How do you incorporate growth or decline factors into the NPV of an annuity or perpetuity?

1 NPV of an annuity

The NPV of an annuity is the sum of the present values of each payment, discounted at a given interest rate. The formula for the NPV of an annuity is:

NPV = C * [(1 - 1 / (1 + r)^n) / r]        

where C is the amount of each payment, r is the interest rate per period, and n is the number of periods. This formula assumes that the payments are made at the end of each period, and that the interest rate is constant.

2 NPV of a perpetuity

The NPV of a perpetuity is the present value of an infinite stream of equal payments, discounted at a given interest rate. The formula for the NPV of a perpetuity is:

NPV = C / r        

where C is the amount of each payment, and r is the interest rate per period. This formula assumes that the payments are made at the end of each period, and that the interest rate is constant.

3 Growth or decline factors

In some cases, the amount of each payment in an annuity or a perpetuity may not be constant, but may grow or decline at a constant rate over time. For example, a bond may pay coupons that increase by a certain percentage every year, or a lease may charge rents that decrease by a certain percentage every year. How do you incorporate these growth or decline factors into the NPV of an annuity or a perpetuity?

4 Adjusted interest rate

One way to incorporate growth or decline factors into the NPV of an annuity or a perpetuity is to adjust the interest rate by subtracting or adding the growth or decline rate. For example, if an annuity pays $1000 per year for 10 years, and the payments grow by 5% per year, and the interest rate is 10%, you can use the following formula:

NPV = C * [(1 - 1 / (1 + r - g)^n) / (r - g)]        

where g is the growth rate per period. In this case, the adjusted interest rate is 10% - 5% = 5%, and the NPV is:

NPV = 1000 * [(1 - 1 / (1 + 0.05)^10) / 0.05] = $7722.89        

Similarly, if a perpetuity pays $1000 per year, and the payments decline by 2% per year, and the interest rate is 10%, you can use the following formula:

NPV = C / (r - g)        

where g is the decline rate per period. In this case, the adjusted interest rate is 10% - (-2%) = 12%, and the NPV is:

NPV = 1000 / 0.12 = $8333.33        

5 Annuity or perpetuity with growth

Another way to incorporate growth or decline factors into the NPV of an annuity or a perpetuity is to treat the cash flow as an annuity or a perpetuity with growth. This method is useful when the growth or decline rate is not constant, but varies over time. For example, if an annuity pays $1000 per year for 10 years, and the payments grow by 5% in the first year, 4% in the second year, 3% in the third year, and so on, you can use the following formula:

NPV = C * [(1 + g1) / (1 + r) + (1 + g1) * (1 + g2) / (1 + r)^2 + ... + (1 + g1) * ... * (1 + gn) / (1 + r)^n]        

where gn is the growth rate in the nth period. In this case, the NPV is:

NPV = 1000 * [(1 + 0.05) / (1 + 0.1) + (1 + 0.05) * (1 + 0.04) / (1 + 0.1)^2 + ... + (1 + 0.05) * ... * (1 + (-0.01)) / (1 + 0.1)^10] = $7539.66        

Similarly, if a perpetuity pays $1000 per year, and the payments grow by 5% in the first year, 4% in the second year, 3% in the third year, and so on, you can use the following formula:

NPV = C * [1 / r + g1 / r^2 + (g1 + g2) / r^3 + ...]        

where gn is the growth rate in the nth period. In this case, the NPV is:

NPV = 1000 * [1 / 0.1 + 0.05 / 0.1^2 + (0.05 + 0.04) / 0.1^3 + ...] = $10000        

6 Here’s what else to consider

This is a space to share examples, stories, or insights that don’t fit into any of the previous sections. What else would you like to add?