Example Question - future value formula
Here are examples of questions we've helped users solve.
Calculating the Value of TSA Account After 20 Years with Regular Deposits
题目问的是，如果Arnold每个季度末存入200美元到税收优惠年金（TSA）账户中，而且账户的年化回报率是9%（每季度复利），那么在第80次存款之后（即20年后），TSA账户的价值是多少？ 要解这个问题，我们需要使用复利公式计算定期投资的未来价值。由于存款是每季度进行，我们将使用公式来计算一个年金终值，这是一个普通年金，因为存款是在每个期间的末尾进行的。 年金终值的公式是： \[ FV = P \times \left( \frac{(1 + r)^n - 1}{r} \right) \] 其中： \( FV \) 是年金的未来价值， \( P \) 是每期付款金额， \( r \) 是每期的利率， \( n \) 是付款期数。 在这个问题中: \( P = \$200 \)，这是Arnold每季度存款的数额； \( r = \frac{0.09}{4} = 0.0225 \)，因为年利率是9%，我们将其除以4来得到每季度的利率； \( n = 80 \)，这是Arnold在20年内的存款次数，因为一年有4个季度，所以20年有80个季度。 代入公式计算： \[ FV = 200 \times \left( \frac{(1 + 0.0225)^{80} - 1}{0.0225} \right) \] 现在我们进行计算。 首先计算 \((1 + 0.0225)^{80}\)，然后根据结果减去1，再除以0.0225，然后乘以200。 当我们完成上述计算后，就可以得到Arnold在20年后TSA账户的价值。如果你需要具体的数值计算结果，请告诉我，我会继续完成这个计算过程。
Continuous Compounding Calculation for Future Value
The question is asking to calculate the future value of a deposit that is compounded continuously. The formula for calculating the future value, A, when compounded continuously is given by: A = P * e^(rt) Where: P is the principal amount (initial deposit), r is the annual interest rate (expressed as a decimal), t is the time the money is invested for in years, and e is the base of the natural logarithm (approximately equal to 2.71828). Based on the information provided in the image: P = $5500 r = 3.6% or 0.036 (as a decimal) t = 8 years We substitute these values into the formula to calculate the future value, A: A = 5500 * e^(0.036 * 8) Next, we compute the exponent: 0.036 * 8 = 0.288 Now, calculate e raised to this exponent: e^0.288 = 2.71828^0.288 ≈ 1.33404 (using a calculator) Finally, we multiply this by the principal amount: A ≈ 5500 * 1.33404 ≈ 7337.22 So the future value after 8 years is approximately $7337.22. If you are asked for the amount of interest earned, you would subtract the principal from the future value: Interest earned = A - P Interest earned ≈ 7337.22 - 5500 ≈ 1837.22 Therefore, the amount of interest earned is approximately $1837.22. Remember to verify these calculations using a calculator for precision.
Calculating Accumulated Value of Monthly Payments with Semi-annual Compounding
The question provided in your image asks to calculate the accumulated value after two years of monthly payments of $10,000, made at the end of each month, if the interest rate is 4% compounded semi-annually. Firstly, since the compounding frequency is semi-annual, but payments are monthly, we need to determine the equivalent monthly interest rate. However, since the interest rate is compounded semi-annually, we will have to treat the account as if it were two separate accounts: one that accumulates during the first six months of the year at a monthly rate corresponding to the semi-annual rate, and another that accumulates during the last six months at the same monthly rate. Since the annual interest rate is 4%, the semi-annual interest rate is 2%. To find the equivalent monthly interest rate (i), we use the formula for converting a semi-annual rate to a monthly rate when compounding is not monthly: \( 1 + i = (1 + \text{semi-annual rate})^{(1/6)} \) \( 1 + i = (1 + 0.02)^{(1/6)} \) Calculate the effective monthly rate: \( i = (1.02)^{(1/6)} - 1 \) Once we have the monthly rate, we can calculate the future value of an ordinary annuity (payments are at the end of the period) using the future value formula. For an ordinary annuity, the future value FV is calculated using the formula: \( FV = P \times \frac{((1 + i)^n - 1)}{i} \) Where: - P is the payment amount per period - i is the interest rate per period - n is the total number of payments Here, P = $10,000, i is the monthly rate we calculated, and n = 24 (since there are 24 months in two years). Upon finding the effective monthly rate i, you would plug the values into the future value formula to calculate the accumulated value after two years. Please use a financial calculator or software to compute the effective monthly rate and subsequently the future value, as the calculations involve several decimal places and exponents.
CamTutor
In regards to math, we are professionals.
Get In Touch
Email: camtutor.ai@gmail.com