Question

You are valuing a company that will experience two stages. During the first stage, the revenues and expenses will grow quickly. During the second stage, the revenues and the expenses will grow at slower rates. The revenues will be received annually and the expenses will be paid annually. The discount rate is 12% per year. For this problem, you may find it useful to consult the two-stage valuation diagram (“red and blue”) that was handed out in class.

Stage 1. Revenues at t = 1 will equal $1 million. The revenues will grow at 10% per year until t = 7. That is, the last revenue collected during the first stage is at t = 7. The expenses at t = 1 will equal $0.8 million. The expenses will grow at 12% per year until t = 7. That is, the last expenses paid during the first stage are at t = 7. Hint: Revenues in Stage 1 are a growing annuity. Expenses in Stage 1 are another growing annuity.

[2 points] What is the present value, at t = 0, of all revenues collected during Stage 1?

[2 points] What is the present value, at t =0, of all expenses paid during Stage 1?

[1 point] What is the net value of Stage 1 (at t = 0)?

Stage 2. Remember that the last payment in Stage 1 is payment at t = 7. Immediately after this payment the second stage begins. The second stage lasts forever. The revenues will grow at 3% per year in the second stage. The expenses in the second stage will grow at 2% per year.

[2 points] What is the present value, at t = 0, of all revenues collected during stage 2?

[2 points] What is the present value, at t = 0, of all expenses paid during stage 2?

[1 point] What is the net value of Stage 2 (at t = 0)?

[2 points] Value of the business. You have now computed the present value of revenues in stage 1, expenses in stage 1, revenues in stage 2, and expenses in stage 2. What is the total value of the business today (at t = 0)?

Answer 1

We will find the present value of annuity by using the PV function in excel.

Revenues - Stage 1

Revenues at T=1, is $1 million and it will grow till year 7 at 10%

The discount rate = 12%

Growth rate = 10%

Effective discount Rate = (1+12%) / (1+10%) -1 = 1.82% (while the revenues are growing at 10%, they are getting discounted at a higher rate of 12%)

NPER = 7 periods

PMT = 1 ($1 Million)

FV=0

Type =1 (Cash flows start at beginning of year)

PV at T=1 = PV(1.82%,7,1,0,1) = - $6.64 Million (Negative sign is just a cash flow convention in excel)

To calculate PV at T=0 Rate = Discount rate = 12%, NPER =1, PMT = 0, FV = -$6.64, Type =1

PV at T=0 = PV(12%,1,0,-6.64,1) = $5.92 Million

Costs

Costs will follow the same pattern of calculation as Revenues, except that PMT = $ 0.6 Million

PV of Costs at T=1 = PV(1.82%,7,0.6,0,1) = - $3.98 Million

PV of Costs at T=0 = PV(12%,1,0,-3.98,0) = $3.55 Million

Net Value of stage 1 at T=0 =(5.92 - 3.55) = $2.37 Million

STAGE 2

Revenues

We need to find the value of revenue in Year 7 if it grows at 10%

We will use FV function. Here, rate =10%, NPER = 6 (Since we are talking of period 1 to 7, 6 years in between), PMT =0, PV =1, Type =1

FV = FV(10%,6,0,1,0) = $1.77 Million

If the revenues now grow at 3% for ever,

The terminal value = V0 (1+g) / (k-g), where V0 = Value in current year,  g= Constant growth rate & k = Cost of Capital

The terminal value of revenue at year 7 = 1.77 x (1+3%) / (12% - 3%) = $ 20.27 Million

Costs

Costs will have the same pattern, except $1 million in revenue is 0.6 Million in costs

FV of costs at T=7 = FV(10%,6,0,0.6,0) = $1.06 Million  

The terminal value of costs at year 7 = 1.06 x (1+3%) / (12% - 3%) = $12.16 Million

Net Value of stage 2 at T=7 =(20.27 - 12.16) = $8.11 Million

PV of Net Value of stage 2 at T=0 = 8.11 / (1+12%)^7 = $3.67 Million

Value of Business at T=0 = Net Value of stage 1 at T=0 + Net Value of Stage 2 at T=0

= 2.37 + 3.67 = $6.04 Million