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4 The answer is lOOOv = 1000 $772 ,8 (1.09) From general reasoning, the reader should justify the relative magnitude of the answers to Examples 1.3 and 1.4. 1,7 THE EFFECTIVE RATE OF DISCOUNT In Section 1.3 the effective rate of interest was defined as a measure of interest paid at the end of the period. In this section we define the effective rate of discount, denoted by d, as a measure of interest paid at the beginning of the period. A numerical illustration will help make this distinction clear. If A goes to a bank and borrows $1( for one year at an effective rate of interest of 6%, then the bank will give A $1( . At the end of the year, A will repay the bank the original loan of $1( , plus interest of $6, or a total of $106. However, if A borrows $100 for one year at an effective rate of discount of 6%, then the bank will collect its interest of 6% in advance and will give A only $94. At the end of the year, A will repay $100. Thus, it is clear that an effective rate of interest of 6% is not the same as an effective rate of discount of 6%. In the above example, A paid $6 interest in both cases. However, in the case of interest paid at the end of the year, A had the use of $100 for the year, while in the case of interest paid at the beginning of the year, A had the use of only $94 for the year. Looking at the above illustration in a slightly different manner, in the case of an effective rate of interest, the 6% is taken as a percentage of the balance at the beginning of the year, while in the case of an effective rate of discount, the 6% is taken as a percentage of the balance at the end of the year. Thus, we can formulate a precise definition of the effective rate of discount as follows: The effective rate of discount d is the ratio of the amount of interest (sometimes called the "amount of discount" or just "discount") earned during the period to the amount invested at the end of the period. This definition is analogous to the alternate definition of the effective rate of interest given in Section 1.3. Several observations about the above definition are important: 1. Observations 1, 2, and 3 in Section 1.3 on the definition of the effective rate of interest also apply to the definition of the effective rate of discount. 2. The phrases amount of discount and amount of interest can be used interchangeably in situations involving rates of discount. 3 The definition does not use the word "principal," since the definition of principal refers to the amount invested at the beginning of the period and not at the end of the period. 4. The key distinction between the effective rate of interest and the effective rate of discount can be summarized as follows: a) Interestpaid at the end of the period on the balance at the beginning of the period. b) Discountpaid at the beginning of the period on the balance at the end of the period. Some readers find the use of the word "paid" in connection with rates of discount somewhat confusing, since the borrower does not directly "pay" the interest as with rates of interest. However, the net result of deducting the interest in advance is no different than if the full amount is borrowed and then the borrower immediately pays the interest. In fact, some readers may find the use of the word "paid" confusing in another sense as well. This confusion involves the possible implication that interest is "paid" in installments as it is earned rather than being accumulated to earn additional interest. We do not attach such a connotation to the word "paid," which can be used in either context above. A term such as "credited" might be preferred by those who find the word "paid" ambiguous in this sense. Effective rates of discount can be calculated over any particular measurement period. Let rf„ be the effective rate of discount during the nth period from the date of investment. A formula analogous to (1.4fc) is dn = A(n) A(n  1) n for integral n > 1. (1.13) A(n) Ain) As mentioned above, /„ may be commonly called either the "amount of discount" or the "amount of interest." In general, d may vary from period to period. However, if we have compound interest, in which case the effective rate of interest is constant, then the effective rate of discount is also constant. The proof of this statement is left as an exercise. These situations are referred to as compound discount, a term analogous to "compound interest." The illustration discussed earlier in this section showed that an effective rate of interest of 6% is not the same as an effective rate of discount of 6%. However, there is a definite relationship between effective rates of interest and effective rates of discount. To develop this relationship we need to define a concept of equivalency as follows:
1 d (1.14) (1.15a) This formula expresses / as a function of d. By simple algebra, it is possible to express as a function of i iid = d d{\ + i) = i Formula (1.15a) is obvious, since it is merely a restatement of the definition of the effective rate of discount as the ratio of the amount of interest (discount) that 1 will earn during the period to the amount invested at the end of the period. There are important relationships between d, a rate of discount, and v, a discount factor. One relationship is identical to (1.15a) d = iv. il.lSb) This relationship has an interesting verbal inte retation. Interest earned on an investment of 1 paid at the beginning of the period is d. Interest earned on an investment of 1 paid at the end of the period is i. Therefore, if we discount / from the end of the period to the beginning of the period with the discount factor V, we obtain d. There is another relationship between d and v which is often useful 1 + i = i±i 1 + / = 1  V. 1 + / (1.16) i  d = id. (1.17) This relationship also has an interesting verbal inte etation. A person can either borrow 1 and repay 1 + / at the end of the period or borrow 1  d and repay 1 at the end of the period. The expression i  dis the difference in the amount of interest paid. This difference arises because the principal borrowed differs by d. Interest on amount d for one period at rate i is id. The effective rate of discount, or compound discount, assumes compound interest. However, it is possible to define simple discount in a manner analogous to the definition of simple interest. Consider a situation in which the amount of discount