Nominal and Real Interest Rates
Interest is the amount of money charged for the borrowing of money to compensate the lender for the lost opportunity cost of not having the money and to compensate the lender for the various risks of lending money, such as inflation risk and credit default risk. Indeed, the word risk is derived from the Tuscan word rischio, which the Tuscans considered the amount necessary to compensate for the lending of money. Usually, the amount of interest charged is expressed as an interest rate, a percentage of the principal or balance per unit of time.
The concept of interest has a long history. Aristotle thought interest was evil, and according to the Koran, God condemned the charging of interest. The earliest known examples of interest were in ancient Mesopotamia, beginning in the 3^{rd} millennium B.C., when an interest rate of either 20% or 33% was charged depending on whether the loan was paid in silver or barley. However, the interest rate did not depend on the amount of time. [No doubt this simplified calculations that required using a sexagesimal (base 60) numbering system and pressing wedgeshaped (cuneiform) styles into wet clay tablets.]
Interest and Interest Rates
Interest rates are the rate of growth of money per unit of time. It is a primary factor in investments, since so many financial assets depend on its value. It is used to determine the present and future value of money and of annuities. Many securities either pay interest or the payoff depends on the interest rate. Whether a business will invest in capital or issue securities depends on the interest rate. Hence, the interest rate allocates economic resources more efficiently. Governments control their economies by adjusting key interest rates through monetary and fiscal policies.
Interest is the cost of money, in the form of a loan, and like the price of virtually everything else, it is determined by supply and demand. In the United States and most other developed countries, the government has a major influence on the interest rate, adjusting it higher to cool the economy and adjusting it lower to stimulate it. The government can also increase the money supply by printing money, or through other monetary and fiscal policies. Another source of supply is the savings of people, businesses, and other organizations. The main demand for money is for loans by people and businesses. Demand can also be affected by the monetary policies of the government.
Charging interest on a loan is sometimes called usury, although in more recent times, it has acquired a negative connotation of excessively high or illegal interest rates being charged. In fact, when the usury rate is limited by law, the rate is called a usury ceiling. However, at least 2 states in the United States do not have usury limits: Delaware and South Dakota, which is why many credit card issuers are located in those states.
Nominal and Real Interest Rates
The nominal interest rate is the stated interest rate. If a bank pays 5% annually on a savings account, then 5% is the nominal interest rate. So if you deposit $100 for 1 year, you will receive $5 in interest. However, that $5 will probably be worth less at the end of the year than it would have been at the beginning. This is because inflation lowers the value of money. As goods, services, and assets, such as real estate, rise in price, it takes more money to buy them.
The real interest rate is the nominal rate of interest minus inflation, which can be expressed approximately by the following formula:
Real Interest Rate = Nominal Interest Rate − Inflation Rate = Growth of Purchasing Power
For low rates of inflation, the above equation is fairly accurate. However, the actual growth of your purchasing power = the nominal interest rate divided by the inflation rate:
1  +  R  =  1 + N 1 + I 
R = Real Interest Rate N = Nominal Interest Rate I = Inflation Rate 
The above equation can be solved for the real interest rate.
Solving for the Real Interest Rate  

(1 + R)(1 + I) = 1 + N  Multiply both sides by (1 + I). 
1 + I + R + RI = 1 + N  Multiply out lefthand side to get terms. 
R + RI = R(1 + I) = N  I  Subtract 1 and I from both sides. 
R = (N − I)/(1 + I)  Divide both sides by 1 + I. 
R  =  N  I 1 + I 
R = Real Interest Rate N = Nominal Interest Rate I = Inflation Rate 
Because people invest to earn more purchasing power, they will only invest or lend money that pays more than the expected inflation rate. In this case, the nominal rate = the real interest rate + the expected inflation.
Nominal Interest Rate Equilibrium
Although there are many different interest rates, their differences result mainly from risk, but they all move up or down along with the prevailing rates. Thus, these rates can be abstracted as a single interest rate — the prevailing interest rate. Generally, as interest rates increase, saving increases and borrowing decreases, and vice versa. If investments pay higher interest, then more people, businesses, and other organizations will invest to earn more money. If interest rates decline, then the motivation to invest declines also, but borrowing increases, which increases demand for money. There is a point where supply equals demand — this is the observed or nominal interest rate.
Irving Fisher looked at interest rate equilibrium as the desire for a specific real rate of return + the expected inflation rate:
Nominal Interest Rate

