PENURUNAN PERSAMAAN PERPETUITAS DAN ANUITAS

Summary

This presentation covers key topics in business mathematics, including present value, perpetuities, annuities, growing annuities, and annuities due. The group provides clear examples, demonstrating how to apply formulas to real-world financial situations. The script also includes detailed calculations, such as determining the current value of investments and understanding the impact of interest rates and growth rates on future values. The presenters systematically explain these concepts, offering a comprehensive overview of financial math used in investment and savings planning, aimed at helping viewers grasp the practical applications of these mathematical principles.

Takeaways

• 😀 The group is presenting a topic on the derivation of financial equations related to perpetuities and annuities in business mathematics and finance.
• 😀 The presentation is structured into seven sections, including fundamental equations, perpetuity, annuities, growing perpetuities, growing annuities, and perpetuities due.
• 😀 The basic equation for future value (FV) and present value (PV) in financial calculations is provided, with a formula: FV = PV(1 + i)^n.
• 😀 The example problem calculates the present value (PV) of an investment that is worth 150 million after 8 years, with a return rate of 6% per year. The result is 94,111,855.
• 😀 The perpetuity formula is introduced, where perpetuity represents an infinite series of cash flows, with the formula PV = A / i, where 'A' is the cash flow, and 'i' is the interest rate.
• 😀 An example of a perpetuity investment is shown, where the present value of a lifelong return of 10 million per year at a 6% interest rate results in a PV of 166,666,666.
• 😀 The annuity formula is introduced, which calculates the present value of periodic cash flows, with the formula PF = A * ((1 - (1 + i)^-n) / i).
• 😀 The example for annuities shows the calculation of an investment’s present value, offering 10 million per year starting from the fifth year at a 6% return, resulting in a present value of 132,015,610.5.
• 😀 The concept of growing perpetuities is explained, where cash flows grow at a rate 'g' every period, and the formula used is PV = A / (i - g), where 'g' is the growth rate and 'i' is the discount rate.
• 😀 The growing annuity formula is provided, and an example is solved showing how to calculate the present value of a growing cash flow stream, where a 5% growth in cash flows at a 7% discount rate results in a PV of 1 billion.

Q & A

What is the formula for calculating the present value (PV) from a future value (FV)?

The formula for calculating the present value (PV) from a future value (FV) is: PV = FV / (1 + i)^n, where FV is the future value, i is the interest rate, and n is the number of periods.

How do perpetuities differ from regular annuities?

Perpetuities are cash flows that continue indefinitely, whereas annuities involve cash flows over a limited number of periods. The present value of a perpetuity is calculated using the formula PV = A / i, where A is the annual cash flow, and i is the interest rate.

What is the formula for calculating the present value (PV) of an annuity?

The formula for calculating the present value (PV) of an annuity is: PV = A × (1 - (1 + i)^(-n)) / i, where A is the payment per period, i is the interest rate per period, and n is the number of periods.

What is the key difference between growing perpetuities and growing annuities?

A growing perpetuity involves cash flows that grow at a constant rate indefinitely, while a growing annuity has cash flows that grow at a constant rate but only for a limited period. The present value of a growing perpetuity is calculated as PV = A / (i - g), where g is the growth rate.

How is the present value of a growing perpetuity calculated?

The present value of a growing perpetuity is calculated using the formula: PV = A / (i - g), where A is the initial cash flow, i is the interest rate, and g is the growth rate of the cash flows.

In the provided examples, how is the present value of an investment with constant cash flow calculated?

In the examples provided, the present value of an investment with constant cash flow is calculated using the perpetuity formula PV = A / i. For instance, an investment offering 10 million per year at a 6% return has a present value of approximately 166.67 million.

What is the formula for the present value of an annuity due?

The formula for the present value of an annuity due is: PV = A × (1 - (1 + i)^(-n)) / i × (1 + i), where A is the payment per period, i is the interest rate per period, and n is the number of periods.

How does the growing annuity formula differ from the growing perpetuity formula?

The growing annuity formula takes into account a limited number of periods, and is calculated as PV = A / (i - g) × [1 - (1 + g)^n / (1 + i)^n], where A is the annual cash flow, i is the interest rate, g is the growth rate, and n is the number of periods.

In the case of Doni's lottery decision, what would the present value of the payments be if he receives 2.5 million every three months with a 14% annual return?

The present value of Doni's lottery payment option, where he receives 2.5 million every three months with a 14% annual return, is calculated by adjusting the interest rate for quarterly payments, and the result is approximately 71.43 million.

How do growing annuities impact retirement savings, as demonstrated in the example with Wira's investment?

Growing annuities allow for increased retirement savings by considering periodic increases in the amount saved. In Wira's case, with monthly contributions of 1 million over 25 years at a 6% return, the present value of the investment at the end of 25 years is 692.99 million.