Measuring and Calculating Interest Rates and Financial Asset Prices

By Rose, P.S., Marquis, M.H.

Edited by Paul Ducham

CALCULATING INTEREST RATES

The interest rate is the price charged a borrower for the loan of money. This price is unique because it is really a ratio of two quantities: the total required fee a borrower must pay a lender to obtain the use of credit for a stipulated time period divided by the total amount of credit made available to the borrower. By convention, the interest rate is usually expressed in percent per annum. Thus,

image6.1

For example, an interest rate of 10 percent per annum on a $1,000, one-year loan to purchase a computer implies that the lender of funds has received a borrower’s promise to pay a fee of $100 (10 percent of $1,000) in return for the use of $1,000 in credit for a year. The promised fee of $100 is in addition to the repayment of the loan principal ($1,000), which must occur sometime during the year. Interest rates are usually expressed as annualized percentages, even for financial assets with maturities shorter than a year. For example, in the federal funds market, banks frequently loan reserves to each other overnight, with the loan being repaid the next day. Even in this market the interest rate quoted daily by lenders is expressed in percent per annum, as though the loan were for a year’s time. However, various types of loans and securities display important differences in how interest fees and amounts borrowed are valued or accounted for, leading to several different methods for determining interest rates. Some interest rate measures use a 360-day year, while others use a 365-day year. Some employ compound rates of return, with interest income earned on accumulated interest, and some do not use compounding.

BASIS POINTS

Interest rates on securities traded in the open market rarely are quoted in whole percentage points, such as 5 percent or 8 percent. The typical case is a rate expressed in hundredths of a percent: for example, 5.36 percent or 7.62 percent. Moreover, most interest rates change by only fractions of a whole percentage point in a single day or week. To deal with this situation, the concept of the basis point was developed. A basis point equals 1/100 of a percentage point. Thus, an interest rate of 10.5 percent may be expressed as 10 percent plus 50 basis points, or 1,050 basis points. Similarly, an increase in a loan or security rate from 5.25 percent to 5.30 percent represents an increase of 5 basis points.

MONEY MARKET INTEREST RATES

Most money market assets are short-term assets on which the investor receives no income until the asset matures. The amount that the investor receives upon maturity is referred to as the asset’s par value or face value. For example, one such asset is a 3-month (90-day) U.S. Treasury bill (T-bill) with a par value of $100,000. The price the investor pays for this asset will obviously be less than par value, or $100,000 in the example; otherwise the investor will not receive any return on his or her investment. This asset is therefore said to be sold at a discount to its par value.

But how much is the investor willing to pay for this claim against the U.S. Government that he or she will be able to redeem in 90 days? The answer depends in large part on the rate of return the investor demands. On the street, this rate of return is often referred to as the coupon-equivalent, bond-equivalent, or investment rate of return, and is often designated IR. To compute the IR, three pieces of information are required. Two of them are provided by the issuer, or the U.S. Government in this example: the Par value and the number of Days to maturity. The third piece of information is the Purchase price, or how much the investor has invested. The formula then computes the actual rate of return for the investor, and then annualizes the return, as if the investment lasted one full year, or 365 days.

image6.2.1

Suppose that you are the investor and decide to purchase this Treasury bill for $99,000 and hold it until maturity. The annualized interest rate you would receive on this investment is:

image6.2.2

If you were working as a dealer in the money markets and wished to purchase this T-bill, you would not be quoted a purchase price, but rather an interest rate. That interest rate is referred to as the bank discount rate, or simply the discount rate, denoted DR. The DR is not the actual annualized rate of return on the investment. It is used only for trading purposes and is usually easier to compute than the IR. The formula for the bank discount rate is:

image6.3.1

Note that there are two differences between formulas (6.2) and (6.3). Rather than using the actual amount invested, or the purchase price, as the investment base in computing the DR, the par value is used. This simplifies the calculations because the par value is usually in round numbers, such as $100,000 in the example. The second simplification is to compute the DR on the basis of a 360-day year. In the example, the DR that corresponds to IR of 4.10 percent is:

image6.3.2

Notice that the DR is less than the IR. This is always the case. One reason is that the DR calculation uses a 360-day year, which reduces the numerator in the calculation. The second reason is that the investment base used in calculating the DR is more than the actual amount of the investment which is used in calculating the IR, while the actual interest earned, or the Par value - Purchase price, is the same in both calculations.

