Answer:
- r = .06: $6710.70
- r = .08: $7399.90
- r = .10: $8158.56
Step-by-step explanation:
You want the rate of change of account balance with respect to interest rate when a deposit of $1000 earns interest at annual rate r compounded monthly for 5 years.
Balance
The balance of the account is given by the compound interest formula:
A = P(1 +r/12)^(12·t)
For P = 1000 and t = 5, this is ...
A = 1000(1 +r/12)^(12·5)
Rate of change
The rate of change of the balance is the derivative of this with respect to r:
dA/dr = 60(1000(1 +r/12)^59)(1/12) = 5000(1 +r/12)^59
For the different interest rates involved, this becomes ...
r = .06: $5000(1 +.06/12)^59 ≈ $6710.70
r = .08: $5000(1 +.08/12)^59 ≈ $7399.90
r = .10: $5000(1 +.10/12)^59 ≈ $8158.56
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Additional comment
The nature of the derivative is that this "rate of change" is the amount of change per unit of r, which is per 100 percentage points. That is, a change of 0.1 percentage point would give a change in balance of approximately 1/1000 of the amounts shown.