earned during each period is constant. Then, the original principal which will produce an accumulated value of 1 at the end of t periods is a\t) = ldt for 0 < t < \ld. (1.18) The second part of the inequality is necessary to keep a"(0 > 0. This contrasts with compound discount, in which case the present value is al(r) = v = (1  d) for t > 0. (1.19) It should be noted that formulas (1.14), (1.15), (1.16), and (1.17) assume effective rates of interest and discount and are not valid for simple rates of interest and discount unless the period of investment happens to be exactly one period. The reader is cautioned that simple discount is not the same as simple interest. However, simple discount does have properties analogous, but opposite, to simple interest. The proofs of the following are left as exercises: 1. A constant rate of simple interest implies a decreasing effective rate of interest, as the period of investment increases, while a constant rate of simple discount implies an increasing effective rate of discount (and interest). 2. Simple and compound discount produce the same result over one measurement period. Over a longer period, simple discount produces a smaller present value than compound discount, while the opposite is true over a shorter period. Two rates of interest or discount are said to be equivalent if a given amount of principal invested for the same length of time at each of the rates produces the same accumulated value. As we shall see in Section 1.8, this definition is applicable for nominal rates of interest and discount, as well as effective rates. Assume that a person borrows 1 at an effective rate of discount d. Then, in effect, the original principal is 1  and the amount of interest (discount) is d. However, from the basic definition of / as the ratio of the amount of interest (discount) to the principal, we obtain This relationship also can be inte reted verbally. Written in the form V = \  d, it is immediately seen that both sides of the equation represent the present value of 1 to be paid at the end of the period. There is one other relationship between i and d which is significant: d = iv = /(1  d) = iid
«() Compound interest iimple interest Simple discount Compound discount Figure 1.3 Comparisons of: (a) simple and compound interest, and (b) simple and compound discount Simple discount is used only for shortterm transactions and as an approximation for compound discount over fractional periods. It is not as widely used as simple interest. The word "discount" unfortunately is used in two different contexts with various shades of meaning in each. It is used in connection with present values (discount factor, discount function, discounting, discounted value) and in connection with interest paid at the beginning of the period (effective rate of discount, amount of discount, compound discount, simple discount). Unfortunately, terms involving the word "discount" are often misused in practice. For example, in the process of discounting, i.e. taking present values, the phrase "rate of discount" is often used when really "rate of interest" is correct. Yet another usage of the word "discount" will appear in Chapter 7. The term will be used to refer to bonds for which the price is less than the redemption value. Exacerbating the confusion even fiirther is a fourth use of the word "discount" to refer to price reductions. Although we do not attach this meaning to the word in this book, it commonly has this meaning in business and financial transactions. Needless to say, the reader should be careful in using the term "discount" to keep the meaning completely clear and should not hesitate to seek clarification from others using the term if there is any possibility of ambiguity. It is instructive to illustrate these results graphically. Figure 1.3(a) compares the accumulation function under simple interest and compound interest. Similarly, Figure 13(b) compares the discount function under simple discount and compound discount. Example 1.5 Rework Example 1.3 using simple discount instead of simple interest. The answer is !  (.09)(3)] = $730. Example 1.6 Rework Example 1.3 using compound discount instead of simple interest. The answer is , 1000(.91)3 = $753.57. From general reasoning, the reader should justify the relative magnitudes of the answers to Examples 1.5 and 1.6. 1.8 NOMINAL RATES OF INTEREST AND DISCOUNT In Sections 1.3 and 1.7 effective rates of interest and discount were discussed. The term "effective" is used for rates of interest and discount in which interest is paid once per measurement period, either at the end of the period or at the beginning of the period, as the case may be. In this section, we consider situations in which interest is paid more frequently than once per measurement period. Rates of interest and discount in these cases are called "nominal." Most persons have encountered such situations in practice. For example. Lender A might charge an annual effective rate of 9% on loans; Lender might charge 8% % compounded quarterly; and Lender might charge 8/2% payable in advance and convertible monthly. Most persons probably realize that these rates are not directly comparable, but they would probably not be able to make a valid comparison among them. Lender A is charging an annual effective rate of interest, which has already been discussed. However, Lender is charging what we call a nominal rate of interest, while Lender is charging what we call a nominal rate of discount. Various terms are used in practice to describe situations in which interest is paid more frequently than once per measurement period. Among these are "payable," "compounded," and "convertible," as in "payable quarterly," "compounded semiannually," and "convertible monthly." The frequency with which interest is paid and reinvested to earn additional interest is called the interest conversion period. The three terms "payable," "compounded," and "convertible" are often used interchangeably. However, they do have different connotations to some users. For example, the term "compounded" seems to imply that the interest being earned is reinvested to earn additional interest, while the term "payable" seems to imply instead that the interest is paid out in installments as it is earned. The term "convertible" does not seem to possess either connotation. The reader is
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