5. THE TIME VALUE OF MONEY
A. Interpret interest rates as required rates of return, discount rates, or opportunity costs :
Interest rates are our measure of the time value of money, although risk differences in financial securities lead to differences in their equilibrium interest rates. Equilibrium interest rates are the required rate of return for a particular investment.
Interest rates are also referred to as discount rates and, in fact, the terms are often used interchangeably.
We can also view interest rates as the opportunity cost of current consumption.
B. Explain an interest rate as the sum of a real risk-free rate, and premiums that compensate investors for bearing distinct types of risk :
The real risk-free rate of interest is a theoretical rate on a single-period loan that has no expectation of inflation in it. U.S. Treasury bills (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. The approximate relation here is :
Nominal risk free rate = real risk free rate + expected inflation rate
Securities may have one or more types of risk :
- Default risk: The risk that a borrower will not make the promised payments in timely manner.
- Liquidity risk: The risk of receiving less than fair value for an investment if it must be sold for cash quickly.
- Maturity risk: Prices of longer-term bonds are more volatile than those of shorter-term bonds.
Required interest rate on a security = nominal risk-free rate + default risk premium + liquidity risk premium + maturity risk premium
C. Calculate and interpret the effective annual rate, given the stated annual interest rate and the frequency of compounding :
D. Solve time value of money problems for different frequencies of compounding :
Financial institutions usually quote rates as annual interest rates, along with a compounding frequency, as opposed to quoting rates as periodic rates. The rate of interest that investors actually realize as a result of compounding is known as the effective annual rate (EAR). EAR represents the annual rate of return actually being earned after adjustments have been made for different compounding periods.
EAR may be determined as follows :
where :
periodic rate = stated annual rate/m
m = the number of compounding periods per year
The greater the compounding frequency, the greater the EAR will be in comparison to the stated one.
E. Calculate and interpret the future value (FV) and present value (PV) of a single sum of money, an ordinary annuity, an annuity due, a perpetuity and a series of unequal cash flows :
Future Value of a Single Sum
Future value (FV) is the amount to which a current deposit will grow over time when it is placed in an account paying compound interest.
The formula for the FV of a single cash flow is :
where :
PV = amount of money invested today (the present value)
I/Y = rate of return per compounding period
N = total number of compounding periods
The factor (1 + I/Y)N represents the compounding rate on an investment and is frequently referred to as the future value factor or the future value interest factor, for a single cash flow at I/Y over N compounding periods.
Present Value of a Single Sum
The Present value (PV) of a single sum is today’s value of a cash flow that is to be received at some point in the future. In other words, 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 of finding the PV of a cash flow is known as discounting. The interest rate used in the discounting process is commonly referred to as the discount rate, the opportunity cost, required rate of return and the cost of capital.
Rewriting the FV equation in terms of PV, we get :
For a single future cash flow, PV is always less than the FV whenever the discount rate is positive.
The quantity 1 / (1 + I/Y)N in the PV equation is frequently referred to as the present value factor, present value interest factor, or discount factor for a single cash flow at I/Y over N compounding periods.
Annuities
An annuity is a stream of equal cash flows that occurs at equal intervals over a given period. There are two types of annuities : ordinary annuities and annuities due.
The ordinary annuity is the most common type of annuity. It is characterized by cash flow at the end of each compounding period.
The other type of annuity is called annuities due, where payments or receipts occur at the beginning of each period (the first payment is today at t = 0).
Future Value of an Annuity Due
Sometimes it is necessary to find the FV of an annuity due (FVAD). Fortunately, our financial calculators can be used to do this, but one slight modification : on the TI BA II, press [2nd] [BGN] [2nd] [SET].
Another way to compute the FV of an annuity is to calculate the FV of an ordinary annuity, and simply multiply the resulting FV by ( 1 + I/Y ). Symbolically : FVAD = FVAO * ( 1 + I/Y )
Present Value of an Annuity Due
There are two ways to compute the PV of an annuity due. The first is to put the calculator in the BGN mode and then input all the relevant variables. The second is to treat the cash flow stream as an ordinary annuity over N compounding periods, and simply multiply the resulting PV by (1 + I/Y). Symbolically : PVAD = PVAO * ( 1 + I/Y ).
Present Value of a Perpetuity
A perpetuity is a financial instrument that pays a fixed amount of money at set intervals over an infinite period of time. A perpetuity is a perpetual annuity. British consol bonds and most preferred stocks are perpetuities since they promise fixed interest or dividend payments forever.
PV perpetuity = PMT / (I/Y)
PV and FV of 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 a cash flow stream, all we need to do is sum the PVs and FVs of the individual cash flows.
