Chapter 1

**BASIC CONCEPTS OF**

**FINANCIAL MATHEMATICS**

**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: |
FV_{1} = PV (1 + i) |

Second: |
FV_{2} = FV_{1} (1 + i) = P (1 + i)^{2} |

Third: |
FV_{3} = FV_{3} (1 + i) = P (1 + i)^{3} |

N-m: |
FV_{n} = 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:**

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:**

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**

i_{e}: |
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 |

i_{e}: |
(1 + 0.36 / 4)^{4} - 1 |

i_{e}: |
(1 + 0.009)^{4} - 1 |

i_{e}: |
1.411582 - 1 = 0.411582 |

i_{e}: |
41.1582% |