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Mathematics as a Tool The Mathematics of Finance
Target Outcomes
At the end of the lesson, you are expected to
- understand and apply concepts related to borrowing and loan repayment; and
- differentiate accumulation from discounting;
Abstraction
Lesson 8. Accumulation and Discounting
8.1 ACCUMULATION
Accumulation is the process of determining the amount (F) of a given principal (P) due at a specified time (t). To accumulate a principal (P) for t years means to solve for the final amount by applying the following formula we have discussed earlier:
For simple interest: F = P (1 + r t)
For compound interest: F = P (1 + ๐)๐^ or F = ๐ (1 + ๐๐)๐๐ก
The term P(1 + i) is often called the accumulation factor , since it accumulates the value of an investment (P) at the beginning of a period to its value at the end of the period. As you may notice, we have already illustrated and discussed accumulation in the previous lesson. Solving for the final amount, final value, or compound amount in compound interest, is โaccumulatingโ a principal (P) , to achieve the amount (F) in t years at a given interest rate.
8.2 DISCOUNTING
Discounting , on the other hand, is the process of determining how much money must be invested today in order for an investment to have a specific value at a future date. That is, determining the present value (P) of an investment before it earns any interest. To discount the amount (F) for t years means to solve for (P) by applying the following formula:
For simple interest: P = (^) 1+ rtF
For compound interest: P = (^) (1+i)FN or P = (^) (1+F r n)nt
A discount is a deduction from the final amount (F) or maturity value of a loan or obligation. A simple discount in banks is often called bank discount or interest-in- advance (Ia). The amount of money that the borrower receives is called proceeds.
To illustrate these concepts, let us consider the case of Mr. Santos, who wants to buy his son a new phone. Because money is tight, he will borrow โฑ20 000 for a year from a lender who charges 12% simple discount. The lender will deduct 12% of โฑ20 000 or โฑ2 400, which is called the amount of discount or interest in advance. The money left from this discount, which is โฑ17 600, is called the proceeds and this is how much Mr. Santos will receive. Mr. Santos will pay โฑ20 000 at the end of the year. With this, the computation of bank discount is based on the final amount or maturity value rather than the present value.
On the other hand, if Mr. Santos borrows โฑ20 000 for 1 year from a lender who charges 12% simple interest, then the lender will give him โฑ20 000. At the end of the year, Mr. Santos will repay the bank the original loan of โฑ20 000, plus interest of โฑ2 400, or a total of โฑ22 400.
Thus, it is clear that an accumulation is not the same as discounting. In the above illustration, Mr. Santos paid โฑ2 400 in both cases. However, in the case of interest paid at the end of the year (accumulation) , he had the use of โฑ20 000 for the year, while in the case of interest paid at the beginning of the year (discounting) , he had the use of only โฑ17 600 for the year.
Here is a timeline to illustrate and differentiate accumulation and discounting. The start of the line at the left is the beginning of the year and the end of the line at the right is the end of the year:
Looking at the above illustration in a slightly different manner, in the case of an accumulation, the 12% is taken as a percentage of the balance at the beginning of the year; while in the case of a discount, the 12% is taken as a percentage of the balance at the end of the year.
It is often necessary to determine how much a person must invest initially so that the balance will be (P) at the end of the period. The answer is๐น(1 + ๐)โ1, since this amount will accumulate to (P) at the end of the period. The term (1 + ๐)โ1^ is called the discount factor , since it discounts the value of an investment at the end of the period to its value at the beginning of the period.
We generalize the above results to periods of time other than one period, i.e. to find the amount which the person must invest in order to accumulate an amount of (P)
The original amount you borrowed is โฑ369 565.2174 and hence should be paid at the end of the second year. You only received โฑ255 000 because the interest deducted in advance is โฑ114 465. Ia = Fdt = 369 565.2174 (0.155) (2) = 114 465.
COMPOUND DISCOUNT
Example 1:
You thought of putting up a small to-go food business and applied for a business loan. Your loan amount is โฑ45 400 to be paid at the end of 5 years and 7 months at 10% compounded monthly. Find the interest and the proceeds from the loan.
Solution:
We first find the proceeds using the following formula: Proceeds = (^) (1+i)FN or Proceeds = F(1 + i)โN
Substitute the following values to the given formula: F = โฑ45 400 r = 10% or 0. t = 5 years and 7 months or (^6712) n = 12 i = (^) nr = 0.10 12 N = nt = 12 (^6712 ) Proceeds = ๐
(๐ + ๐ข)โ๐^ = 45 400 (๐ + ๐.๐๐๐๐ )โ๐๐ = 26 036.
Hence, you will only receive โฑ26 036.3156 and paid an interest of โฑ19 363.6844 by
Ia = P[ (๐ + ๐ข)๐^ โ ๐ ] = 26 036.3156 [(๐ + ๐.๐๐๐๐ )
๐๐ โ ๐] = 19 363.6844.
Example 2:
If you want to have โฑ15 500 after 10 months, what amount should you invest today if your money earns 8% compounded quarterly?
Solution:
We are asked to solve for the present value of โฑ15 500 given the following values: F = 15 500 r = 8% or 0. t = 10 months =^1012 n = 4 i = ๐๐ = 0.08 4 N = nt = 4 (^1012 ) =^4012 P = ๐ญ(๐ + ๐)โ๐ต = ๐๐ ๐๐๐(๐ + ๐.๐๐๐ )โ
40 (^12) = 14 509. 9013
Therefore, you need โฑ14 509.9013 to have โฑ15 500 after 10 months.
Supplementary Materials
Read chapter 10 of Aufmann, R.N. Clegg, D.K. et.al. (2010). Mathematical Excursion (2nd ed.). Brooks/Cole, Cengage Learning.
Read Chapter 5 of Johnson, D.B. and Mowry, T.A.(2012,2007). Mathematics: A practical Odyssey, (7th^ ed.). Brooks/Cole, Cengage Learning.
