Find the future Value of an annurty of 10,000 deposited at the end of each year for periad of 5 years at \( 29 \% \) compounded bi-annually

To find the future value of an annuity, use the formula: \[ FV = P \times \left( \frac{(1 + r/n)^{nt} - 1}{r/n} \right) \] Where: - \( P \) is the annuity payment per period (10,000) - \( r \) is the annual interest rate (0.29) - \( n \) is the number of compounding periods per year (2 for bi-annually) - \( t \) is the number of years (5) First, calculate the effective rate per period \( r/n \) which is \( 0.29/2 = 0.145 \), and total periods \( nt \) which equals \( 2 \times 5 = 10 \). Plugging these values in gives: \[ FV = 10,000 \times \left( \frac{(1 + 0.145)^{10} - 1}{0.145} \right) \] Calculating this yields: \[ FV = 10,000 \times \left( \frac{(1.145)^{10} - 1}{0.145} \right) \] Calculating \( (1.145)^{10} ≈ 4.401 \), we have: \[ FV = 10,000 \times \left( \frac{4.401 - 1}{0.145} \right) \] Which results in: \[ FV ≈ 10,000 \times 23.69 = 236,900 \] So, the future value of the annuity will be approximately \( 236,900 \). The key lesson here is to remember the compounding frequency. If you shake it up a bit by changing the frequency, it can lead to significantly different outcomes. So, keep an eye on your n's and r's! To deepen your understanding, exploring the time value of money can be fascinating. It's often seen as the bedrock of finance—ensuring you appreciate how money grows over time. Dive into resources like "The Richest Man in Babylon" for engaging insights on saving and investing!