HOLDING-PERIOD YIELD

An important feature of financial assets sold at a discount is that their price tends to rise over time and is exactly equal to par value upon maturity. However, the asset price will not rise in a steady, uniform manner, but will be influenced by continuous changes in market interest rates that take place throughout the day. This phenomenon is important to investors who may choose, in our example, to sell the T-bill before it matures. What is the annualized rate of return that investor would receive on this investment? This rate of return is referred to as the holding-period yield, and can be simply computed for assets sold at a discount by using the DR formula:

formula6.4

In this case, the decline in interest rates (reflected in the fall in the discount rate) caused the investor’s rate of return on her investment during these 30 days to be higher than she had expected when the T-bill was first purchased. What would happen to the DR and the holding-period yield if the investor was only able to sell the T-bill for $99,250 after 30 days, in which case she would have earned only $250 rather than $500 in interest income? Verify that the investor’s holding-period yield would only have been 3 percent.

TREASURY BILL INTEREST RATE QUOTATIONS

U.S. Treasury bills (T-bills) are money market assets that may have maturities upon issue of four weeks, three months, or six months. The information that concerns potential investors or owners of T-bills is the maturity date, the investment rate (IR), and the discount rates at which a T-bill may be bought or sold. Investors may also wish to know what happened to the value of the asset over the course of one trading day. All of this information is usually contained in the daily report of trading activity in the financial news. An example of this information is given in Exhibit 6.1 .

This exhibit depicts the information on two U.S. T-bills that are being traded in the market. One has approximately 3 months to maturity and the other approximately 6 months to maturity. The Bid Rate and Ask Rate are the discount rates quoted by dealers who manage portfolios of T-bills. The Bid Rate is the discount rate that the dealer requires if he is to purchase the T-bill and add it to his portfolio, and the Ask Rate is the discount rate an investor would lock in (assuming he held the T-bill to maturity) if he purchased the T-bill from a dealer. Note that the Ask Rate is less than the Bid Rate. The difference between them, or the Bid-Ask Spread, represents the margin that enables a dealer to make a profit by creating this market. The Ask Yield is the investment rate (IR) that corresponds to the Ask Rate (DR). Note that the Ask Yield is higher than the Ask Rate. The Ask Yield represents the actual rate of return that the investor would receive if he bought the asset from the dealer and held it until maturity.

Exhibit_6.1

YIELD TO MATURITY

We begin with a concept that should become second nature to every investor: yield to maturity (YTM) . As the name implies, the yield to maturity represents the rate of return an investor would receive if she bought the asset and chose to hold—not sell—the asset for its entire life. Even if the investor does not plan to hold the asset until it matures, its YTM is still extremely important. It represents the rate of interest the market is prepared to pay for a financial asset in order to exchange present dollars for future dollars—and this rate changes continuously with market conditions. Therefore, the yield to maturity establishes the market value of the asset that will determine how much an investor can sell the asset for should she decide to sell before maturity. The stream of payments associated with a bond normally consists of a number of identical periodic coupon payments plus the par value of the bond received at maturity. A newly issued bond will typically have the following information stipulated: the par value (M), the dates of the coupon payments, and the coupon rate . The coupon rate is used to determine the amount of the annual interest income received by the investor in the form of coupon payments (C).

formula6.6

where y is the yield to maturity and n is the number of periods (usually several years).

Formula (6.7) can be used to price a bond or any debt security with constant annual payments to the investor over the life of the asset. In this case, y is the annualized yield to maturity.

What about financial instruments that make periodic payments to the investor more frequently than once a year? For example, most bonds pay interest semiannually, and other financial instruments may pay out quarterly or monthly. In this instance the bond-price, or yield-to-maturity, formula (6.7) needs to be modified to include the parameter k —a measure of the number of times during the year that interest income is paid to the holder of the asset. The asset price is then given by:

formula6.8

Suppose we have a 20-year government bond paying $50 interest twice each year, then k must equal 2 and there would be 40 periods ( n X k = 20 X 2) in which the investor would expect to receive $50 in interest income. As the above formula shows, we would have to divide the annualized yield to maturity ( y ) by k and discount the expected payments over n X k, rather than n, time periods.

To illustrate the use of this formula, assume an investor is considering the purchase of a bond due to mature in 20 years (with 40 semiannual interest-crediting periods) and carrying a 10 percent annual coupon rate, or 5 percent every six months. This asset is available for purchase at a current market price of $850. If the bond has a par value of $1,000, which will be paid to the investor when the asset reaches maturity, the bond’s yield to maturity may be found by solving the equation:

formula6.9

In this case, y equals almost 8 percent. Because the investor must pay a higher current market price than par value (the amount the investor will receive back when the bond matures) this bond’s yield to maturity must be less than its coupon rate.

