# Study Notes: Time Value of Money (TVM) Problems The following notes break down key concepts and examples related to solving Time Value of Money (TVM) problems using financial formulas and calculators. These problems involve variables such as present value (PV), future value (FV), interest rate (I%), number of periods (N), and payment (PMT). --- ## Key Concepts ### 1. Time Value of Money (TVM) The **Time Value of Money** is the idea that money available today is worth more than the same amount in the future due to its earning potential. This principle underpins many financial calculations. ### 2. Key Variables in TVM - **N**: Number of periods (e.g., years, quarters, months) - **I%**: Interest rate per period (expressed as a percentage) - **PV**: Present Value (the value of money today) - **FV**: Future Value (the value of money at a future date) - **PMT**: Payment (recurring cash flows, such as monthly deposits or withdrawals) - **P/Y**: Payments per year - **C/Y**: Compounding periods per year ### 3. TVM Formula The general formula for TVM is: $$ FV = PV \times (1 + \frac{I}{C/Y})^{N \times C/Y} $$ Where: - $FV$ = Future Value - $PV$ = Present Value - $I$ = Annual interest rate (as a decimal) - $C/Y$ = Compounding periods per year - $N$ = Number of years --- ## Solving TVM Problems ### Example 1: Future Value with Monthly Compounding **Problem:** - $N = 5$ years - $I\% = 3.8\%$ annual interest rate - $PV = -8000$ (negative because it's an outflow) - $PMT = 0$ (no recurring payments) - $FV = ?$ - $P/Y = 1$ (payments per year) - $C/Y = 12$ (compounding monthly) **Solution:** Using the formula: $$ FV = PV \times (1 + \frac{I}{C/Y})^{N \times C/Y} $$ Substitute the values: - $PV = -8000$ - $I = 0.038$ - $C/Y = 12$ - $N = 5$ $$ FV = -8000 \times (1 + \frac{0.038}{12})^{5 \times 12} $$ After calculation: $$ FV = 9671 \, \text{(to the nearest dollar)} $$ When solving for $FV$, ensure you correctly handle the negative sign for $PV$, as it represents an outflow. --- ### Example 2: Solving for Number of Periods (N) **Problem:** - $N = ?$ - $I\% = 6.1\%$ annual interest rate - $PV = -1000$ - $PMT = 0$ - $FV = 2000$ - $P/Y = 1$ - $C/Y = 4$ (compounding quarterly) **Solution:** Rearrange the formula to solve for $N$: $$ N = \frac{\log(\frac{FV}{PV})}{C/Y \cdot \log(1 + \frac{I}{C/Y})} $$ Substitute the values: - $FV = 2000$ - $PV = -1000$ - $I = 0.061$ - $C/Y = 4$ $$ N = \frac{\log(\frac{2000}{1000})}{4 \cdot \log(1 + \frac{0.061}{4})} $$ After calculation: $$ N = 11.5 \, \text{(to the nearest quarter)} $$ For example, if you invest $1000 at 6.1% interest compounded quarterly, it will take approximately 11.5 years to double your money to $2000. --- ### Example 3: Future Value with Semiannual Compounding **Problem:** - $N = 4$ years - $I\% = 5.4\%$ annual interest rate - $PV = -2000$ - $PMT = 0$ - $FV = ?$ - $P/Y = 1$ - $C/Y = 2$ (compounding semiannually) **Solution:** Using the formula: $$ FV = PV \times (1 + \frac{I}{C/Y})^{N \times C/Y} $$ Substitute the values: - $PV = -2000$ - $I = 0.054$ - $C/Y = 2$ - $N = 4$ $$ FV = -2000 \times (1 + \frac{0.054}{2})^{4 \times 2} $$ After calculation: $$ FV = 2475.10 $$ Compounding frequency ($C/Y$) significantly impacts the future value. Semiannual compounding results in higher $FV$ compared to annual compounding for the same interest rate. --- ### Example 4: Solving for Present Value (PV) **Problem:** - $N = 4$ years - $I\% = 3\%$ annual interest rate - $PV = ?$ - $PMT = 0$ - $FV = 2475.10$ - $P/Y = 1$ - $C/Y = 1$ (compounding annually) **Solution:** Rearrange the formula to solve for $PV$: $$ PV = \frac{FV}{(1 + \frac{I}{C/Y})^{N \times C/Y}} $$ Substitute the values: - $FV = 2475.10$ - $I = 0.03$ - $C/Y = 1$ - $N = 4$ $$ PV = \frac{2475.10}{(1 + \frac{0.03}{1})^{4 \times 1}} $$ After calculation: $$ PV = 2199.09 $$ Many students forget to divide the interest rate by the compounding frequency ($C/Y$). Always adjust the interest rate to match the compounding periods. --- ## Key Takeaways 1. **Understand the Variables**: Clearly identify $N$, $I\%$, $PV$, $FV$, $PMT$, $P/Y$, and $C/Y$ before solving. 2. **Adjust for Compounding**: Always divide the annual interest rate by $C/Y$ and multiply $N$ by $C/Y$. 3. **Use a Financial Calculator**: For complex problems, a financial calculator can save time and reduce errors. 4. **Check Units**: Ensure consistency in time periods (e.g., years vs. months) and compounding frequency. Practice solving problems with different compounding frequencies to build confidence and intuition. --- ## Practice Problems 1. If you invest $5000 at an annual interest rate of 4.5% compounded monthly, what will the future value be after 6 years? 2. How long will it take for $1500 to grow to $3000 at an annual interest rate of 5% compounded annually? 3. What is the present value of $10,000 to be received in 8 years if the annual interest rate is 7% compounded quarterly? Try solving these problems step-by-step using the formulas provided. Double-check your calculations!