Question

Problem 6.23 (Solution Video) Cullumber Energy Company owns several gas stations. Management is looking to open a new station in the western suburbs of Baltimore. One possibility that managers at the company are evaluating is to take over a station located at a site that has been leased from the county. The lease, originally for 99 years, currently has 73 years before expiration. The gas station generated a net cash flow of $87,970 last year, and the current owners expect an annual growth rate of 6.3 percent. If Cullumber Energy uses a discount rate of 13.9 percent to evaluate such businesses, what is the present value of this growing nnuity? (Round factor values to 6 decimal places, e.g. 1.521253 and final answer to 2 decimal places, e.g. 15.21.) Present value

Answer 1

Net cash flows generated last year, C₀ = $ 87,970

Let's assume C_i represent the cash flows generated in the year i

C₁ = C₀ x (1 + g)

C₂ = C₀ x (1+g)² and so on....

Growth rate in cash flows = g = 6.3%

Discount rate, k = 13.9%

Time to maturity, N = 73

Present value of the growth annuity

$=\frac{C_{1}}{(1+k)} + \frac{C_{2}}{(1+k)^2} + \frac{C_{3}}{(1+k)^3} + .....+\frac{C_{N}}{(1+k)^N}$

$=\frac{C_{0}\times (1+g)}{(1+k)} + \frac{C_{0}\times (1+g)^{2}}{(1+k)^2} + \frac{C_{0}\times (1+g)^{3}}{(1+k)^3} + .....+\frac{C_{0}\times (1+g)^{73}}{(1+k)^{73}}$

Please note that this a geometric progression (GP) of the form: A + AR + AR² + .....AR^N-1

Where A = First term of GP = C₀ x (1 + g) / (1 + k) = 87,970 x (1 + 6.3%) / (1 + 13.9%) =  82,100.18

R = Common ratio = Factor value = (1 + g) / (1 + k) = (1 + 6.3%) / (1 + 13.9%) =  0.933275

and N = number of terms in GP = 73

Such a GP can be added using the formula: A + AR + AR² + .....AR^N-1 = A x (1 - R^N) / (1 - R)

Hence, Present value of the growth annuity

= A x (1 - R^N) / (1 - R) = 8,100.18 x (1 - 0.933275⁷³) / (1 - 0.933275) =  $ 1,222,465.48