Types of Interest

Simple interest: simple interest does not take compounding into account, and is determined by multiplying the principal by the interest rate (per period) by the number of time periods.

To calculate: Add up all the interest paid/payable in a period. Divide that by the principal at the beginning of the period. E.g. on $100 (principal):

  • credit card debt where $1/day is charged. 1/100 = 1%/day.
  • corporate bond where $3 is due after six months, and another $3 is due at year end. (3+3)/100 = 6%/year.
  • certificate of deposit (GIC) where $6 is paid at year end. 6/100 = 6%/year.

There are three problems with simple interest:

  • The time periods used for measurement can be different, making comparisons wrong. You cannot say the 1%/day credit card interest is ’equal’ to a 365%/year GIC.
  • The time value of money means that $3 paid every six months hurts more than $6 paid only at year end. So you cannot ’equate’ the 6% bond to the 6% GIC.
  • When interest is due, but not paid, it must be clear what happens. Does it remain ’interest payable’, like the bond’s $3 payment after six months? Or does it get added to the original principal, like the 1%/day on the credit card? Each time it is added to the principal it ’compounds’. The interest from that time forward is calculated on that (now larger) principal. The more frequent the compounding, the faster the principal grows, and the greater the interest amount is.

Compound interest: In order to solve these three problems, there is a convention that interest rates will be disclosed as if the term is one year and the compounding is yearly. The discussion at compound interest shows how to convert to and from the different measures of interest.

Real interest: This is calculated as (nominal interest rate) - (inflation). It attempts to measure the value of the interest in units of stable purchasing power. See the discussion at real interest rate.

Cumulative interest/return: This calculation is (FV/PV)-1. It ignores the ’per year’ convention and assumes compounding at every payment date. It is usually used to compare two long term opportunities. Since the difference in rates gets magnified by time, so the speaker’s point is more clearly made.

Other Exceptions:

  • US and Canadian T-Bills (short term Government debt) have a different convention. Their interest is calculated as (100-P)/P where ’P’ is the price paid. Instead of normalizing it to a year, the interest is prorated by the number of days ’t’: (365/t)*100. (See also: Day count convention). The total calculation is ((100-P)/P)*((365/t)*100)
  • Corporate Bonds are most frequently payable twice yearly. The amount of interest paid is the simple interest disclosed divided by two (multiplied by the face value of debt).

Rule of 78: Some consumer loans calculate interest by the "Rule of 78" or "Sum of digits" method. Seventy-eight is the sum of the numbers 1 through 12, inclusive. And the practice enabled quick calculations of interest in the pre-computer days. In a loan with interest calculated per the Rule of 78, the total interest over the life of the loan is calculated as either simple or compound interest and amounts to the same as either of the above methods. Payments remain constant over the life of the loan; however, payments are allocated to interest in progressively smaller amounts. In a one-year loan, in the first month, 12/78 of all interest owed over the life of the loan is due; in the second month, 11/78; progressing to the twelfth month where only 1/78 of all interest is due. The practical effect of the Rule of 78 is to make early pay-offs of term loans more expensive. Approximately 3/4 of all interest due on a one year loan is collected by the sixth month, and pay-off of the principal then will cause the effective interest rate to be much higher than than the APY used to calculate the payments.

The United States outlawed the use of "Rule of 78" interest in loans over five years in term. Certain other jurisdictions have outlawed application of the Rule of 78 in certain types of loans, particularly consumer loans.

Rule of 72: The "Rule of 72" is a "quick and dirty" method for finding out how fast money doubles for a given interest rate. For example, if you have an interest rate of 6%, it will take 72/6 or 12 years for your money to double, compounding at 6%. This is an approximation that starts to break down above 10%.

Market Interest Rates:

There are markets for investments which include the money market, bond market, as well as retail financial institutions like banks, which set interest rates. Each specific debt takes into account the following factors in determining its interest rate:

Opportunity cost: This encompasses any other use to which the money could be put, including lending to others, investing elsewhere, holding cash (for safety, for example), and simply spending the funds.

