Time Value of Money: Practice Problems

The time value of money (TVM) is a core concept in finance. It states that money available at the present time is worth more than the same amount in the future due to its potential earning capacity. Understanding TVM is crucial for making informed investment decisions, evaluating loan options, and planning for long-term financial goals. Let’s explore some practice problems to solidify your understanding. Problem 1: Future Value Suppose you deposit $1,000 into a savings account that earns an annual interest rate of 5%, compounded annually. How much will you have in the account after 3 years? Solution: We use the future value (FV) formula: FV = PV * (1 + r)^n Where: * PV = Present Value ($1,000) * r = Interest rate (5% or 0.05) * n = Number of years (3) FV = $1,000 * (1 + 0.05)^3 FV = $1,000 * (1.05)^3 FV = $1,000 * 1.157625 FV = $1,157.63 Therefore, you will have approximately $1,157.63 in the account after 3 years. Problem 2: Present Value You want to have $5,000 in 5 years. If you can earn an annual interest rate of 7%, compounded annually, how much do you need to deposit today? Solution: We use the present value (PV) formula: PV = FV / (1 + r)^n Where: * FV = Future Value ($5,000) * r = Interest rate (7% or 0.07) * n = Number of years (5) PV = $5,000 / (1 + 0.07)^5 PV = $5,000 / (1.07)^5 PV = $5,000 / 1.40255 PV = $3,564.97 Therefore, you need to deposit approximately $3,564.97 today to have $5,000 in 5 years. Problem 3: Compound Interest Frequency Consider an investment of $2,000 that earns 6% annual interest. Calculate the future value after 2 years if the interest is compounded: (a) Annually, (b) Quarterly, (c) Monthly. Solution: (a) Annually: FV = $2,000 * (1 + 0.06)^2 = $2,000 * 1.1236 = $2,247.20 (b) Quarterly: FV = $2,000 * (1 + (0.06/4))^(2*4) = $2,000 * (1 + 0.015)^8 = $2,000 * 1.12649 = $2,252.98 (c) Monthly: FV = $2,000 * (1 + (0.06/12))^(2*12) = $2,000 * (1 + 0.005)^24 = $2,000 * 1.12716 = $2,254.32 Notice how more frequent compounding leads to a slightly higher future value. Problem 4: Annuity You plan to save $500 per month for the next 10 years for retirement. If you expect to earn an annual interest rate of 8%, compounded monthly, how much will you have saved at the end of 10 years? Solution: This requires the future value of an annuity formula: FV = PMT * [((1 + r)^n – 1) / r] Where: * PMT = Payment per period ($500) * r = Interest rate per period (8%/12 = 0.08/12 = 0.006667) * n = Number of periods (10 years * 12 months/year = 120) FV = $500 * [((1 + 0.006667)^120 – 1) / 0.006667] FV = $500 * [(2.2196 – 1) / 0.006667] FV = $500 * [1.2196 / 0.006667] FV = $500 * 182.95 FV = $91,475 Therefore, you will have approximately $91,475 saved at the end of 10 years. These examples demonstrate the fundamental principles of TVM. Remember to carefully identify the variables and choose the correct formula for each problem. Practice with various scenarios to become proficient in applying these concepts.