Conversión de tasas de interés

Introduction to Financial Engineering

Overview of the Class

• The class focuses on economic engineering or financial mathematics, continuing from previous discussions about nominal and effective interest rates.

• Today's topic is the conversion of interest rates, specifically how to change nominal rates to periodic rates, effective rates, and vice versa.

Converting Nominal Rates

Changing Nominal to Periodic Rates

• To convert a nominal rate to a periodic rate, divide the nominal rate by 100 and then by the number of compounding periods in a year. For example, a 24% nominal rate compounded monthly results in a 2% monthly rate.

• In another case with a 12% nominal rate compounded quarterly, dividing gives us a 3% quarterly rate after considering there are four quarters in a year.

• A 30% nominal rate compounded semi-annually converts to a 15% semi-annual rate using similar calculations (30/100 = 0.3; then divided by 2).

• For bimonthly compounding at a nominal rate of 30%, it results in a 5% bimonthly rate since there are six bimonths in one year.

Converting Nominal Rates to Effective Rates

Formula for Conversion

• The formula for converting from nominal (j) to effective (i) involves capitalizations per year (m) and the period (n). This is crucial when calculating effective annual rates from monthly or other periodic rates.

Example Calculations

1. Example with Monthly Compounding:

• A nominal interest of 24% compounded monthly leads us through calculations yielding an effective annual interest of approximately 26.82%. The steps involve substituting values into the formula correctly.

2. Example with Quarterly Compounding:

• Converting a 12% nominal interest compounded quarterly into an effective semiannual interest requires understanding that n equals two for semestral periods; this yields an effective semestral interest of about 6.09%.

3. Daily Compounding Example:

• A daily compounding scenario at an annualized nominal rate of 36%, where we consider commercial days (360), leads us through complex calculations resulting in an approximate effective bimestral interest of around 6.18%. This highlights different approaches based on whether daily compounding is considered commercial or real time-based on days per year used for calculation purposes.

Practice Exercises

Proposed Exercises

• Students are encouraged to practice conversions with given scenarios:

• Convert 16% capitalizable quarterly into an effective monthly.

• Convert 18% capitalizable semiannually into an effective quarterly.

• Convert 20% capitalizable weekly into an effective daily commercial.

These exercises will help reinforce understanding and application of concepts discussed throughout the lesson while providing answers for self-assessment purposes at the end of the session.

Conversion of Effective to Nominal Interest Rates

Understanding the Conversion Process

• The discussion begins with converting an effective interest rate to a nominal rate, using the established formula where j represents the nominal rate and m indicates the frequency of capitalizations.

• For a 3% effective quarterly interest rate, we substitute known values into the formula. Here, m is set to 12 for monthly capitalizations since there are 12 months in a year.

• With all necessary data at hand, including the effective rate (3% or 0.03), we calculate how many quarters fit into a year (4). This leads us to compute (1 + 0.03/4)^4 = 1.1255 .

• After performing calculations, we derive that subtracting one from our result and multiplying by 12 gives us an approximate nominal interest rate of 11.88% for monthly capitalization.

Further Examples and Calculations

• Next, we convert a 7.5% effective interest rate to a nominal quarterly capitalized rate. We identify that m = 3 , as there are three four-month periods in a year.

• The calculation involves determining how many semesters fit into a year (2). Following similar steps as before leads us to find that this results in an approximate nominal interest rate of 14.81%.

Proposed Exercises

• The video presents exercises requiring conversion of different rates: converting a 1% effective monthly rate to a bimonthly nominal rate and changing a 6% annual effective rate to one that is quarterly capitalized.

Conclusion and Future Topics

• The session concludes with an invitation for viewers to ask questions regarding financial mathematics topics such as equivalence of rates—how to convert between different types of nominal and effective rates will be discussed in future videos.