In general, the interest rate is the percentage of capital or principal, expressed in hundredths, that is paid for the use of it in a certain unit of time (normally one year).
The current or market interest rate is calculated primarily on the basis of the relationship between the supply of money and the demand of borrowers. When the supply of money available for investment increases faster than the needs of borrowers, interest rates tend to fall. Similarly, interest rates tend to rise when the demand for investment funds grows faster than the supply of available funds faced by those demands.
Returning to the issue that concerns us, the existence of three different classes of capitalization requires the use of different interest rates that are adapted to each class in particular, defining them based on previously established assumptions. Thus we find:
I. Periodic capitalization
- Nominal rate (i): Also known as both by one or simply as an interest rate, it is the profit generated by a capital of $ 1 in one year; that is, it is equal to one hundredth of the ratio or as a percentage (profit produced by a capital of $ 100 in one year).
We can also define it as the annual interest rate that governs during the period of the financial operation; This means that compounding occurs in the period in which the rate is indicated.
Generalizing, when time n and the period in which rate i is expressed coincide with capitalization, rate i is said to be nominal.
It appears in the compound interest amount formula M1 = C (1 + i) n.
- Effective rate (i '): It is the amount times one that, applied to a capital C in n periods, produces an amount M2 equal to the one obtained using the proportional rate m times in each of the n periods with subperiodic capitalization.
It appears in the amount formula M2 = C (1 + i ') n, so that M2 = M3. Starting from this last equality, we can express the effective rate as a function of the proportional rate:
M2 = M3
C (1 + i ') n = C (1 + i / m) nm
1 + i' = (1 + i / m) m (We simplify C and n.) I
'= (1 + i / m) m - 1 (We solve for i '.)
II. Subperiodic capitalization
- Proportional rate (i / m): When compounding is done every fraction of time m times less than the period considered n, a rate m times less is also taken; the latter results from the quotient between the nominal rate i and the number of sub-periods m, and is the rate commonly called proportional. Thus, for example, the rates proportional to i per 1 per year are: for the semester, i / 2; for the quarter, the i / 4; for the month, i / 12 times 1; etc.
It is applied in the formula of amount M3 = C (1 + i / m) n m.
- Equivalent rates (im): They are those that, corresponding to different capitalization periods, make the capital acquire the same definitive values, also equal, after the same time.
They can also be defined as subperiodic rates that, capitalizing m times in the period, produce at the end of the same amounts as with periodic capitalization and nominal rate.
The equivalent rate is used in the amount formula M4 = C (1 + im) nm, so that M1 = M4. This last equality allows us to express the equivalent rate as a function of the nominal rate:
M4 = M1
C (1 + im) nm = C (1 + i) n
(1 + im) m = 1 + i (We simplify C and n.)
Im = (1 + i) 1 / m - 1 (We isolate im.)
III. Continuous capitalization
- Nominal rate (i): Same nominal rate, periodic compounding. However, in this case it does not appear in the base of the capitalization factor, but in the exponent of the formula of amount at continuous interest M5 = C ei n. The maximum possible amount is thus obtained Instantaneous rate (d): It is the one that, applied to a capital C in n periods with continuous capitalization, produces the same amount (M6) as that obtained when using the nominal rate i in the same time and with the same capital but with periodic capitalization (M1).
Using Differential Calculus terms, we also say that it is the change of $ 1 in an instant.
It is used in the formula of amount at continuous interest M6 = C edn, so that M1 = M6. We can express the instantaneous (constant) rate as a function of the nominal rate:
M6 = M1
C edn = C (1 + i) n
ed = 1 + i (We simplify C and n.)
D ln e = ln (1 + i) (We apply ln member to member.)
D = ln (1 + i) (Since ln e = 1, d is cleared.)
Consulted bibliography
- José González Galé, "Interests and Certain Annuities", Macchi.Murioni-Trossero Editions, "Financial Calculation Manual", Macchi Editions, 1999. Miguel M. Tajani, "Financial Mathematics", Cesarini Hnos. - Editors, 1986. Notes from class Various economics texts.
