Perpetuity calculator

Each payment is worth less than the one before it, and the shrinkage is geometric. Discounted at 5 percent, the payment in year 100 is worth less than a cent on the dollar, so the tail of the series contributes almost nothing and the whole thing converges to $20,000.

The algebra is the neatest in finance: an infinite sum collapses to one division. Payment over rate, and no term appears anywhere, because there is no term.

Value moves inversely with the rate, so small changes are large. At 5 percent, $1,000 a year is $20,000. In the second example $2,000 at 4 percent is $50,000: twice the payment and a rate one point lower produces two and a half times the value.

That sensitivity is why terminal values in valuation models deserve suspicion. A perpetuity sits at the end of most discounted cash flow models, and a small change in the assumed rate moves the answer more than years of forecasting detail.

If the payment grows at a steady rate, the denominator becomes the discount rate minus the growth rate, which is the Gordon growth model. The dividend discount calculator uses exactly that form to price a share from its dividend.

The constraint is that growth must stay below the discount rate. If it does not, the sum does not converge and no finite value exists, which is the arithmetic reason a business cannot be assumed to grow faster than its discount rate forever.

Terminal values in valuation, preferred shares with no maturity, and some endowment and ground rent structures. A very long annuity is close enough to one that the annuity present value calculator and this page converge as the term grows. Time value of money covers the discounting behind both, and present value covers the term. This is educational material, not financial advice.