Inflation: Since the lender is deferring his consumption, he will at a bare minimum, want to recover enough to pay the increased cost of goods due to inflation. Because future inflation is unknown, there are three tactics.

  • Charge X% interest ’plus inflation’. Many governments issue ’real-return’ or ’inflation indexed’ bonds. The principal amount and the interest payments are continually increased by the rate of inflations. See the discussion at real interest rate.
  • Decide on the ’expected’ inflation rate. This still leaves both parties exposed to the risk of ’unexpected’ inflation.
  • Allow the interest rate to be periodically changed. While a ’fixed interest rate’ remains the same throughout the life of the debt, ’variable’ or ’floating’ rates can be reset. There are derivative products that allow for hedging and swaps between the two.

Default: There is always the risk the borrower will become bankrupt, abscond or otherwise default on the loan. The risk premium attempts to measure the integrity of the borrower, the risk of his enterprise succeeding and the security of any collateral pledged. Loans to developing countries have higher risk premiums than those to the US government. An operating line of credit to a business will have a higher rate than a mortgage.

worthiness of businesses is measured by bond rating services and individual’s credit scores by credit bureaus. The risks of an individual debt may have a large standard deviation of possibilities. The lender may want to cover his maximum risk. But lenders with portfolios of debt can lower the risk premium to cover just the most probable outcome.

Deferred consumption: Charging interest equal only to inflation will leave the lender with the same purchasing power, but he would prefer his own consumption NOW rather than later. There will be an interest premium of the delay. See the discussion at time value of money. He may not want to consume, but instead would invest in another product. The possible return he could realize in competing investments will determine what interest he charges.

Length of time: Time has two effects.

  • Shorter terms have less risk of default and inflation because the near future is easier to predict than events 20 year off.
  • Longer terms allow for investments in larger projects with higher eventual returns. Contrast this to the lender’s preference for readily available cash for contingencies. This is why banks pay higher interest on non-redeemable GICs than on checking account balances.
  • Long-term interest rates fell in much of the developed world in the second half of 2006.

Other: Borrowers and lenders may face individual tax rates, transaction costs and foreign exchange rate risks. In a liquid market they cannot exert their personal preferences. It is the sum total of the participants who determine rates. The market for financial instruments has moved from the local, to the national, and is now international.

Interest Rates in Macroeconomics

Output and Unemployment:

Interest rates are the main determinant of investment on a macroeconomic scale. Broadly speaking, if interest rates increase, then investment decreases due to the higher cost of borrowing (all else being equal).

Interest rates are generally determined by the market, but government intervention - usually by a central bank- may strongly influence short-term interest rates, and is used as the main tool of monetary policy. The central bank offers to buy or sell money at the desired rate and, due to their control of certain tools (such as, in many countries, the ability to print money) they are able to influence overall market interest rates.

Investment can change rapidly to changes in interest rates, affecting national income, and, through Okun’s Law, changes in output affect unemployment.

Open Market Operations in the United States:

The Federal Reserve (often referred to as ’The Fed’) implements monetary policy largely by targeting the federal funds rate. This is the rate that banks charge each other for overnight loans of federal funds, which are the reserves held by banks at the Fed.

Open market operations are one tool within monetary policy implemented by the Federal Reserve to steer short-term interest rates. Using the power to buy and sell treasury securities, the Open Market Desk at the Federal Reserve Bank of New York can supply the market with dollars by purchasing T-notes, hence increasing the nation’s money supply. By increasing the money supply or Aggregate Supply of Funding (ASF), interest rates will fall due to the excess of dollars banks will end up with in their reserves. Excess reserves may be lent in the Fed funds market to other banks, thus driving down rates.

Money and Inflation:

Loans, bonds, and shares have some of the characteristics of money and are included in the broad money supply.

Through the quantity theory of money, increases in the money supply lead to inflation. This means that interest rates can affect inflation in the future.