Finance - IX
Time Value of Money
9 Jan 2025
Simple interest
The time value of money is a principle which states that money received in the future is worth less than money received today. This is because money received today can be invested to earn interest. For example, $100 today is worth more than $100 a year from now. Suppose that we can earn 5% interest on our money. Then, $100 today will be worth $105 a year from now, which is more than $100 a year from now. The time value of money is a fundamental concept in finance and is used to make investment decisions.
The present value of a savings/investment is its amount or value today. The future value of a savings/investment is its amount or value at some future date. Since money has a time value:
Future value = Present value * (1 + Interest rate * n)
Typically the interest rate is expressed relative to a period of time, such as 5% per year, or 1% per month. Simple interest is the interest paid on the principal (i.e. the initial investment) alone.
Compound interest
Compount interest is interest paid on both the principal and the interest that has been added to the principal:
Future value = Present value * (1 + Interest rate)n
The difference between simple and compound interest can be significant over long periods of time (for non near-zero interest rates).
So far we haven't considered inflation, which can erode the purchasing power of money over time. For example, with zero inflation, $1 growing at a 10% interest rate for 10 years will be worth $2.59. However, with 5% inflation, the same investment will be worth only $1.63 in today's money.
Discounting
Discounting is the process of finding the present value of a future sum of money. It is the inverse of compounding.
Annuity
An annuity is a series of equal payments made at regular intervals. An ordinary annuity is one in which payments are made at the end of each period, while an annuity due is one in which payments are made at the beginning of each period.
With annuities, interest is paid only on payments that have been made. For example, suppose we have an ordinary annuity of $1000 per year for 3 years at an interest rate of 8%. The future value of the annuity at the end of the second year is
$1000 * (1 + 0.08) + $1000 = $2180
Notice that interest is paid on the $1000 payment made at the end of the first year, but not on the $1000 payment made at the end of the second year.
Amortization
An amortized loan is a loan that is repaid in equal installments. Amortized loans are preferable because they are easier for the borrower to track and budget for, since the payments are the same each period.
Annual percentage rate and effective annual rate
The annual percentage rate (APR) is the annual rate that is charged for borrowing or earned through an investment. APR is the product of the periodic interest rate and the number of periods in a year. Various laws require that the APR be disclosed to the borrower.
APR, however, does not reflect the true cost of borrowing or the true return on an investment, because it does not take into account the effect of compounding. For example, a loan with an APR of 10% may actually have an effective annual rate of 10.47%. In other words, the borrower is actually paying 10.47% interest on the loan, not 10%. The effective annual rate (EAR) is the annual rate that is actually earned or paid after compounding.