Analyzing A Bank's "Perpetual Annuity" Offer: A 6% Interest Rate Calculation

A local bank advertises the following deal: “Pay us $100 a year for 10 years and then we will pay you (or your beneficiaries) $100 a year forever.” Is this a good deal if the interest rate is 6%?

Solution:

To determine if this is a good deal, we need to compare the present value (PV) of what you pay and the present value of what you receive using a discount rate of 6%.

Step 1: Calculate the PV of Payments (Outflows)

You will pay $100 per year for 10 years. The present value of this annuity (PV of payments) is given by the annuity formula:

$$PV_{\text{payments}} = C \times \left( 1 - \frac{1}{(1 + r)^t} \right) \div r$$

where:

• $C = 100$ (annual payment),
• $r = 0.06$ (6% interest rate),
• $t = 10$ (number of years of payment).

$$PV_{\text{payments}} = 100 \times \left( 1 - \frac{1}{(1.06)^{10}} \right) \div 0.06$$
$$PV_{\text{payments}} = 100 \times \left( 1 - \frac{1}{1.7908} \right) \div 0.06$$
$$PV_{\text{payments}} = 100 \times \left( 1 - 0.5584 \right) \div 0.06$$
$$PV_{\text{payments}} = 100 \times 0.4416 \div 0.06 = 100 \times 7.36 = 736$$

So the present value of your payments is $736.

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Step 2: Calculate the PV of Receipts (Inflows)

After 10 years, the bank pays $100 per year forever. The present value of a perpetuity starting in year 11 is:

$$PV_{\text{perpetuity at year 10}} = \frac{C}{r} = \frac{100}{0.06} = 1666.67$$

Since this amount is received starting in year 11, we discount it back 10 years to today:

$$PV_{\text{perpetuity today}} = \frac{1666.67}{(1.06)^{10}}$$ $$PV_{\text{perpetuity today}} = \frac{1666.67}{1.7908} \approx 931$$

So the present value of what you receive today is $931.

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Step 3: Compare Inflows and Outflows

• PV of Payments (Outflows): $736
• PV of Receipts (Inflows): $931
• Net Present Value (NPV):

$$NPV = PV_{\text{receipts}} - PV_{\text{payments}}$$ $$NPV = 931 - 736 = 195$$

Since the NPV is positive ($195$), this is a good deal!

Final Answer:

Yes, this is a good deal because the present value of what you receive ($931) is greater than what you pay ($736), giving a positive net benefit of $195.