When you rent an apartment, usually you are required to pay rent for each month at the beginning of the month. In other words, you pay the rent before you have the use of the apartment. We said [Section (1.1)] that interest may be thought of as a rent for the use of the investor’s money. It is therefore not surprising that there are financial arrangements in which the interest must be paid by the borrower before the borrowed money becomes available.
3A certificate of deposit (CD) requires the investor to deposit money to the issuing bank or savings and loan for a fixed term. Should the investor decide to withdraw deposited funds before the end of the term, there is usually a substantial penalty — perhaps one quarter’s interest payment — but withdrawals of interest are usually allowed. Liquid CDs may allow one or more partial withdrawals without penalty before the CD matures. With a traditional CD, the interest rate is fixed at the time the account is opened.
However, the CD market has expanded to include market-linked CDs and CDs for which the investor may request one-or-more adjustments to the interest rate should interest rates go up. Most often, additional funds may not be added to a CD once the account has been opened; however, some CDs allow the customer to make a limited number of additional deposits.
When money is borrowed with interest due before the money is re-leased, we describe the relationship using discount rates. If an investor lends $K for one basic period at a discount rate D, then the borrower will have to pay $KD in order to receive the use of $K. Therefore, in-stead of having the use of an extra $K, the borrower only has the use of an extra $K $KD D .1 D/$K. The quantity $KD is called the amount of discountfor the loan.
Note that in Section (1.3), we defined the amount of interest on an interval, and now we have defined the amount of discount on an interval. In any cashflow with a beginning and ending balance and no withdrawals or deposits, the amount of interest and the amount of discount are the same; they are both equal to the change in the balance.
EXAMPLE 1.6.1
Problem:Chan borrows $1,000 at a discount rate of 7%. How much extra money does he have the use of?
Solution In order to get the $1,000, he must first pay .:07/$1;000 D $70. Chan therefore has the use of an extra $1;000 $70 D $930:
In our example, the discount rate is 7% D $70
$1;000D $1;000 $930
$1;000 :
In other words, the discount rate for the loan period may be obtained by first calcu-lating the difference between the stated amount of the loan and the amount of extra money actually available, then dividing this difference by the stated amount of the loan.
We wish to define an effective discount rate analogous to the effective interest rate [defined by (1.3.5)]. Suppose that the loan period is the interval Œt1; t2from time t1to t2. Also suppose that at time t1the borrower will have the use of an extra $a.t1/.
At time t2 this debt will have grown to $a.t2/. Therefore, the amount of discount for the interval Œt1; t2 is $a.t2/ $a.t1/;and the discount per dollar borrowed at a discount is $
a.t2/ a.t1/ a.t2/
. We define the effective discount rate for the interval Œt1; t2to be
(1.6.2) dŒt1;t2D a.t2/ a.t1/ a.t2/ :
Comparing definition (1.6.2) with definition (1.3.5), we see that the definitions for iŒt1;t2 and dŒt1;t2 have the same numerators but different denominators. To
compute the interest rate iŒt1;t2, your denominator is the accumulated amount a.t1/ at the beginning of the interval Œt1; t2. To compute the discount rate dŒt1;t2, you divide by the accumulated amount a.t2/at the end of the interval Œt1; t2.
If, as is commonly the case, AK.t / D Ka.t/, then
(1.6.3) dŒt1;t2D AK.t2/ AK.t1/ AK.t2/ :
Recall that if n is a positive integer, the interval Œn 1; nis called the n-th time period. We agree to write dnfor dŒn 1;n. Thus
(1.6.4) dnD a.n/ a.n 1/
a.n/ and a.n 1/ D a.n/.1 dn/:
Compare this definition with the definition of ingiven in Equation (1.3.7).
EXAMPLE 1.6.5 Computing interest and discount rates; Solution includes important calculator information on using stored intermediate results
Problem:Suppose that the growth of money is governed by the accumulation func-tion a.t/ D .1:05/2t.1 C :025t/. Find d4and i4.
Solution Note that a.4/ D .1:05/2.1:1/ D 1:21275 and a.3/ D .1:05/32.1:075/
1:156624568. Therefore d4 D a.4/ a.3/a.4/ :046279474 and i4 D a.4/ a.3/a.3/ :048525195.
Note that in our computations, we will always use the stored values resulting from previous calculations and these may include more places of accuracy than we have reported; we usually only report the displayed value resulting from 9-formatting. For example, the calculator stores 1.156624567708 fora.3/ rather than the announced 1.156624568. However, if you use the number 1.156624568 (displayed for a.3/ if you have 9-formatting) instead of the stored value 1.156624567708, your calculated value fori4will be
approximately .048525194.
As was the case in Example (1.6.5), usually iŒt1;t2and dŒt1;t2 are not equal.
However, they are clearly related. With an eye to pursuing this relationship, we define what it means for a rate of interest and a rate of discount to be equivalent.
IMPORTANT DEFINITION 1.6.6
A rate of interest and a rate of discount are said to be equivalent for an interval Œt1; t2if for each $1 invested at time t1, the two rates pro-duce the same accumulated value at time t2. More generally, two dif-ferent methods of specifying an investment’s growth (over a given time period) are called equivalent if they correspond to the same accumula-tion funcaccumula-tion.
Focus on a loan lasting from time t1to time t2. If the loan is for an amount $L and the interest rate is iŒt1;t2, then we must repay $L.1 C iŒt1;t2/. On the other hand, if the loan is made at a discount with discount rate dŒt1;t2and the repayment amount is $L.1 C iŒt1;t2/, then the borrower walked away at the beginning of the loan period with $L.1 C iŒt1;t2/.1 dŒt1;t2/of the lender’s money. Consequently, on the interval Œt1; t2, an interest rate of iŒt1;t2is equivalent to a discount rate of dŒt1;t2precisely when $L D $L.1 C iŒt1;t2/.1 dŒt1;t2/for all loan amounts $L. It follows that the rates are equivalent if and only if
(1.6.7) 1 D .1 C iŒt1;t2/.1 dŒt1;t2/:
Equation (1.6.7) may also be algebraically derived using Equations (1.3.5) and (1.6.2), but this demonstration is less instructive from an interest theory point of view.
Equation (1.6.7) is equivalent to
0 D iŒt1;t2 dŒt1;t2 iŒt1;t2dŒt1;t2 which in turn gives rise to
(1.6.8) iŒt1;t2D dŒt1;t2
1 dŒt1;t2
and
(1.6.9) dŒt1;t2D iŒt1;t2
1 C iŒt1;t2
:
If we have a positive interest rate iŒt1;t2, it follows from equation (1.6.9) that the discount rate dŒt1;t2is less than iŒt1;t2.
Recalling (1.6.4) and that we write dnfor dŒn 1;n, (1.6.7), (1.6.8), and (1.6.9) respectively, give us the equations
(1.6.10) .1 C in/.1 dn/ D 1;
(1.6.11) inD dn
1 dn
;
and
(1.6.12) dnD in
1 C in:
EXAMPLE 1.6.13 Discount rates and compound interest
Problem:Suppose that an account is governed by compound interest at an annual effective interest rate of 8%. Find an expression for dn, the discount rate for the n-th year.
Solution Since the account is governed by compound interest at an annual effective rate of 8%, in D :08 for all positive integers n. Therefore (1.6.12) yields dn D
:08
1C:08 D 1:08:08 D 272;a constant. We will return to constant dnin Section (1.9).