What is the present value of a $1,000 perpetuity growing at 2% annually with a 10% discount rate?

What is the present value of a $1,000 perpetuity growing at 2% annually with a 10% discount rate?

• A. $10,000
• B. $12,500
• C. $15,000
• D. $11,500

Answer: (B)

To calculate the present value of a perpetuity that grows at a constant rate, you can use the Gordon Growth Model, which is expressed as: \[ PV = \frac{C}{r - g} \] where: - PV is the present value, - C is the cash flow of the perpetuity, - r is the discount rate, and - g is the growth rate. In this scenario: - The cash flow \( C \) is $1,000. - The growth rate \( g \) is 2% (or 0.02 as a decimal). - The discount rate \( r \) is 10% (or 0.10 as a decimal). Plugging these values into the formula gives: \[ PV = \frac{1000}{0.10 - 0.02} \] This simplifies to: \[ PV = \frac{1000}{0.08} \] \[ PV = 12500 \] Thus, the present value of this growing perpetuity is $12,500. This method accurately accounts for the fact that future cash flows will increase over time due to the growth factor, making the investment more valuable today as compared to a non-growing perpetuity.

To calculate the present value of a perpetuity that grows at a constant rate, you can use the Gordon Growth Model, which is expressed as:

[ PV = \frac{C}{r - g} ]

where:

• PV is the present value,

• C is the cash flow of the perpetuity,

• r is the discount rate, and

• g is the growth rate.

In this scenario:

• The cash flow ( C ) is $1,000.

• The growth rate ( g ) is 2% (or 0.02 as a decimal).

• The discount rate ( r ) is 10% (or 0.10 as a decimal).

Plugging these values into the formula gives:

[ PV = \frac{1000}{0.10 - 0.02} ]

This simplifies to:

[ PV = \frac{1000}{0.08} ]

[ PV = 12500 ]

Thus, the present value of this growing perpetuity is $12,500. This method accurately accounts for the fact that future cash flows will increase over time due to the growth factor, making the investment more valuable today as compared to a non-growing perpetuity.