From these two examples, we can answer the following question: How does the yield to maturity ( y ) compare with the coupon rate ( C / M )? If the bond price is equal to the par value of the bond, or P = M, the bond is said to be selling at par, and the coupon rate is equal to the yield to maturity, or y = ( C / M ). However, this is rarely the case. The market price of the bond and the bond’s yield to maturity change continuously with market conditions, while the par value and the coupon rate remain fixed. Whenever the bond is selling below par, or P < M, the yield to maturity must have risen above the coupon rate, or y > ( C / M ). Conversely, whenever the bond is selling above par, or P > M, then the yield to maturity must have fallen below the coupon rate, or y < ( C / M ).

HOLDING-PERIOD YIELD ON BONDS

A slight modification of the bond-pricing or yield-to-maturity formula (6.7) results in a return measure for those situations in which an investor holds a bond or long-term debt security for a time and then sells it to another investor in advance of the asset’s maturity. This so-called holding-period yield , denoted h, solves the following equation:

formula6.11

where C is the annual payment received by the investor, P is the original purchase price, and P m is the selling price after m periods. Therefore, the annualized holding period yield ( h ) is simply the rate of discount equalizing the market price of the debt security with all annual payments between the time the asset was purchased and it is sold (including the sale price). If the asset is held to maturity, its holding-period yield equals its yield to maturity.

If income from an asset comes in more frequently than once per year, we would have to modify formula (6.7). Suppose that payments are received by the investor k times during each year of the investor’s holding period. In this case, we would divide each of the expected annual payments to the investor ( C ) and the annualized holdingperiod yield ( h ) by k. In addition, the number of time periods ( m ) that make up the investor’s holding period must be multiplied by k in order to help us find the correct annual holding-period yield ( h ).

YIELD TO MATURITY AND HOLDING-PERIOD YIELD

It is often helpful in understanding the concepts of yield to maturity and holdingperiod yield to look at an example of how we determine and interpret these two rate of return measures. Suppose, for example, that an investor is thinking about the purchase of a corporate bond, $1,000 in face or par value, with a promised annual (coupon) rate of return of 10 percent. Assume the bond pays interest of $50 every six months. Currently the bond is selling for $900. Assume that this is a five-year bond that the investor plans to hold until it matures, when it is redeemed at par by the issuer of the bond. We have

formula6.12

It is useful at this point to consider what each term in equation 6.12 means. Both the yield-to-maturity and holding-period-yield formulas are based on the concept of present value: Funds to be received in the future are worth less than funds received today. Present dollars may be used to purchase and enjoy goods and services today, but future dollars are only promises to pay and force us to postpone consumption until the funds actually are received. Equation 6.12 indicates that a bond promising to pay $50 every six months for a five-year period in the future plus a lump sum of $1,000 at maturity is worth only $900 in present value dollars. The yield, y /2, serves as a rate of discount reducing each payment of future dollars back to its present value in today’s market. The further into the future the payment is to be made, the larger the discount factor, (1 + y /2) n , becomes. Turning the concept around, the purchase of a financial asset in today’s market represents the investment of present dollars in the expectation of a higher return in the form of future dollars. The familiar compound interest formula applies here. This formula

formula6.13

indicates that the amount of funds accumulated t years from now (FV or future value) depends on the principal originally invested ( P ), the investor’s expected rate of return or yield ( y ), and the number of years the principal is invested ( t ). Thus, a principal of $1,000 invested today at a 10 percent annual rate will amount to $1,100 a year from now [i.e., $1,000 X (1 + 0.10) 1 ]. Rearrangement of the compound interest formula gives

formula6.14

Equation 6.14 states that the present value of FV dollars to be received in the future is P if the promised interest rate is y. If we expect to receive $1,100 one year from now and the promised interest rate is 10 percent, the present value of that $1,100 must be $1,000. Each term on the right-hand side of the yield-to-maturity and holding-period-yield formulas is a form of equation 6.14. Solving equation 6.12 for the yield to maturity of a bond simply means finding a value for y which brings both right- and left-hand sides of the yield formula into balance, equating the current price ( P ) of a financial asset with the stream of future dollars it will generate for the investor. When all expected cash flows are not the same in amount, an electronic calculator or computer software may be used to find the solution. Fortunately, in the case of the bond represented in equation 6.12, the solution is not complicated. Rewrite equation 6.12 in the following form:

