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 calculate the **Future Value (FV)** of an annuity where \( \text{USD }10,000 \) is deposited at the end of each year for a period of 5 years at an annual interest rate of \( 29\% \) compounded bi-annually, follow these steps: ### **1. Determine the Effective Annual Interest Rate** Since the interest is compounded semi-annually (twice a year), we first need to convert the nominal annual interest rate to an **Effective Annual Rate (EAR)**. \[ \text{EAR} = \left(1 + \frac{r}{m}\right)^m - 1 \] where: - \( r = 0.29 \) (29%) - \( m = 2 \) (compounding periods per year) \[ \text{EAR} = \left(1 + \frac{0.29}{2}\right)^2 - 1 = (1.145)^2 - 1 = 1.299025 - 1 = 0.299025 \text{ or } 29.9025\% \] ### **2. Apply the Future Value of an Ordinary Annuity Formula** Since the deposits are made **at the end of each year**, we use the **Ordinary Annuity** formula: \[ \text{FV} = \text{PMT} \times \left( \frac{(1 + i)^n - 1}{i} \right) \] where: - \( \text{PMT} = \$10,000 \) - \( i = 0.299025 \) (Effective Annual Rate) - \( n = 5 \) (number of years) \[ \text{FV} = 10,000 \times \left( \frac{(1 + 0.299025)^5 - 1}{0.299025} \right) \] ### **3. Calculate the Compound Growth** First, compute \( (1 + 0.299025)^5 \): \[ (1.299025)^5 \approx 3.700 \] Now, plug this back into the FV formula: \[ \text{FV} = 10,000 \times \left( \frac{3.700 - 1}{0.299025} \right) = 10,000 \times \left( \frac{2.700}{0.299025} \right) \approx 10,000 \times 9.03 = 90,300 \] ### **4. Conclusion** The **Future Value** of the annuity is approximately: \[ \boxed{\$90,\!300} \]