= Real Interest Rate

+ Expected Inflation Rate.
If the expected inflation rate was high, then people would demand a higher nominal rate for their investments; for why would anyone invest if they did not expect a real return? Although no one knows future interest rates, the nominal interest rate somewhat indicates expected interest rates.
The Taxation of Nominal Interest Rates
Most earned interest, or any positive return from investments, is taxed. However, taxes currently apply to the nominal rate of return, not the real rate — thus, the tax rate on the real rate of return exceeds the published tax rate.
Real Rate of Return

= Nominal Interest Rate

× (1 − Your Tax Bracket)

− Inflation Rate
Example: Calculating the Real Interest Rate after Taxes
If you earned 5% nominal interest on your money with 3% inflation, and you are in the 25% tax bracket, what is your real interest rate after taxes?
Solution:
Using the above formula:
Real Rate of Return = 5% × .75  3%. = .75%
As you can see from the above, if you are in a high tax bracket, you must earn significantly more than 5% to earn a decent real return. At the 35% bracket, given the above nominal interest rate and inflation rate, the real interest rate would be 0! You can see why the wealthy invest in taxfree municipal bonds.
Simple Interest
Simple interest, often called the nominal annual percentage rate (APR), is uncompounded interest, which is calculated by multiplying the principal times the interest rate. The earned interest is not added to the principal, so the amount of interest earned is always the same for a given interest rate.
Simple Interest = Principal × Interest Rate
A good example of simple interest is the interest earned by bonds. Most bonds pay a coupon rate, which is simply the stated interest rate of the bond when it is first issued. When the interest is earned, it is sent to the bondholder — it is not added to the bond's principal and the interest earns no additional interest unless the bondholder reinvests the interest in another investment, such as a savings account.
Compounding Interest
Compounded interest is calculated using the principal + previously earned interest. For instance, if you deposit $100 in a savings account paying 6% interest, compounded semiannually, then your money is earning 3% every 6 months, so that at the end of 6 months, you would have $103. But in the next 6 months, there would be $103 earning interest instead of just $100, so $103 × 3% = $3.09. Add this to the 1^{st} $3 already earned will yield a total of $6.09 for the 1^{st} year, 9¢ more than if the interest rate was simple interest. This is equivalent to a simple interest rate of 6.09% per year. Because money earns interest, it has a future value that exceeds its present value by the amount of the interest earned — called the future value of money or the future value (FV) of a dollar. The future value can be expressed as:
FV  =  P(1 + r)^{n} 
FV = Future Value P = Principal r = interest rate per compounding period n = number of compounding periods 
Using the above example: $100 × (1+.03)^{2} = $106.09. Interest rates are often used to compare investments, but not all investments have the same compounding period, or it may not be compounded at all, as is the case for a zero coupon bond, which pays no interest. The interest is earned by buying the bond at a discount and receiving face value at maturity. However, an effective compounded interest rate can be found even for a discounted bond, because it is possible to convert compounding interest rates into other rates with different periods of compounding. Most investments that pay interest normalize the interest rate to an annual rate — the APR. Thus, using the above example, a savings deposit that pays 6% compounded semiannually is equivalent to 6.09% compounded annually. By normalizing interest rates to an effective annual percentage rate, different investments can be easily compared.
Rule of 70 and the Rule of 115 — Quick Methods to Find Doubling and Tripling Times
The Rule of 70 is a simple method to find how quickly a principal that is earning a compounding interest rate will take to double: divide 70 by the interest rate for the compounding period.
Time to Double = 70 / Interest Rate