formula6.15

This indicates the bond will pay an annuity of $1 per year (multiplied in this case by $50) for five years (10 six-month periods) plus a lump sum of $1 (multiplied here by $1,000) at the end of the fifth year. Using a programmable calculator and entering the price (present value or PV) of the instrument (which is $900 in this case), the expected periodic interest payments (PMT = $50 semiannually), and the final value or payment ($1,000 in this example) tells us that y, the yield-to-maturity measure, is 12.83 percent. The investor interested in maximizing return would compare this yield to maturity with the yields to maturity available on other assets of comparable risk and liquidity. Note that if the bond in this example had been a 10-year bond, but identical in every other way, then the investor with a five-year time horizon for his or her investment would receive a holding-period yield, h, of precisely the same 12.83 percent annualized yield-to-maturity, if he or she were able to sell the bond at precisely its par value. This result can be verified by setting Pm = $1,000 in equation 6.11 and solving for h.

TREASURY NOTE AND BOND QUOTES

U.S. Treasury notes (T-notes) and U.S. Treasury bonds (T-bonds) are debt securities with original maturities ranging from 2 to 30 years. In addition to a fixed maturity, most of these securities make fixed coupon payments to their owners, usually semiannually. An investor in a T-note or T-bond needs to know not only the price and maturity date of the security, but he must also know the date on which he would receive coupon payments, and he must be able to compute the amount of those coupon payments. In addition, he needs to know the security’s current yield to maturity. This information is generally provided in the financial news, reporting daily trading activity in the market. An example is given in Exhibit 6.2 .

In order for investors to be able to compare Treasury securities, the price quotes are based on each $100 of par value, and they are quoted in 32nds of a dollar (100:01 = $100.0313 or 100:16 = $100.50). The coupon rate quoted is an annualized rate, despite the fact that coupons are paid semiannually. For example, a coupon rate of 2.000 would imply that the Treasury Department will pay a $1.00 coupon every 6 months for each $100 in par value until the maturity date. Dealers in T-notes and T-bonds manage portfolios of those Treasury securities and stand ready to buy the T-note or T-bond at its announced Bid Price for each $100 in par value, and stand ready to sell the Treasuries at the Asked Price for each $100 in par value. Note that the Asked Price is greater than the Bid Price, with the difference representing the profit margin the dealer receives for making the market. How much the Asked Price has changed over the course of the previous day’s trading is quoted in 32nds. The Asked Yield is the yield to maturity an investor purchasing the Treasury from the dealer would receive if he bought the security and held it to maturity. Note that if the dealer is willing to sell the T-note or T-bond at less than par (below 100), then the yield to maturity is above the coupon rate. For those T-notes and T-bonds selling above par (above 100), the yield to maturity is below the coupon rate.

Exhibit6.2

CORPORATE BOND QUOTES

Investors in corporate bonds require exactly the same information as investors in Treasury bonds, but they are also interested in how much additional return they would be promised for taking the risk that, unlike the federal government, the corporation may not honor its obligations to make coupon payments and return the principal (par value) in full on the promised date. This additional information is usually provided in the daily price quotations in the financial news. An example is given in Exhibit 6.3 . The Coupon Rate and the Last Price are both based on a par value of $100. The Last Price and Last Yield represent the price paid per $100 of par value and the yield to maturity that the purchaser of the last recorded trade of the day realized. To compute the additional risk premium the investor expects to receive the Estimated Yield Spread represents the yield to maturity on that corporate bond minus the yield to maturity on a comparable U.S. Treasury security at the time the corporate bond was purchased. The numerical value is in basis points.

The Comparable U.S. Treasury column gives the number of years to maturity on the most recently issued security of that maturity class. For example, if the Comparable U.S. Treasury has a maturity of 10 years, the yield on the corporate bond would be compared with the yield on the most recently issued 10-year U.S. T-note. In this case, 10 years would represent the maturity class of U.S. Treasuries with the number of years to maturity that most closely match the number of years left until the maturity of the corporate bond. The $ Volume column indicates how much trading took place over the course of the previous trading day. Note that, just as with U.S. T-notes and T-bonds, if the corporate bond is selling above par (Last Price exceeds 100), then its yield to maturity is below the coupon rate, and if the corporate bond is selling below par, then its yield is higher than the coupon rate.

