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Chapter 1

BASIC CONCEPTS OF
FINANCIAL MATHEMATICS

 

by josavere

1. VALUE OF THE MONEY THROUGH TIME

The money is an asset that cost with the pass of time; the interest rates are received periodic (monthly, quarterly, etc.). In finances, is understood the composed interest: interests produced periodically turn automatically into capital.

Example:

If I place $1,000,000 (PV) to an interest rate (i) 3% monthly, when finishing the first period, the capital is equal to $1,030,000 and the new interest will be 3% of this number, and thus, successively.

PV: initial capital
i: interest rate
n: number of periods (in the same ones that appears the rate)
FV: value of the PV plus the gained interests

Periods:

First: FV1 = PV (1 + i)
Second: FV2 = FV1 (1 + i) = P (1 + i)2
Third: FV3 = FV3 (1 + i) = P (1 + i)3
N-m: FVn = FV (n - 1) (1 + i) = PV (1 + i)n

KEY FOR THE ANALYSIS:

To correctly process a graph that indicates the investments () and the outcome
(­) in the exact date that are considered display.

Using Excel is possible to solve any situations that may appear.

 

2. BASIC PROBLEMS

A. TO CALCULATE A FUTURE VALUE: knowing a real value now (present value), the interest rate and the number of periods (expressed in the same unit that i is defined).

Example: PV: $1.000.000; i: 2.5%; n: 24.

FV = PV (1 + i)n = 1000000 (1 + 0.025)24 = 1.808.725,95

Now, using Excel; steps to follow:

a. Open Excel.

b. Click in functions (fx).

c. Select in the left menu in "Financial" category. In the menu of the right, it names of function "VF".

d. Click in button to accept (it appears a window)

e. In the box that ask for the following information:

  • Interest rate(2.5%).
  • Number of periods: (24).
  • Payments (to place 1, in this model is unique payment).
  • VA "present value" (- 1.000.000)

f. Click in button to accept

g. To evaluate the answer (1.808693.601)

B. DEFINED A FUTURE VALUE: to calculate a present value that must invest now for accumulate a sum in a defined time with a known interest rate.

Example: if I need to accumulate $5.000.000 at the end of the third year, how much money I must deposit today if 1,5% interests pay to me to monthly?

Now, calculate the same using Excel; steps to follow:

a. Open Excel.

b. Click in functions (fx).

c. Select in the left menu in "Financial" category. In the menu of the right, it names of function "VF future value".

d. Click in button to accept (it appears a window).

e. In the box that ask for the following information:

•      Interest rate (1.5%).
•      Number of periods: (36).
•      Payments (to place 1, in this model is unique payment).
•      VA "present value" (5.000.000)

f. Click in button to accept

g. To evaluate the answer (- 2.925.476.337)

C. TO CALCULATE THE FUTURE VALUE (FV) WITH PERIODIC PAYMENTS:

One appears when equal and periodic payments become (payments), knows the number of periods (n) and the interest rate by every period.

 

Example: if saving monthly a defined number, to one appraises previously agreed, during a number of periods decided how much money I reach to reunite?

Numerical exercise: what capital will be at the end of 15 months, if monthly $50,000 is deposited in an investment that recognizes 1,5% by month?

Now,using Excel; steps to follow:

a. Open Excel.

b. Click in functions (fx).

c. Select in the left menu in "Financial" category. In the menu of the right, the function "VF future value".

d. Click in button to accept (it appears a window)

e. In the box that ask for the following information:

  • Interest rate (1.5%).
  • Number of periods: (15).
  • Payments (50.000)

f. Click in button to accept.

g. To evaluate the answer (- 834.106.8888)

D. TO CALCULATE THE PERIODIC PAYMENTS (PMT): when is wanted to accumulate a future value (well-known), is defined a period of time to make it (n) and the interest rate (i), expressed in the same period of the schedule, how much one must contribute each pay?

Example: how much I must save during 10 months to have $1.500.000 in the end, if  the interest  rate is 1,5% monthly cash.

Now, we calculate the same, but using Excel, steps to follow:

a. Open Excel.

b. Click in functions (fx).

c. Select in the left menu in "Financial" category. In the menu of the right, the function "VF".

d. Click in button to accept (it appears a window).

e. In the box that ask for the following information:

  • Interest rate (1.5%).
  • Number of periods: (10).
  • Future value: (1.500.000) (number to accumulate)

f. Click in button to accept.

g. To evaluate the answer (- 2.925.476.337)

E. TO CALCULATE A PRESENT VALUE (PV) OF PERIODIC PAYMENTS:

To calculate a present value (PV) that gain interest rate(i), which allows receiving a defined periodic sum (well-known), during a previously decided time. If, I want to receive a monthly amount previously defined (n); the interest rate suitable, how much must give today?

 

Example: to have a rent of $500,000 monthly, during 60 months, which must be the initial investment, if the interests are recognized by month is 1,5%?

Steps to follow:

a. Open Excel.

b. Click in functions (fx).

c. Select in the left menu in "Financial" category. In the menu of the right, take "Payment".

d. Click in button to accept (it appears a window).

e. In the box that ask for the following information:

  • Interest rate: (1.5%)
  • Number of periods: (60)
  • Future value: 500.000 (number to accumulate)

f. Click in button to accept.

g. To evaluate the answer (- 19.690.134.44)

F. TO CALCULATE INTEREST RATE AND THE PERIODS: knowing the other elements: periodic payments, number of periods, interest rate and a present or future value, according to the case. Using the computer or the Excel, the i (interest rate) and the n can be found (periods).

Let us illustrate of once with examples:

I. Interest rate: as it is the interest rate that duplicates 1.000.000 of pesos in 24 months.

Steps to follow:

a. Open Excel.

b. Click in functions (fx).

c. Select in the left menu in "Financial" category. In the menu of the right,
it names of function "Rate".

d. Click in button to accept (it appears a window).

e. In the box that ask for the following information:

  • Interest rate (1.5%)
  • Number of periods (24)
  • VA (- 1.000.000)
  • VF (2.000.000)

f. Click in button to accept.

g. To evaluate the answer (2.9302237%)

II. To calculate the number of periods: as soon as time I accumulate 2.000.000 dollars, investing 1.000.000 to r = 2,9302237% the monthly one.

Steps to follow:

a. Open Excel.

b. Click in functions (fx).

c. Select in the left menu in "Financial" category. In the menu of the right, it names of function "Nper" that it means I number of periods.

d. Click in button to accept (it appears a window).

e. In the box that ask for the following information:

  • Interest rate (2.9302237%)
  • VA (- 1.000.000)
  • VF (2.000.000)

f. Click in button to accept.

g. To evaluate the answer 24 months.

G. FORMULA TO CALCULATE THE EFFECTIVE RATE OF INTEREST

ie: Effective interest rate
i: Nominal interest rate
t: Periods of capitalization

Example: if the nominal rate is of annual 36% with quarterly capitalization, how much is the effective interest?

t: 4 trimesters
ie: (1 + 0.36 / 4)4 - 1
ie: (1 + 0.009)4 - 1
ie: 1.411582 - 1 = 0.411582
ie: 41.1582%

 

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