Question

If the interest rate increases, what happens to the present value of a future amount?

• A. It increases
• B. It remains the same
• C. It decreases
• D. It becomes zero
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Hint:

Present value and interest rates are inversely related. Think of the interest rate as a discount rate: a higher discount rate reduces the present value of future cash.

Answer 1

The Correct Option is C

Step 1: Recalling the Present Value Formula:
The present value ($PV$) of a single future cash flow ($FV$) received $n$ periods in the future discounted at an interest rate ($r$) is calculated as: $$PV = \frac{FV}{(1 + r)^n}$$

Step 2: Analyzing the Mathematical Relationship:
In this equation, the interest rate variable ($r$) is located in the denominator. This setup creates an inverse relationship between the interest rate and the present value: $$\text{As } r \uparrow \implies (1 + r)^n \uparrow \implies \frac{FV}{(1 + r)^n} \downarrow \implies PV \downarrow$$

Step 3: Differential Proof:
Taking the first derivative of $PV$ with respect to the discount rate $r$: $$\frac{d(PV)}{dr} = -n \cdot FV \cdot (1 + r)^{-(n+1)} < 0 \quad (\text{since } n, FV, (1+r) > 0)$$ Since the first derivative is strictly negative, an increase in the interest rate causes the present value to decrease (C).