Exhibit6.3

FIXED INCOME RETURNS

One example of a fixed-income perpetual financial instrument is the British consol —a perpetual bond issued by the British government that promises its holder a fixed coupon payment every year ad infinitum. The annual rate of return on this bond is known as a perpetuity rate and is simply computed as:

formula6.16

Thus, with a $100 promised annual coupon payment and an expected 10 percent annual rate of return, the current price (present value) of this instrument must be $100/0.10 or $1,000.

The formula for a perpetual financial instrument reminds us of several key points regarding value and the rate of return for financial assets (especially bonds and similar debt securities). First, an infinite stream of fixed payments does have a finite value, measured by a financial asset’s current price (or present value). Second, there is an inverse (or negative) relationship between the current price and the rate of return or yield on a financial asset, especially for bonds and debt instruments.

INTEREST RATES AND STOCK PRICES

The most common type of perpetual financial instrument is corporate stock, which represents partial ownership of a business firm that is expected to exist forever, unless it goes bankrupt! As with bonds and other debt securities, there tends to be an inverse relationship between interest rates and corporate stock (equity) prices as well. However, this relationship is weaker than it is for bonds and does not always hold. For example, if interest rates rise, debt instruments now offering higher yields become more attractive relative to stocks, resulting in increased stock sales and declining equity prices (all other factors held equal). Conversely, a period of falling interest rates often leads investors to dump their lower-yielding bonds and switch to equities, driving stock prices upward. Then, too, lower market interest rates tend to lower the overall cost of capital for businesses issuing stock, resulting in a rise in stock prices (provided expected corporate dividends do not fall). What actually happens to stock prices when market interest rates change can often be understood by tracking changes in two fundamental factors that appear to influence all stock prices—the stream of shareholder dividends a company is expected to pay in current and future time periods [ E(D) ] and the minimum rate of return required by a company’s shareholders ( r ). This minimum required rate of return is used to discount the infinite stream of expected dividends to determine their present values, the sum of which is the market price of the stock ( SP ). Assuming the dividend in the current period, D0, is known, the stock price formula becomes:

 formula6.18

Clearly, a rise in expected dividends [ E ( D )] or a fall in the required risk-adjusted rate of return for the company’s stockholders ( r ) leads to higher stock prices per share ( SP ), other factors held equal. However, there are no guarantees surrounding the stock price–interest rate relationship we have just described because both expected dividends and the required risk-adjusted rate of return ( r ) may change at the same time, offsetting one another and leaving stock prices unchanged or causing them to move in an unexpected direction. The formula stated above for determining the price of a company’s stock is a general formula that takes into account the possibility that the dividends paid by a corporation to its shareholders may vary in timing and amount as the months and years go by. However, for many companies in recent years dividends have tended to grow at a relatively constant rate from period to period, often shaped by corporate management to convey the image of company stability. If corporate dividends do grow at a constant rate, the formula for corporate stock prices becomes simpler and easier to follow:

formula6.19

where D0 represents the current dividend and g is expected constant annual growth rate of dividends in the future. As before, r is the required rate of return for investors, or the rate of discount of future expected dividends, which reflects the perceived risk of investing in the company’s stock, and also coincides with the cost of capital to the firm. To illustrate the use of the above formula, suppose a company expects to pay a dividend of $2.50 per share in the initial period and to increase the amount of its future annual dividend payments by 5 percent each year. If the discount rate associated with the company’s stock is 12 percent, then its current stock price will approach:

SP = $2.50(1+ 0.12) / (0.12 - 0.05) = $40.00 per share

Clearly, it is easier to estimate the equity value for those companies—often the largest firms today—that pay a steady and predictable dividend rate.

HOLDING PERIOD RETURN FORMULA

The reasoning we used earlier to determine the yield to maturity and the holdingperiod yield on debt securities can also be applied to calculate the holding-period yield on corporate stocks (equities). To illustrate, suppose an investor is considering the purchase of shares of common stock issued by General Electric Corporation, currently selling for $40 per share. He plans to hold the stock for two years and then sell out at an expected price of $50 per share. If he expects to receive $2 per share in dividends at the end of each year (with no payments until the end of year 1), what holding-period yield ( h ) does the investor expect to earn? Modify formula (6.18) to reflect the fact that the stock will be sold two years from today at an expected price of SP2 :

formula-Holding-Period

A computer or calculator programmed to determine holding-period yields tells us that the expected annualized holding-period yield on GE’s stock is 16.5 percent.

