Question

Problem 1 Suppose there is a series of cashflows that lasts n1 peri ods, {at}, 0 < t < n, and that is growing at constant rate g, (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 i.e. at a1 = aot 1+r an (1+r)"

Answer 1

We need to find a formal expression for the present discounted value of the cashflow.

The cash flow is given as :

$S= a_0 &plus;\frac{a_1}{1&plus;r} &plus;...&plus;\frac{a_n}{(1&plus;r)^{n}}$

Now a₁ = (1+g)* a₀ ,...., a_n = (1+g)ⁿ *a₀ , where g is the constant growth rate each year. The initial cash flow is a₀ in the 0th period. Also, r is the discount rate.

This can be written as :

$S= a_0 &plus;\frac{(1&plus;g)a_0}{1&plus;r} &plus;...&plus;\frac{(1&plus;g)^na_0}{(1&plus;r)^{n}}$



or,

$S= a_0 [1&plus;\frac{(1&plus;g)}{1&plus;r} &plus;...&plus;\frac{(1&plus;g)^n}{(1&plus;r)^{n}}]$

or,

$S= a_0 [\frac{1-\frac{(1&plus;g)^n}{(1&plus;r)^n}}{1-\frac{1&plus;g}{1&plus;r}}]$

[This is the formula of a GP series with common difference equals to (1+g)/(1+r) ]

Or,

$S= a_0 [\frac{1-\frac{(1&plus;g)^n}{(1&plus;r)^n}}{\frac{r-g}{1&plus;r}}]$

Or,

$S= [a_0\frac{(1&plus;r)}{(r-g)}]*[{1-\frac{(1&plus;g)^n}{(1&plus;r)^n}}]$

Thus this is the calculated expression for the present discounted value of the cash flow.