Financial math notes. simple interest, compound interest, annuities and amortizations

In the following booklet some topics from unit 1 are developed (subtopics are pending because they are topics that will be dealt with more fully in another 1.1, 1.1.1 and 1.1.3), such as: importance of financial mathematics, booklet.

financial-mathematical-notes

Concept and calculation of the simple interest rate as well as the compound interest; Unit 2 (sub-topic 2.1) includes the definition and application of the commercial discount; from unit 3 (subtopics 3.1, 3.1.1, 3.1.2, 3.1.3, 3.2, 3.2.1 and 3.2.2) simple annuities are dealt with in their different modalities: expired, anticipated and deferred as well as definition and preparation of the amortization tables. The program corresponds to the subject of Financial Mathematics (CPC-1032), which is part of the third semester of the Public Accounting degree.

The included examples have been compiled from various texts of Financial Mathematics; however, most of them have been modified, in terms of their wording, with the intention of making them easier to understand and interpret. It is expected in another semester to develop an exercise booklet that serves to exercise the topics presented.

It should be noted that in this booklet, the Simple Interest and Compound Interest subtopics as well as the annuity subtopics are developed more widely, since they establish the bases for developing mathematical thinking that allows understanding the following subtopics.

It is important to clarify that the issues related to Cetes and the Stock Market are pending as they are topics that will be dealt with in greater detail in another booklet.

1.1 IMPORTANCE OF FINANCIAL MATHEMATICS IN THE PROFILE OF THE PUBLIC ACCOUNTANT

COMPETENCE TO DEVELOP: In this unit, the competence that the student manages to develop is that of knowing, analyzing and evaluating the foundations of financial mathematics for decision-making. and the impact that the value of money has over time and its equivalence through the various capitalization factors

Financial mathematics is a type of applied mathematics that aims to achieve the maximum benefit as a buyer and the most attractive returns as an investor. As a buyer, maximum benefit when getting money borrowed, in cash, goods or services and to those who have capital, to lend it, that is, invest it if it generates interest and other benefits.

In addition, by applying the methods of valuation of money over time, the results can be interpreted to make effective decisions that yield the maximum benefits of the economic and financial interests and objectives of economic entities.

SIMPLE INTEREST

It is the amount that is paid for the use of other people's money, or the money that is earned by making our money available to third parties (banks, personal loans) through deposits in savings or loan accounts. It should also be noted that in this type of interest, only the capital earns interest for the entire duration of the transaction.

Interest is the amount paid for using the money requested as a loan, or the amount obtained from the investment of some capital.

If we designate C to a certain amount of money at a given date, which we will call moment zero, whose value increases to S at a later date, then we have to

? = ???

Where

K = It is the initial capital that serves as a basis to generate interest, either for a loan or for an investment. I = It is the amount paid for the use of money. T = Time. It is the number of periods (years, months, days, etc.) that the capital remains borrowed or invested. I = Interest rate. It is the ratio of the interest accrued with respect to the initial capital; that is, it is the amount that when multiplied by the initial capital results in the interest accrued in a given period of time.

FORMULAS

? = ???

If we clear K, i and t we will have the following formulas:

? =

? ?? ? ??

? =

NOTE. To apply the above formulas, the data on the interest rate and time must refer to the same unit of measurement, that is, if the interest is annual, the time will be expressed annually; if the time is expressed monthly, the interest must be obtained per month.

i = 12% ????? So to use them in the formulas it will be from the

_{t = 4 ?????} Following way:

i =.12 ?????

t ==.33? ñ ??

Both i and t remain

expressed in the same unit of measurement, that is, in years.

SIMPLE EXACT AND ORDINARY INTEREST

Regarding this point, we must point out that the exact simple interest, considers, to make its calculations, a time base of 365 for one year and 30 and 31 days as marked on the calendar for each month. For its part, ordinary simple interest, which is the most used, considers a time base of 360 days for the year and 30 days for the months.

EXAMPLE. Determine the exact and ordinary simple interest over $ 2,000.00 at 5% for 50 days.

EXACT TIME Convert days to years: Resolution

Data:

K = 2,000 ^{t =} ^{? = ???} _i = 5% per year

Compound interest is used primarily for deposits in banks and in savings and loan associations. These companies use the money deposited to make loans to individuals or businesses. When money is deposited in a bank, the depositor is lending money to the bank for an indefinite time, in order to earn interest.

BASIC CONCEPTS

Capitalization or conversion period. It is the agreed time interval, in the obligation, to capitalize the interests; This interval can be annual, semi-annual, quarterly, monthly, etc.

Capitalization or Conversion Frequency. Number of times that interest is added to principal in a year.

