with a TVM problem. A cash flow that occurs in the present (today) is put at time zero. Cash outflows (payments) are given a negative sign, and cash inflows (receipts) are given a positive sign. Once the cash flows are assigned to a time line, they may be moved to the beginning of the investment period to calculate the PV through a process called "discounting" or to the end of the period to calculate the FV using a process called "compounding".

Perpetuity

A financial instrument that pays a fixed amount of money at set intervals over an infinite period of time. A perpetuity is essentially a perpetual annuity. **British consol bonds and most preferred stocks are examples of perpetuities since they promise fixed interest or dividend payments forever.

Real Risk-Free Rate of Interest

A theoretical rate on a single-period loan that has no expectation of inflation in it. When we speak of a "real rate of return", we are referring to an investor's increase in purchasing power (after adjusting for inflation). Since expected inflation in future periods is not zero, the rates we observe on T-bills , for example, are "risk-free rates" but not "real" rates of return. T-bill rates are "nominal risk-free rates" because they contain an "inflation premium". ** Nominal Risk-Free Rate= Real Risk-Free Rate + Expected Inflation Rate

Risk Premium

Added to the "nominal risk-free rate" to adjust for greater default risk, less liquidity, and longer maturity.

Key Chapter Concept VI

An annuity is a series of equal cash flows that occurs at evenly spaced intervals over time. "Ordinary Annuity" cash flows occur at the end of each time period. "Annuity Due" cash flows occur at the beginning of each time period. Perpetuities are annuities with infinite lives: PV of a perpetuity= PMT/I/Y The present (future) value of any series of cash flows is equal to the sum of the present (future) values of the individual cash flows.

Annuities

An annuity is a stream of "equal cash flows" that occurs at "equal intervals" over a given period. ** Receiving $1,000 per year at the end of each year for the next eight years is an example of an annuity.

Key Chapter Concept I

An interest rate can be interpreted as the rate of return required in equilibrium for a particular investment, the discount rate for calculating the present value of future cash flows, or as the opportunity cost of consuming now, rather than saving and investing.

Interest Component of Loan

Beginning Balance of the period x periodic interest rate

Calculating EAR

EAR= (1+ periodic rate)^m -1 where: Periodic Rate= stated annual rate/m m= the number of compounding periods per year ** Whenever compound interest is being used, the stated rate and the actual (effective) rate of interest are equal only when interest is compounded annually. Otherwise, the greater the compounding frequency, the greater the EAR will be in comparison to the stated rate. ** The computation of EAR is necessary when comparing investments that have different compounding periods. It allows for an "apples-to-apples" rate comparison.

Future Value of a Single Sum (Cash Flow)

FV= PV (1 + I/Y)^N, where: PV= Amount of money invested today I/Y= Rate of return per compounding period N= Total number of compounding periods

Key Chapter Concept V

FV= PV(1+I/Y)^N PV= FV/(1+I/Y)^N

PV of a Deferred Perpetuity

First, find the PV of the perpetuity. Second, discount that PV for the number of periods the perpetuity is deferred.

Key Chapter Concept IV

For non-annual time value of money problems, divide the stated annual interest rate by the number of compounding periods per year, "m", and multiply the number of years by the number of compounding periods per year.

Rate of Return for a Perpetuity

I/Y= PMT/ PV of the perpetuity

PV and FV of an Uneven Cash Flow Series

In essence, a series of uneven cash flows is nothing more than a stream of annual single sum cash flows. Thus, to find the PV or FV of an uneven cash flow stream, all you need to do is find the sum of the PV's or FV's of the individual cash flows. **When calculating and finding the sum of these individual cash flow, you MUST preserve the +/- signs of each cash flow.

Opportunity Cost

Interest rates can also be viewed as the "opportunity cost" of current consumption . ** If the current market rate of interest on 1-year securities is 5%, earning an additional 5% is the opportunity forgone when current consumption is chosen rather than saving (postponing consumption).

Maturity Risk

Longer maturity bonds have more maturity risk than shorter-term bonds and require a maturity risk premium because the prices of longer-term bonds are more volatile than shorter-term bonds.

Present Value of a Single Sum

Today's value of a cash flow that is to be received at some point in the future. It is the amount of money that must be invested today, at a given rate of return over a given period of time, in order to end up with a specified FV. The process for finding the PV of a cash flow is called "discounting". ** The interest rate used in the discounting process is referred to as the "discount rate" but may also be referred to as the "opportunity cost", "required rate of return", and the "cost of capital". This rate represents the annual compound rate of return that can be earned on an investment.

Compound Interest

When an investment is subject to compound interest, the growth in the value of the investment from period to period reflects not only the interest earned on the original principal amount but also on the interest earned on the previous period's interest earnings--the interest on interest.