Question

# Someone offers you a security which pays \$n at the end of the nth year until...

Someone offers you a security which pays \$n at the end of the nth year until forever (i.e., it pays \$1 at the end of the first year, \$2 at the end of the second year, and so on). If the annually compounded interest rate is 10% per year, what is the fair price of such security?

ANS: \$110

Writing down the Present Value (PV) of the perpetuity:

------------------------------------------- Eq. 1

Multiplying by 1.1

...............................................Eq. 2

Substracting Eq1 from Eq2, we get

Evaluate the sum of the terms on right as an infinite Geometric Progression with d = 1/1.1

Using sum of infinite GP formula

,

we get

Thus, the fair price of the security is \$110

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