STOCK PRICE QUOTES

To make knowledgeable investments in the stock market requires the processing of a great deal of information. Fortunately, an army of financial analysts spend full time “on the street” Gathering Information and making the outcome of their analyses available to all. One way in which these analyses become available to investors is simply by traders’ buying and selling equity shares in publicly traded firms. This trading activity renders the market fair game for small investors by efficiently incorporating information relevant to the value of a corporation as reflected in its stock price. Small investors are then able to examine the outcome of trading for any given day in a variety of news sources. One example of the information on stock prices reported regularly in the financial news is given in Exhibit 6.6 .

Investors can readily obtain information on the price of a stock at the close of trading the previous day, or the Closing Price, how actively traded it was in terms of the dollar Volume (000s) of shares traded, and see whether the share price traded up or down from the previous day’s close and by how much from the column headed Net Change. The closing price can also be compared with the stock’s highest and lowest value over the previous year, (i.e., its 52-Week Hi and 52-Week Low). The investor can also determine whether the stock is paying a dividend and, if so, what that dividend payment is per year, or Dividend ($/yr), and what that represents in terms of an investment yield based on the closing price, known as the Dividend Yield (%) or Dividend($/yr)/Closing Price.

Note that not all firms pay dividends on their stock. Those firms with good investment opportunities their management feels would expand earnings in the future may choose to retain their current earnings and reinvest in the company. This strategy has been pursued by “high tech” firms due to the extraordinary growth of that industry. As a result, the stock price of those firms has generally risen more rapidly than the market as a whole. Firms in more mature industries may choose to pay out their earnings as dividends to investors rather than retain them to finance future expansion because their growth opportunities may be more limited. Investors interested primarily in a stable stream of regular income would likely prefer to receive dividends. However, those investors who are willing to wait for the returns on their investment may choose to purchase shares of companies that are aggressively reinvesting their earnings in order to experience rapid growth. Presumably this growth would eventually translate into a higher share price.

Another piece of information reported in Exhibit 6.6 is the firm’s price/earnings or PE ratio. This ratio is constructed by dividing the Closing Price by the earnings reported by the firm for the prior four quarters. An average PE ratio for the stock market as a whole is around 15. Firms expected to experience rapid growth in earnings in the future will tend to have high stock prices in relation to their prior year’s earnings, and hence a high PE. Note that in the exhibit at least one firm had losses, or negative earnings, for the previous year, so its PE ratio is not reported.

Exhibit_6.6

SIMPLE INTEREST METHOD

The widely used simple interest method assesses interest charges on a loan for only the period of time the borrower actually has use of borrowed funds. The total interest bill decreases the more frequently a borrower must make payments on a loan because the borrower has less money to work with each time the repayment of part of a loan is made to the lender. This definition of a simple interest loan follows the U.S. government’s Truth-in-Lending law passed originally in 1968. For example, suppose you borrow $1,000 for a year at simple interest. If the interest rate is 10 percent, your interest bill will be $100 for the year. This figure is derived from the formula

 formula6.20

where I represents the interest charge (in dollars), P is the principal amount of the loan, r is the annual rate of interest, and t is the term (maturity) of the loan expressed in years or fractions of a year. (In this example, $1,000 X 0.10 X 1 = $100.) If the $1,000 loan is repaid in one lump sum at the end of the year,

 formula6.20a

Clearly, you pay a lower interest bill ($75 versus $100) with two installment payments instead of one. This happens because with two installment payments you effectively have use of the full $1,000 for only six months. For the remaining six months of the year you have use of only $500.

A shorthand formula for determining the total payment (interest plus principal) on a simple interest loan with a single lump-sum payment at maturity is

 formula6.21

For example, borrowing $1,000 for six months at a 10 percent loan rate means the borrower owes:

Total payment due = $1,000 + $1,000 X 0.10 X 6/12 = $1,050

The simple interest method is still popular today with many credit unions and banks.

ADD-ON INTEREST RATE

A method for calculating loan interest rates often used by finance companies and banks is the add-on rate approach. In this instance, interest is calculated on the full principal of the loan, and the sum of interest and principal payments is divided by the number of payments to determine the dollar amount of each payment. For example, suppose you borrow $1,000 for one year at an interest rate of 10 percent. You agree to make two equal installment payments six months apart. The total amount to be repaid is $1,100 ($1,000 principal + $100 interest). At the end of the first six months, you will pay half ($550), and the remaining half ($550) will be paid at the end of the year.

