QuestionFind the market value of Lawrence's shares given a \$1.80 dividend and an 11% return with varying growth rates.

Studdy Solution

STEP 1

Assumptions1. The most recent annual dividend, $0$ , is $1.80. . The required return is$ 11\% $.<br />3. The dividends are expected to grow at$ 8\% $annually for3 years.<br />4. The dividends are expected to grow at a constant annual growth rate in years4 to infinity, which can be$ 5\% $,$ 0\% $, or$ 10\%$ depending on the scenario.

STEP 2

We need to calculate the dividends for the first years using the growth rate of $8\%$ . The formula for the dividend in year $n$ is $_n = D_{n-1} \times (1 + g)$ where $g$ is the growth rate.

STEP 3

Calculate the dividend for the first year $1 = D0 \times (1 + g) = \$1.80 \times (1 +0.08)$

STEP 4

Calculate the dividend for the second year $2 = D1 \times (1 + g)$ We will substitute the value of $1$ from the previous step into this equation.

STEP 5

Calculate the dividend for the third year $3 = D2 \times (1 + g)$ We will substitute the value of $2$ from the previous step into this equation.

STEP 6

Now, we need to calculate the present value of these dividends. The formula for the present value of a dividend in year $n$ is $PV(D_n) = \frac{_n}{(1 + r)^n}$ where $r$ is the required return.

STEP 7

Calculate the present value of the dividends for the first3 years $PV(D1) = \frac{1}{(1 + r)}$ $PV(D2) = \frac{2}{(1 + r)^2}$ $PV(D3) = \frac{3}{(1 + r)^3}$ Substitute the values of $1$ , $2$ , $3$ , and $r$ into these equations.

STEP 8

The total present value of the dividends for the first3 years is the sum of the present values calculated in the previous step.
$PV(D_{1-3}) = PV(D1) + PV(D2) + PV(D3)$

STEP 9

For years4 to infinity, the dividends grow at a constant rate. This is a perpetuity, and its present value in year3 is given by the formula $PV(Perpetuity) = \frac{4}{r - g}$ where $4$ is the dividend in year4, $r$ is the required return, and $g$ is the growth rate.

STEP 10

Calculate $4$ using the formula $4 = D3 \times ( + g)$ Substitute the value of $3$ from step5 and the growth rate for each scenario into this equation.

STEP 11

Substitute the values of $4$ , $r$ , and $g$ into the formula for the present value of the perpetuity for each scenario.

STEP 12

The present value of the perpetuity in year needs to be discounted back to year0. The formula for this isPV(Perpetuity0) = \frac{PV(Perpetuity)}{( + r)^}Substitute the values of $PV(Perpetuity)$ and $r$ into this equation.

STEP 13

The market value of the shares is the sum of the present value of the dividends for the first3 years and the present value of the perpetuity.
$Market\, Value = PV(D_{-3}) + PV(Perpetuity0)$ Substitute the values of $PV(D_{-3})$ and $PV(Perpetuity0)$ into this equation to find the market value for each scenario.

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