Continuous Annuities | Exam FM | Financial Mathematics Lesson 15 - JK Math

Summary

This script explores the concept of continuous annuities, a theoretical financial model where payments are made continuously over time rather than at discrete intervals. It explains how to calculate the future and present values of such annuities using calculus, specifically integrals of accumulation and present value factors. The script also discusses the force of interest and provides formulas for continuous annuities when interest is defined with a force of interest. An example problem illustrates the application of these formulas, offering a practical understanding of continuous annuities in financial scenarios.

Takeaways

• 📚 The lesson introduces continuous annuities, which are theoretical financial instruments where payments are made continuously over time rather than at specific points.
• 🤔 The concept of continuous payments is hard to conceptualize in real life, as actual payments cannot be made continuously, but it serves as a useful theoretical scenario for understanding annuity calculations.
• 🧮 To calculate the future value of continuous annuity payments, calculus is used, specifically integrating the accumulation function \(1 + i\)^t from 0 to \(n\).
• 📉 The integral of the accumulation function results in a formula involving the force of interest, which is the natural logarithm of \(1 + i\), where \(i\) is the interest rate per payment period.
• 🔢 The future value of a continuous annuity is given by the formula \((1 + i)^n / \text{force of interest} - 1\), which simplifies to a comparison with the future value of an annuity immediate.
• 📉 The present value of a continuous annuity is calculated similarly by integrating the present value factor, resulting in a formula that also involves the force of interest.
• 🔄 The formulas for continuous annuities can be derived from the formulas for annuities with discrete payments by considering the limit as the payment frequency approaches infinity.
• 💡 The force of interest is a key concept in continuous annuity calculations, representing the exponential growth or decay of money over time when interest is compounded continuously.
• 📝 The script provides a detailed example of calculating both the present and future values of a continuous annuity with a given set of parameters.
• 🔗 The lesson concludes with a comparison of continuous annuity formulas to those of annuities immediate, highlighting the difference in the denominators of the formulas.
• 📚 Additional formulas for continuous annuities are presented when the interest is defined with a force of interest, involving exponential functions and integrals.

Q & A

What is a continuous annuity?

A continuous annuity is a theoretical financial concept where payments are made continuously over a period of time, rather than at specific points in time. It's a useful model for theoretical purposes but is not practical for real-life scenarios.

Why are continuous annuities considered theoretical rather than practical?

Continuous annuities are theoretical because actual payments cannot be made continuously in real life. They serve as a useful mathematical model for understanding financial concepts over time.

How can the future value of continuous payments be determined?

The future value of continuous payments can be determined using calculus, specifically by integrating the accumulation function (1 + i)^t with respect to t from 0 to n.

What is the formula for the future value of a continuous annuity?

The formula for the future value of a continuous annuity is (1 + i)^n / (natural log of (1 + i)) - 1, where n is the number of payment periods and i is the annual effective interest rate.

What is the force of interest and how is it related to the formulas for continuous annuities?

The force of interest is the natural logarithm of (1 + i), where i is the interest rate. It is used in the formulas for continuous annuities as the denominator in both the future and present value calculations.

How is the present value of a continuous annuity calculated?

The present value of a continuous annuity is calculated by integrating the present value factor, which is (1 / (1 + i))^t, with respect to t from 0 to n.

What is the relationship between the formulas for continuous annuities and annuity immediates?

The formulas for continuous annuities and annuity immediates are similar, with the main difference being the denominator: for continuous annuities, it is the force of interest, while for annuity immediates, it is the interest rate i.

Can the formulas for annuity immediates be converted to formulas for continuous annuities?

Yes, by multiplying the formulas for annuity immediates by i/delta (where delta is the force of interest), you can obtain the equivalent formulas for continuous annuities.

What is the example problem presented in the script, and what is the solution?

The example problem involves calculating the present and future values of a continuous annuity that pays $100 continuously throughout the year for 12 years with an annual effective interest rate of 7%. The present value is calculated to be $821.75, and the future value is $1850.75.

What is the significance of the integral in the formulas for continuous annuities?

The integral is used to sum up the effects of all the continuous payments over the period of the annuity. It allows for the calculation of the accumulated value or the future value at a specific time n for continuous payments.

How does the script mention the use of exponential notation in the formulas?

The script mentions exponential notation, exp(x), as e raised to the power of x, where e is the base of the natural logarithm. This notation is used in the formulas for continuous annuities when integrating the accumulation and present value factors.