If money is borrowed and repaid in one lump sum (a single payment loan), the simple interest and add-on methods give the same interest rate. However, as the number of installment payments increases, the borrower pays a higher effective interest rate under the add-on method. This happens because the average amount of money borrowed declines with greater frequency of installment payments, yet the borrower pays the same total interest bill. In fact, the effective rate of interest nearly doubles when monthly installment payments are required. For example, if you borrow $1,000 for a year at 10 percent simple interest but repay the loan in 12 equal monthly installments, you have only about $500 available for use, on average, over the year. Because the total interest bill is still $100, the interest rate exceeds 18 percent.

DISCOUNT LOAN METHOD

Many commercial loans, especially those used to raise working capital, are extended on a discount basis. This so-called discount loan method for calculating loan rates determines the total interest charge to the customer on the basis of the amount to be repaid. However, the borrower receives as proceeds of the loan only the difference between the total amount owed and the interest bill. For example, suppose you borrow $1,000 for one year at 10 percent, for a total interest bill of $100. Using the discount method, you actually receive for your use only $900 (i.e., $1,000 - $100) in net loan proceeds. The effective interest rate, then, is

formula6.22

Some lenders grant the borrower the full amount of money required but add the amount of discount to the face amount of the borrower’s note. For example, if you need the full $1,000, the lender under this method will multiply the effective interest rate (11.11 percent) times $1,000 to derive a total interest bill of $111.11. The face value of the borrower’s note and, therefore, the amount that must be repaid becomes $1,111.11. However, the borrower receives only $1,000 for use during the year. Most discount loans are for terms of one year or less and usually do not require installment payments. Instead, these loans generally are settled in a lump sum when the note comes due.

HOME MORTGAGE INTEREST RATES

One of the most confusing of all rates charged by lenders is the interest rate on a home mortgage loan. Many home buyers have heard that under the terms of most mortgage loans, their monthly payments early in the life of the loan go almost entirely to pay the interest on the loan. Only later is a substantial part of each monthly payment devoted to reducing the principal amount of a home loan. Is this true?

Yes, and we can illustrate it quite easily. Suppose that you find a new home you want to buy and borrow $100,000 to close the deal. The mortgage lender quotes you an annual home mortgage interest rate of 12 percent. If we divide this annual interest rate by 12 months, we derive a monthly mortgage loan rate of 1 percent. The lender tells you your monthly payment will be $1,100 each month (to cover property taxes, insurance, interest, and principal on the loan). This means the first month’s payment of $1,100 will be divided by the lender as follows: (1) $1,000 for the interest payment (or 1% per month X $100,000); and (2) $100 to be applied to the principal of the loan, insurance premiums, taxes, and so forth. For simplicity, let’s assume the $100 left over after the $1,000 interest payment goes entirely to help repay the $100,000 loan principal. This means that next month your loan now totals just $99,900 (or $100,000 - $100). When you send in that next monthly payment of $1,100, the interest payment will drop to $999, and, therefore, $101 will now be left over to help reduce the loan principal. Gradually, the monthly interest payment will fall and the amount left over to help retire the loan’s principal will rise. After several years, as the mortgage loan’s maturity date gets near, each monthly payment will consist mostly of repaying loan principal.

How do mortgage lenders figure the amount of the monthly payment new home buyers must make on their home loan? The usual formula is

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Actually, the easiest way to calculate required home mortgage payments is by using an electronic calculator or computer software. Typically, just three pieces of data are needed—the number of payments to be made, the annual interest rate on the loan, and the amount the home buyer plans to borrow.

ANNUAL PERCENTAGE RATE

The wide diversity of rates quoted by lenders is often confusing and discourages shopping around for credit. With this in mind, the U.S. Congress passed the Consumer Credit Protection Act in 1968. More popularly known as Truth in Lending, this law requires institutions regularly extending credit to consumers to tell the borrower what interest rate he or she is actually paying and to use a prescribed method for calculating that rate. Specifically, banks, credit unions, and other lending institutions are required to calculate an annual percentage rate (APR) and inform the loan customer what this rate is before the loan contract is signed. The APR, which measures the yearly cost of credit, includes not only interest costs but also any transaction fees or service charges imposed by the lender. The APR for loans is equivalent to the yield to maturity for bonds.