It is also possible to compute PV of an uneven cash flow stream by using the cash flow (CF) keys and the net present value (NPV) function on your calculator. The Fn variable indicates how many times a particular cash flow amount is repeated.
Solving Time Value of Money Problems When Compounding Periods Are Other Than Annual
Since an increase in the frequency of compounding increases the effective rate of interest, it also increases the FV of a given cash flow and decreases the PV of a given cash flow.
There are two ways to use your financial calculator to compute PVs and FVs under different compounding frequencies :
F. Demonstrate the use of a time line in modeling and solving time value of money problems :
Loan Payments and Amortization
Loan amortization is 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. 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.
Funding a Future Obligation
There are many TVM applications where it is necessary to determine the size of the deposit(s) that must be made over a specified period in order to meet a future liability. Two common examples are (1) setting up a funding program for future college tuition and (2) the funding of a retirement program.
The Connection Between Present Values, Future Values, and Series of Cash Flows
The cash flow additivity principle refers to the fact that present value of any stream of cash flows equals the sum of the present values of the cash flows.
PV = amount of money invested today (the present value)
I/Y = rate of return per compounding period
N = total number of compounding periods
The factor (1 + I/Y)N represents the compounding rate on an investment and is frequently referred to as the future value factor or the future value interest factor, for a single cash flow at I/Y over N compounding periods.
Present Value of a Single Sum
The Present value (PV) of a single sum is today’s value of a cash flow that is to be received at some point in the future. In other words, 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 of finding the PV of a cash flow is known as discounting. The interest rate used in the discounting process is commonly referred to as the discount rate, the opportunity cost, required rate of return and the cost of capital.
Rewriting the FV equation in terms of PV, we get :
For a single future cash flow, PV is always less than the FV whenever the discount rate is positive.
The quantity 1 / (1 + I/Y)N in the PV equation is frequently referred to as the present value factor, present value interest factor, or discount factor for a single cash flow at I/Y over N compounding periods.
Annuities
An annuity is a stream of equal cash flows that occurs at equal intervals over a given period. There are two types of annuities : ordinary annuities and annuities due.
The ordinary annuity is the most common type of annuity. It is characterized by cash flow at the end of each compounding period.
The other type of annuity is called annuities due, where payments or receipts occur at the beginning of each period (the first payment is today at t = 0).
Future Value of an Annuity Due
Sometimes it is necessary to find the FV of an annuity due (FVAD). Fortunately, our financial calculators can be used to do this, but one slight modification : on the TI BA II, press [2nd] [BGN] [2nd] [SET].
Another way to compute the FV of an annuity is to calculate the FV of an ordinary annuity, and simply multiply the resulting FV by ( 1 + I/Y ). Symbolically : FVAD = FVAO * ( 1 + I/Y )
Present Value of an Annuity Due
There are two ways to compute the PV of an annuity due. The first is to put the calculator in the BGN mode and then input all the relevant variables. The second is to treat the cash flow stream as an ordinary annuity over N compounding periods, and simply multiply the resulting PV by (1 + I/Y). Symbolically : PVAD = PVAO * ( 1 + I/Y ).
Present Value of a Perpetuity
A perpetuity is a financial instrument that pays a fixed amount of money at set intervals over an infinite period of time. A perpetuity is a perpetual annuity. British consol bonds and most preferred stocks are perpetuities since they promise fixed interest or dividend payments forever.
PV perpetuity = PMT / (I/Y)
PV and FV of 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 a cash flow stream, all we need to do is sum the PVs and FVs of the individual cash flows.
It is also possible to compute PV of an uneven cash flow stream by using the cash flow (CF) keys and the net present value (NPV) function on your calculator. The Fn variable indicates how many times a particular cash flow amount is repeated.
Solving Time Value of Money Problems When Compounding Periods Are Other Than Annual
Since an increase in the frequency of compounding increases the effective rate of interest, it also increases the FV of a given cash flow and decreases the PV of a given cash flow.
There are two ways to use your financial calculator to compute PVs and FVs under different compounding frequencies :
- Adjust the number of periods per year (P/Y) mode on the calculator to correspond to the compounding frequency.
- Keep the calculator in the annual compounding mode (P/Y = 1) and enter I/Y as the interest rate per compounding period, and N as the number of compounding periods.
F. Demonstrate the use of a time line in modeling and solving time value of money problems :
Loan Payments and Amortization
Loan amortization is 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. 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.
Funding a Future Obligation
There are many TVM applications where it is necessary to determine the size of the deposit(s) that must be made over a specified period in order to meet a future liability. Two common examples are (1) setting up a funding program for future college tuition and (2) the funding of a retirement program.
The Connection Between Present Values, Future Values, and Series of Cash Flows
The cash flow additivity principle refers to the fact that present value of any stream of cash flows equals the sum of the present values of the cash flows.
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