Interest Rates


Remember in chapter 4, we discussed how to compute the present and future values with a given market interest rate. In this chapter, we take a deeper look into how interest rates are determined and what affects them.

Interest Rate Quotes and Adjuestments

We are bombarded with interest rates daily. Banks use them on loans and savings accounts. Newspapers are used to get rates out to the public about savings and auto loan rates. It is imporant to understand that interest rates are the price of money. This price is how much you pay to get money now or how much you will be paid to get money in the future. These rates are set by market forces; when supply (savings) is high, rates are low. On the other hand, when demand is high (loans) interest rates are high. Interest rates can be quoted in many different terms: annually, quarterly, monthly, and daily are the most common. As will be shown later, it is necessary to adjust the interest rate to a time period that matches the cash flow.

Effective Annual Rate

This EAR is what interest rates are often quoted in, also called annual percentage yield (APY). This rate is that which will be earned at the end of one year. Chapter 4 is where we first saw EAR as the discount rate in the time value of money. Remember, that the interest rate is added to one and then multiplied by the principle to find the future value.

Different Time Periods

In simple terms, to find an equivalent interest rate for different periods of time, take (1 + r) ^n -1. n is the appropriate power that reflects the number of compoundings per year. For example, if interest rates are quarterly, there are 4 compounds in that year; so n = (4/12). If rates are semiannual, n = (6/12). n can be larger than 1 also, this would give rates that are longer than 1 period.
Remember: when computing present or future values, adjust the discount rate to match that of the time period of cash flows.

Annual Percentage Rates

Although EAR is a way to quote interest rates, the most common way they are quoted is in annual percentage rates (APR). This is the simple interest earned in one year; interest earned without the effect of compounding. This means that APR are typically less than the actual amount of interest that will be earned. Because APR doesn't reflect the true amount of earnings in a year, it CANNOT be used as a discount rate. APR is a way of quoting the actualy interest earned each compounding period.

Interest rate per compounding period = APR/m (m = number of compounding periods per year.)

Once the rate per compounding period is determined, interest rates for other time intervals can be calculated. This involves taking the APR and converting it to an EAR. This is done by the following equation:

1 + EAR = ((1+APR)/m)^m

Using this equation, we can determine the present value of an investment opportunity.

Determinants of Interest Rates

Fundamentally, interest rates are determined by market forces based on supply and demand for the funds. The supply and demand is determined by the populations, banks, and firm's willingness to borrow, save, and lend funds. Inflation is another important factor in the determination of the interest rates.

Inflation measures how purchasing power of a given sum of money declines due to increasing prices. This rate affects how one evaluates the interest rates being quoted by banks.

Nominal interest rates are also used in determining the real interest rate. Nominal rates indicate the rate at which money will grow if invested for a given period of time. If inflation is increasing, the the nominal rate no longer represents the true increase in purchasing power.

The real interest rate is the growth of purchasing power after adjusting for inflation. This rate gives a more accurate picture of the power of moeny. The real interst rate is determiend by:

(nominal rate - inflation rate) / (1+ inflation rate)

The Yeild Curve

Interest rates that bank offer depend of the horizon, or term, of the investmetn/loan. The term of an investment is the time until maturity. The relationship between the investment term and the interest rate is known as the term structure. This term structure relationship can be plotted on a yield curve with the term on the x-axis and the interest rate on the y-axis. Some yield curves plot this information for the U.S. Treasury securities. Because the government will never default on its loans, they are considered to be risk-free. A risk-free interest rate is therefore the rate money can be borrowed or lent without risk over a given period. Because a security is risk-free, the present value of its cash flow can be determined, whether they have the same maturity date or different maturity dates. This uses the following equations:

PV(same maturity date) = Cn / (1 + rn)^n
PV(different maturity date) = C1/(1 + r1) + C2 / (1 +r2)^2 ... + Cn / (1 + rn)^n


The shape of the yield curve is strongly influenced by interest rate expectation. A steep curve tends to indicate that interest rates are expected to rise in the future, while an inverted curve generally signals an expected decline in future interest rates. An inverted curve tends to be interpreted as a negetive forecast for economic growth; each of the last 6 recessions were preveded by an inverted curve.

Applications

The discount rates can be used to compute loan payments or find the opportunity cost of capital.

Many loans have monthly payments that must be made, causing these types of loans to be called amortizing loans. These monthly payments include interest plus some part of the principle. The more monthly payments taht are made, the more the payments go towards paying off the principle. In other words, the closer the maturity date, the lower the interest payments and the higher the principle payments. The amount of a loan paymetn can be determined by:

C = P / ((1/r) * (1 - ((1 + r)^n))

Chapter 3 argued that the market interest rate should be used to compute present values. But this rate is too ambiguous for real application, so the opportunity cost of capital will be used form here forward. The opportunity cost of capital is the best available expected return offered int eh market on an investement of camparable risk and term to the cash flow being discounted. This is the return the investor forgoes when the investor takes on a new investemtn. It is trying to determine which investment alternative will be better for the investor.