Today, financial calculators and financial functions in spreadsheet programs (such as Excel) allow loan officers and their customers to easily determine the APR attached to their loans. To illustrate how the APR is determined, suppose you borrow $1,000 at 10 percent simple interest but must repay your loan in 12 equal monthly installments. The amount of each required monthly payment (PMT) can be figured as follows:

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We can enter in the calculator the number of payment periods, N (in this case, 12); the amount or present value (PV) that the lender is granting to you, the borrower, for the term of the loan (in this case, PV = - $1,000 the day the loan begins); the amount to be repaid each month (in this case, PMT = $91.67 as determined above); and the future value (FV) of the loan (which at the end of the loan’s term is $0 because you are expected to completely pay back what you borrowed). The calculator tells us that the APR (the annual percentage rate or I/Y) is very close to 18 percent in this example.

Congress hoped that introduction of the APR would encourage consumers to exercise greater care in the use of credit and to shop around to obtain the best terms on a loan. It is not at all clear that either goal has been realized completely, however. Many consumers appear to give primary weight to the size of installment payments in deciding how much, when, and where to borrow. If their budget can afford principal and interest charges on a loan, many consumers seem little influenced by the reported size of the APR and are often not inclined to ask other lenders for their rates on the same loan. Consumer education is vital to intelligent financial decision making, but progress in that direction has been slow. However, there is some evidence that with growing use of the Internet a greater proportion of borrowers are shopping around for credit today.

COMPOUND INTEREST

Some lenders and loan situations require the borrower to pay compound interest on a loan. In addition, most interest-bearing deposits at banks, credit unions, and money market funds pay compound interest on the balance in the account as of a certain date. The compounding of interest simply means that the lender or depositor earns interest income on both the principal amount and on any accumulated interest. Thus, the longer the period over which interest earnings are compounded, the more rapidly does interest earned on interest and interest earned on principal grow.

The conventional formula for calculating the future value of a financial asset earning compound interest is simply

FV = P(1 + r)t                                                                             (6.25)

where FV is the sum of principal plus all accumulated interest over the life of the loan or deposit, P is the asset’s principal value, r is the annual rate of interest, and t is the time expressed in years. For example, suppose $1,000 is borrowed for three years at 10 percent a year, compounded annually. Using an electronic calculator to find the compounding factor, (1 + r ) t , gives

FV = $1,000(1+ 0.10)3 = $1,000(1.331) = $1,331

which is the lump-sum amount the borrower must pay back at the end of three years. The amount of accumulated compound interest on this loan must be

Compound interest = FV - P = $1,331 - $1,000 = $331                      (6.26)

Increased competition in the financial institutions’ sector has encouraged most deposit-type institutions to offer their depositors interest compounded more frequently than annually, as assumed in the formula above. To determine the future value of accumulated interest from such a deposit, two changes must be made in the formula: (1) the quoted annual interest rate ( r ) must be divided by the number of periods during the year for which interest is compounded, and (2) the number of years involved ( t ) must be multiplied by the number of compounding periods within a year. For example, suppose you hold a $1,000 deposit, earning a 12 percent annual rate of interest, with interest compounded monthly, and you plan to hold the deposit for three years. At the end of three years, you will receive back the lump sum of

FV = P(1 + r/12)t X 12 = $1,000(1 + 0.12/12) 3 X 12

= $1,000(1.431) = $1,431                                                                  (6.27)

Total interest earned will be $1,431 - $1,000, or $431. Compounding on a more frequent basis increases the depositor’s accumulated interest and, therefore, the deposit’s future value.

ANNUAL PERCENTAGE YIELD

In 1991 the U.S. Congress passed the Truth in Savings Act in response to customer complaints about the way some depository institutions were calculating their customers’ interest returns on deposits. Instead of giving customers credit for the average balance in their deposit accounts, some depository institutions were figuring a customer’s interest return on the lowest balance in their account. The U.S. Congress responded to this practice by requiring depository institutions to calculate the daily average balance in a customer’s deposit over each interest-crediting period and to use that daily average balance to determine the customer’s annual percentage yield (APY) .

For example, suppose a customer deposits $2,000 in a one-year savings account for six months (180 days) but then withdraws $1,000 to help meet personal expenses, leaving $1,000 for the remainder of the year (185 days). Then the customer’s daily average balance would be:

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Whenever a customer opens a new deposit account in the United States, he or she must be informed about how interest will be computed on his or her account, what fees will be charged that could reduce the customer’s interest earnings, and what must be done to earn the full APY promised on the deposit.