?? Where:

?? = ^{# ??} fc = conversion frequency # mc = number of months spanning the conversion period

EXAMPLE: What is the conversion frequency (fc) for a bank deposit that pays 5% interest compounded quarterly?

Data: 12

?? = = 4

# mc = 3 ³

Interest rate per period

Where:

? i = annual interest rate

? = fc = conversion frequency

EXAMPLE: What is the conversion frequency and the interest rate per period (r) at 60% per year capitalized monthly, of any operation?

Data: ?? = = 12? =. 60 =.05 i = 60% ¹²

NOTES: It is very important that for the solution of compound interest problems, the annual interest be converted at the corresponding rate according to the capitalization period established.

Whenever it is indicated that the interest rate is capitalizable, the annual rate must be converted to the interest rate per period, that is, the formula of interest rate per period must be applied

Total periods: Total number of periods covered by the operation, that is, the number of times interest will be capitalized during the entire duration of the operation.

? = (????? ??? ñ ??) (??) Where:

????? ????? n = Total periods

? =

# ??

EXAMPLE: Determine the interest rate per period (r) and the number of compounding periods (n) for an investment at 9% compound annual interest, for 10 years.

Data:

i = 9% per year t = 10 years ?? = 12 = 1 _{? =}. ₌.09? = (10) (1)

? = 10

? = 120

12? = 10

NOTE: Every time they calculate n, specify if they are semesters, quarters, etc. Do the same for the case of r.

COMPOUND AMOUNT

Deduction of the formula

Year 1 K + Ki = K (1 + i)

Year 2 K (1 + i) + {K (1 + i)} i = K (1 + i) * (1 + i) = K (1 + i) ²

Year 3 K (1 + i) + {K (1 + i)} i + i = K (1 + i) ² * (1 + i) = K (1 + i) ³ So at the end of n years we will have:

Where:

? =? (? +?) ? S = Compound amount

C = Capital or compound present value r = interest rate per period

n = total number of periods

Deduction of the compound amount formula with example

EXAMPLE: A principal of $ 1,000.00 is deposited at an interest rate of 3% per year, if the deposit is not withdrawn and the interest is reinvested, every year, for 3 years. What will be the compound amount at the end of those 3 years and what amount represents the interest?

Data: Conversions Resolution

C = 1,000.00

PRESENT VALUE OF A DEFERRED ANNUITY

Formula:

? _?? = ?? - (? +?) ^-? (? +?) _-? Where:

? ? _?? = present value of a deferred annuity

EXAMPLE: Calculate the present value of a rent of $ 5,000.00 semiannual, if the first payment is to be received within 2 years and the last payment within 6 years.

Consider the 8% interest rate convertible semi-annually.

Data: Conversions Resolution

i R = 5,000.00 = 8% conv / wk ?? = 126 = 2 ??? = ?? - (?? +?) (? +?) -? t =

# mc = 6? ==.04? _?? = 5,000 1 - (1 +.04) ⁻⁹ (1 +.04) ₋₃

? _?? =?. 04

m? == 94-1 = 3 ??? = 5,000 (. 8889)

? _?? = 5,000 (. 8889)

? _?? = 5,000 {7.43} (. 8889)

? _?? = 5,000 (. 6.60)

? _?? = ??, ???. ??

AMOUNT OF AN ANNUITY DIRECTED

The amount can be calculated as that of a past due annuity (to know how to calculate it, refer to the exercise on page 24), and in this case, postponing it no longer has any effect on the behavior of the annuity. That is why the consideration of whether the annuity is deferred or immediate, is of no interest when what is required to determine is the amount.

3.2 AMORTIZATION

Amortization is a way to liquidate or gradually reduce a debt through periodic payments, generally equal, that cover both part of the interest and part of the total value of the debt (original capital).

EXAMPLE: If today you acquire a debt of $ 5,000.00 with interest at 5% convertible semi-annually that will be amortized in 6 semi-annual payments in the next 3 years, the first at the end of 6 months.

It indicates what type of annuity it is: Since the first payment is made after the first 6 months of the operation, it is deduced that it is past due Record the data Find the value of the partial payments and


		INTEREST CONTAINED IN THE PAYMENT *E r**	CAPITAL CONTAINED IN THE PAYMENT AB	ACCUMULATED PAID CAPITAL C + D *	BALANCE INSOLUTO
Initial capital -D
0					5,000.00
one	907.75	125.00	782.75	782.75	4,217.25
two	907.75	105.43	802.32	1,585.07	3,414.93
3	907.75	85.37	822.38	2,407.45	2,592.55
4	907.75	64.81	842.94	3,250.38	1,749.62
5	907.75	43.74	864.01	4,114.39	885.61
6	907.75	22.14	885.61	5,000.00	0.00