The time value of money (TVM) refers to the fact that $1 today is worth more than $1 in the future. This is because the $1 today can be invested to earn interest immediately. The TVM reflects the relationship between present value, future value, time, and interest rate. The time value of money underlies rates of return, interest rates, required rates of return, discount rates, opportunity costs, inflation, and risk. It reflects the relationship between time, cash flow, and interest rate.

There are three ways to interpret interest rates:

• Required rate of return is the return required by investors or lenders to postpone their current consumption.
• Discount rate is the rate used to discount future cash flows to allow for the time value of money (that is, to bring a future value equivalent to present value).
• Opportunity cost is the most valuable alternative investors give up when they choose what to do with money.

In a certain world, the interest rate is called the risk-free rate. For investors preferring current to future consumption, the risk-free interest rate is the rate of compensation required to postpone current consumption. For example, the interest rate paid by T-bills is a risk-free rate of interest.

In an uncertain world, there are a few factors that complicate interest rates:

• Inflation Premium (I): When prices are expected to increase, lenders charge not only an opportunity cost for postponing consumption but also an inflation premium that takes into account the expected increase in prices. The nominal cost of money consists of the real rate (a pure rate of interest) and an inflation premium.

• Default Risk Premium: Companies exhibit varying degrees of uncertainty concerning their ability to repay lenders. Lenders therefore charge interest rates that incorporate default risk (D).

• Liquidity Premium (L) compensates the investor for the extra cost of converting an investment to cash relatively quickly. Government securities typically carry little-to-no liquidity premium because they are traded in active markets with many willing buyers and sellers. Yields on corporate debt, particularly securities issued by smaller firms, reflect a liquidity premium because investors are more likely to be forced to accept a discount to market value when selling their positions.

• Maturity Premium (M) is earned by investors who are willing to lend over longer periods, which requires greater exposure to interest rate risk. Even longer-dated government bonds that are free of default risk will have yields that reflect a maturity premium relative to the yields on short-term government securities.

r = r_f + I + D + L + M

The nominal risk-free rate reflects the real risk-free rate and the premium for expected inflation.

(1 + Nominal risk-free rate) = (1 + Real risk-free rate)(1 + Inflation premium)

In practice: Nominal risk-free rate = Real risk-free rate + Inflation premium

Compounding is the process of accumulating interest over a period of time. A compounding period is the number of times per year that interest is paid. Continuous compounding occurs when the number of compounding periods becomes infinite; interest is added continuously.

Discounting is the calculation of the present value of some known future value. Discount rate is the rate used to calculate the present value of some future cash flow. Discounted cash flow is the present value of some future cash flow.