Time preference can be measured by auctioning off a risk free security—like a US Treasury bill.
This is because money can be put in a bank account or any other (safe) investment that will return interest in the future.
Most actuarial calculations use the risk-free interest rate which corresponds to the minimum guaranteed rate provided by a bank's saving account for example, assuming no risk of default by the bank to return the money to the account holder on time.
The reverse operation—evaluating the present value of a future amount of money—is called a discounting (how much will $100 received in 5 years—at a lottery for example—be worth today?).
This is because if $100 is deposited in a savings account, the value will be $105 after one year, again assuming no risk of losing the initial amount through bank default.
Interest is the additional amount of money gained between the beginning and the end of a time period.
In this case, the bank is the borrower of the funds and is responsible for crediting interest to the account holder.
Similarly, when an individual invests in a company (through corporate bonds, or through stock), the company is borrowing funds, and must pay interest to the individual (in the form of coupon payments, dividends, or stock price appreciation).
[1] The interest rate is the change, expressed as a percentage, in the amount of money during one compounding period.
A compounding period is the length of time that must transpire before interest is credited, or added to the total.
There are several types and terms associated with interest rates: The operation of evaluating a present sum of money some time in the future is called a capitalization (how much will 100 today be worth in five years?).
The reverse operation—evaluating the present value of a future amount of money—is called discounting (how much will 100 received in five years be worth today?).
For example, if you are to receive $1000 in five years, and the effective annual interest rate during this period is 10% (or 0.10), then the present value of this amount is The interpretation is that for an effective annual interest rate of 10%, an individual would be indifferent to receiving $1000 in five years, or $620.92 today.
A cash flow is an amount of money that is either paid out or received, differentiated by a negative or positive sign, at the end of a period.
, of a stream of cash flows consists of discounting each cash flow to the present, using the present value factor and the appropriate number of compounding periods, and combining these values.
Many financial arrangements (including bonds, other loans, leases, salaries, membership dues, annuities including annuity-immediate and annuity-due, straight-line depreciation charges) stipulate structured payment schedules; payments of the same amount at regular time intervals.
: The present value of an annuity immediate is the value at time 0 of the stream of cash flows: where: The above formula (1) for annuity immediate calculations offers little insight for the average user and requires the use of some form of computing machinery.
A perpetuity refers to periodic payments, receivable indefinitely, although few such instruments exist.
The present value of a perpetuity can be calculated by taking the limit of the above formula as n approaches infinity.
: A corporation issues a bond, an interest earning debt security, to an investor to raise funds.
[2] The purchase price can be computed as: The purchase price is equal to the bond's face value if the coupon rate is equal to the current interest rate of the market, and in this case, the bond is said to be sold 'at par'.
Finally, if the coupon rate is greater than the market interest rate, the purchase price will be greater than the bond's face value, and the bond is said to have been sold 'at a premium', or above par.
These calculations must be applied carefully, as there are underlying assumptions: (In fact, the present value of a cashflow at a constant interest rate is mathematically one point in the Laplace transform of that cashflow, evaluated with the transform variable (usually denoted "s") equal to the interest rate.
The full Laplace transform is the curve of all present values, plotted as a function of interest rate.
For discrete time, where payments are separated by large time periods, the transform reduces to a sum, but when payments are ongoing on an almost continual basis, the mathematics of continuous functions can be used as an approximation.)
Whenever there will be uncertainties in both timing and amount of the cash flows, the expected present value approach will often be the appropriate technique.
With Present Value under uncertainty, future dividends are replaced by their conditional expectation.
The project claims to return the initial outlay, as well as some surplus (for example, interest, or future cash flows).
[2] The traditional method of valuing future income streams as a present capital sum is to multiply the average expected annual cash-flow by a multiple, known as "years' purchase".
This was the method used for example by the English crown in setting re-sale prices for manors seized at the Dissolution of the Monasteries in the early 16th century.