Examples: How long will a savings account paying 5% compounded annually take to double? 70/5 = 14 years. As a check, using part of the formula for future value listed above, (1.05)^{14} ≈ 1.98, so the Rule of 70 is a close approximation. Note, however, that the Rule of 70 approximation becomes less accurate for higher interest rates. For instance, if the interest rate is 14%, then 70/14 = 5, but (1.14)^{5} ≈ 1.93
The Rule of 115 is the triple version of the Rule of 70. Dividing 115 by the interest rate yields the number of years required to triple an investment. So a 10% rate of return will take about 11.5 years to triple the principal.
Effective Interest Rate
An effective interest rate is calculated for a standard time, usually 1 year, in which case, it is called an effective annual rate. Investments can more easily be compared using effective annual rates.
1 + i = (1 + r/n)^{n}  Equalize the annual rate (i) to the compounded rate (r), with n = number of compounding periods. 
(1 + i)^{1/n} = 1 + r/n  Take the nth root of both sides. 
r/n = (1 + i)^{1/n} − 1  Subtract 1 from both sides and transpose sides. 
r = n × [(1 + i)^{1/n} − 1]  Multiply both sides by n. 
r  =  n × (1 + i)^{1/n} − 1 
i = effective interest rate r = interest rate per compounding period n = number of compounding periods 
Example: So if a bank wants to advertise a 10% interest rate that is compounded quarterly, then what is the nominal interest rate, compounded quarterly, that will yield 10% annually? The solution:
r = 4 × (1 + 10%)^{1/4} − 1 = 4 × [(1.1)^{1/4} − 1] = 4 × [1.024 − 1] = 4 × 0.024 = 0.096 = 9.6%
So 9.6% compounded quarterly will yield 10% compounded annually.
Example: A coupon bond pays 6% annually in semiannual payments. What is the interest rate compounded semiannually?
r = 2 × (1 + 0.06%)^{1/2} − 1 = 2 × [(1.06)^{1/2} − 1] = 2 × [1.0296 − 1] = 2 × 0.0296 = 0.0592 = 5.92%
Continuously Compounding Interest
Many portfolio simulations and pricing models for derivatives use a continuously compounded interest rate formula.
If a savings account paid a nominal interest rate of 6%, that was compounded semiannually, the real compounded rate can be found using the following formula:
C  =  (  1  +  r  n  )  n    1 
=  (  1  +  .06 2  )  2    1  
=  0.0609  =  6.09%  
C = Compounded Interest Rate r = Nominal Interest Rate n = number of compounding periods 
To find the daily compounded rate for a nominal annual interest rate of 6%, divide the interest rate by 365, and raise the quantity in parentheses to the 365^{th} power. We note that as n increases, the increase in the 1^{st} term becomes less and less, reaching a limit as n increases to infinity. This limit is the natural logarithm base e:
As  n  →  ∞,  (  1  +  1  n  )  n  →  e  =  2.718281828... 
n = number of compounding periods 
As a corollary of the above equation, we arrive at Formula 3:
As  n  →  ∞,  (  1  +  r  n  )  n  →  e^{r} 
r = Rate of growth or interest rate n = number of compounding periods 
Thus, by substituting the result of Formula 3 into Formula 1, we see that:
Continuous Compounded Interest Rate = e^{r}  1
Example: Calculating the Continuously Compounded Interest Rate or the Effective Annual Percentage Rate
If a bank advertises a savings account that pays a 6% nominal interest rate that is compounded continuously, what is the effective annual percentage rate?
Solution:
Using the above formula:
Continuously Compounded Interest Rate = e^{.06}  1 = 1.061837  1 ≈ 6.1837%
Although it sounds like you'll make a fortune by having it continuously compounded, it's not much more than the daily compounded rate of:
6% Compounded Daily = (1 + .06/365)^{365} ≈ 0.061831 ≈ 6.1831%
Note that since 1 + Growth Factor = e^{(Growth Factor)}, we can simplify the 1^{st} formula relating real interest rates, nominal interest rates, and inflation rates by the following equation:
e^{R}  =  e^{N} e^{I}  =  e^{(NI)} 
R = Real Interest Rate N = Nominal Interest Rate I = Inflation Rate 
Taking the natural logarithm of both sides, simplifies the above equation even further:
R = N  I
Thus, for continuously compounded rates, the approximation formula for relating the real interest rate to the nominal interest rate and inflation rate becomes exact.