Question

Problem 1 Suppose there is a series of cashflows that lasts n + 1 pery- ods, {at}, 0 <t<n, and that is growing at constant rate 9, i.e. a4 = (1 + g)'ao, Vt. The discount rate is fixed at r and assume g <r. Find an expression for the discounted present value of the cashflows at time 0. Formally, find an expression for S = do + 12 + ... + (147)

Answer 1

Given:

• a_t = (1 + g)^ta₀ ¥t,
• Discount rate fixed at ‘r’, where r > g,
• S₀ = a₀ + [a_1/(1 +r)] + [a_2/(1 +r)²] + …….. + [a_n/(1 +r)ⁿ]

Now,

S₀ = a₀ + [a_1/(1 +r)] + [a_2/(1 +r)²] + …….. + [a_n/(1 +r)ⁿ]

    = a_{0 +} [a₀(1 +g)_/(1 +r)] + [a₀(1 +g)²_/(1 +r)²] + …….. + [a₀(1 +g)ⁿ_/(1 +r)ⁿ]

    = a₀ [1 + (1 +g)_/(1 +r) + (1 +g)²_/(1 +r)² + …….. + (1 +g)ⁿ_/(1 +r)ⁿ]

     =

(1+g)^n- ao [1-(1+r)^n- [1-4

[ Since, sum of finite GP series: a(1 - rⁿ/(1 - r)]

^­Simplifying, we get:

S_{0 =} a₀[(1 – r) – (1 + g)ⁿ(1 + r)^1-n]/(r – g)