Review of Financial Mathematics (CPA FRM)

Summary

This session covers fundamental concepts in financial mathematics, focusing on the valuation of bonds and net present value (NPV). It explains the future and present value of single and series cash flows, using practical examples such as investing $100,000 at 8% annual interest. The session also delves into more complex topics like perpetuities and annuities, showcasing how to calculate their present values. Emphasizing the importance of interest rates and time periods, this session provides valuable insights for anyone looking to understand the principles behind investment and financial decision-making.

Takeaways

• 😀 The future value (FV) of a single cash flow can be calculated using the formula: FV = PV * (1 + r)^n, where PV is the present value, r is the interest rate, and n is the number of periods.
• 😀 To calculate the future value, an example was provided with a $100,000 investment at 8% interest for 5 years, resulting in $146,932.81.
• 😀 Scientific calculators can be used to calculate future value by entering the principal, multiplying by (1 + interest rate), raising to the power of the number of periods, and solving.
• 😀 The present value (PV) of a single cash flow can be calculated by rearranging the future value formula: PV = FV / (1 + r)^n.
• 😀 An example of the present value calculation showed that $10,000 received in 6 years at 10% interest has a present value of $5,645.
• 😀 When calculating the future value of a series of cash flows, the future value of each individual cash flow is calculated separately, then summed.
• 😀 An example with three cash flows ($200 in Year 1, $400 in Year 2, and $600 in Year 3) showed that the future value at Year 3 was $1,265.28.
• 😀 To calculate the present value of a series of cash flows, each cash flow is discounted back to the present using the formula PV = Σ(Cn / (1 + r)^n).
• 😀 For example, three cash flows ($200, $400, and $600) at 10% interest resulted in a present value of $963.19.
• 😀 Perpetuities are cash flows that continue forever. The present value of a perpetuity is calculated as PV = C / r, where C is the cash flow and r is the interest rate.
• 😀 A growing perpetuity is a cash flow series that grows at a constant rate. The present value of a growing perpetuity is calculated using the formula PV = C1 / (r - g), where C1 is the first cash flow, r is the interest rate, and g is the growth rate.

Q & A

What is the formula for calculating the future value of a single cash flow?

The formula for the future value (FV) of a single cash flow is: FV = PV_0 × (1 + R)^n, where PV_0 is the present value, R is the interest rate per period, and n is the number of periods.

How do you calculate the future value of $100,000 invested at 8% interest for 5 years?

To calculate the future value, use the formula: FV = 100,000 × (1 + 0.08)^5. This gives a future value of $146,932.81.

What type of calculator is required to perform the future value calculation described in the transcript?

A scientific calculator is required to perform the calculation of future value using the formula, as it involves exponentiation (raising to a power).

What is the formula to calculate the present value of a single future cash flow?

The formula for the present value (PV) of a future cash flow is: PV_0 = FV_n / (1 + R)^n, where FV_n is the future value, R is the interest rate per period, and n is the number of periods.

How would you calculate the present value of $10,000 due in 6 years at a 10% annual interest rate?

To calculate the present value, use the formula: PV_0 = 10,000 / (1 + 0.1)^6. This gives a present value of $5,645.

What is the method to find the future value of a series of cash flows?

To find the future value of a series of cash flows, you calculate the future value of each cash flow individually, considering the number of periods until the target period, and sum them up. The formula is: FV_n = Σ(C_i × (1 + R)^(n - i)) for each cash flow C_i.

How do you calculate the future value of the cash flows $200, $400, and $600 over three years at 8% interest?

For each cash flow, calculate the future value as follows: $200 × (1 + 0.08)^2 = $233.28, $400 × (1 + 0.08)^1 = $432, and $600 × (1 + 0.08)^0 = $600. The total future value is $1,265.28.

What is the formula for the present value of a series of cash flows?

The formula for the present value (PV) of a series of cash flows is: PV_0 = Σ(C_i / (1 + R)^i), where C_i is each cash flow and i is the respective time period.

How do you calculate the present value of the cash flows $200, $400, and $600 at 10% interest?

To calculate the present value, discount each cash flow: $200 / (1 + 0.1)^1 = $181.82, $400 / (1 + 0.1)^2 = $330.58, and $600 / (1 + 0.1)^3 = $450.79. The total present value is $963.19.

What is the difference between a perpetuity and an annuity in financial mathematics?

A perpetuity is a series of equal cash flows that continues indefinitely, while an annuity consists of a series of equal cash flows occurring over a finite number of periods. The present value of a perpetuity is calculated using PV = C / R, and for an annuity, PV = C × [1 - (1 + R)^-n] / R.