Which one of the following statements correctly defines a time value of money relationship?

• A. The time value of money assumes that money loses value only if interest rates are negative.
• B. The present value of a future sum is always greater than the future sum if interest rates are positive.
• C. The future value of a present sum is always less than the present value due to inflation.
• D. The present value of a future sum is the amount that, if invested today at a given interest rate, will grow to equal the future sum.

Answer: (D)

Verified step by step guidance

1. Step 1: Understand the concept of the time value of money. The time value of money is a fundamental financial principle stating that a sum of money is worth more today than the same sum in the future due to its earning potential. This is primarily influenced by interest rates and investment opportunities.
2. Step 2: Analyze the statement provided in the problem. The correct definition of the time value of money is: 'The present value of a future sum is the amount that, if invested today at a given interest rate, will grow to equal the future sum.' This highlights the relationship between present value, future value, and interest rates.
3. Step 3: Break down the key terms. Present value (PV) refers to the current worth of a future sum of money or cash flows, discounted at a specific interest rate. Future value (FV) is the value of a current sum of money at a future date, given a specific interest rate.
4. Step 4: Use the formula for present value to understand the relationship mathematically. The formula is: ${\mathrm{PV}}_{}={\mathrm{FV}}_{}/(1+r_{})n^{}$, where PV is the present value, FV is the future value, r is the interest rate, and n is the number of periods.
5. Step 5: Recognize that the correct statement emphasizes the relationship between present value and future value, assuming a positive interest rate. It does not involve inflation or negative interest rates, which are separate concepts.