Using a dividend discount model, find the intrinsic value (price at time zero) for a firm whose expected dividend is $1.92 and growth in dividends for year 2, 3, and 4 is 23.2% and constant growth afterwards is 0.54%. The cost of equity is 9.21%. (Answer in 2 decimal places, 6.21 for $6.21).

Question

BlackTom AI · Accepted Answer

We start by restating the problem in our own words to anchor the calculation: we are using a dividend discount model to find the intrinsic value P0 of a firm. The given information is:
- The expected dividend for year 1 is D1 = 1.92.
- Dividends grow at 23.2% for years 2, 3, and 4.
- After year 4, dividends grow at a constant rate g = 0.54% per year.
- The cost of equity (discount rate) is k = 9.21%.
- We will compute the value at time 0 (P0) to two decimal places.

First, compute each of the dividends up to year 4 using the stated growth pattern:
- D1 = 1.92 (given).
- D2 = D1 × (1 + 0.232) = 1.92 × 1.232 = 2.36544.
- D3 = D2 × (1 + 0.232) = 2.36544 × 1.232 = 2.914222.
- D4 = D3 × (1 + 0.232) = 2.914222 × 1.232 = 3.590322.

From year 5 onward, dividends grow at g = 0.54% per year. To value the perpetuity starting in year 5, we first determine D5:
- D5 = D4 × (1 + 0.0054) = 3.590322 × 1.0054 ≈ 3.609711.

The value of the perpetuity at the end of year 4 (P4) equals the present value at t = 4 of all dividends from year 5 onward, which is:
- P4 = D5 / (k − g) = 3.609711 / (0.0921 − 0.0054) = 3.609711 / 0.0867 ≈ 41.64.

Now discount all cash flows back to time 0. The present value at t = 0 is the sum of the present values of D1, D2, D3, and the cash flow in year 4 consisting of D4 plus the terminal value P4, all discounted by (1 + k)^t:
- PV(D1) = D1 / (1 + k) = 1.92 / 1.0921 ≈ 1.76.
- PV(D2) = D2 / (1 + k)^2 = 2.36544 / (1.0921)^2 ≈ 1.99.
- PV(D3) = D3 / (1 + k)^3 = 2.914222 / (1.0921)^3 ≈ 2.24.
- PV of year 4 cash flow: D4 / (1 + k)^4 ≈ 3.590322 / (1.0921)^4 ≈ 2.53.
- PV of the perpetuity value: P4 / (1 + k)^4 ≈ 41.64 / (1.0921)^4 ≈ 29.32.

Summing these present values gives the intrinsic value P0:
- P0 ≈ 1.76 + 1.99 + 2.24 + 2.53 + 29.32 ≈ 37.84.

A couple of important considerations and potential sources of small variation:
- Rounding at intermediate steps (D2, D3, D4, D5, and P4) can shift the final answer by a few tenths of a dollar.
- If D1 were interpreted as a starting point for growth from the present (i.e., D0 = 1.92 and D1 is 1.92 × 1.232), the numbers would differ; in that interpretation, the calculations yield a notably different P0.

In this stream of computation, using D1 = 1.92 and the described growth pattern up to year 4 with a perpetuity starting in year 5 at g = 0.54%, the intrinsic value at time 0 comes out to around 37.84, before any rounding to two decimals. If you need the precise two-decimal value, you can carry the intermediate steps with more exact rounding, but the method remains the same: compute D1–D4 with the specified growth, compute P4 from D5 and (k − g), discount D1–D4 and P4 back to present value, and sum them to obtain P0.

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