Sensitivity (Meta) Math Example 2

Solution

1. 1
   Differentiate with respect to $r$: $\frac{dA}{dr} = P \cdot t(1+r)^{t-1}$.
2. 2
   For $P=1000$, $r=0.05$, $t=10$: $\frac{dA}{dr} = 1000 \times 10 \times (1.05)^9 \approx 1000 \times 10 \times 1.5513 \approx 15513$.
3. 3
   Interpretation: a 1-percentage-point increase in rate ($\Delta r = 0.01$) changes $A$ by approximately 15513 \times 0.01 \approx \155$.
4. 4
   This shows $A$ is quite sensitive to the interest rate over long periods.