Finance-Theory · Unit 9 · Video 8 · Interactive Practice

When Dividends Change Shape: Multi-Stage Valuation, Dividend Policy and the Price of Capital

IKey Formulas

Formula	Name	What it needs
$P_t = \sum_{k=1}^{\infty} \dfrac{\mathsf{E}_t[D_{t+k}]}{(1+r_{t+k})^k}$	General dividend discount model	An expected dividend and a risk-adjusted rate for every future date; it assumes nothing about the path of dividends
$V = \dfrac{C}{r}\left[1 - \dfrac{1}{(1+r)^n}\right]$	A level block (annuity)	Cash flow $C$ for $n$ years at rate $r$ ; the value is one period before the block's first payment
$P_0 = V_1 + d_{10}V_2 + d_{30}V_3$	Blocks assembled by hand	Blocks of 10, 20 and 30 years, each valued as an annuity; $d_m$ is the discount factor for year $m$
$P_0 = \displaystyle\sum_{t=1}^{T} \frac{D_0(1+g)^t}{(1+r)^t} + \frac{1}{(1+r)^T}\cdot\frac{D_T}{r}$	Two stages: growth, then flat	The video's example: $1 just paid, $r = 20\%$ , $g = 6\%$ for $T = 7$ years, then zero growth, giving about 4.39 + 2.10 = $6.49

Key Insight: Cost of capital, required rate of return, risk-adjusted discount rate and user cost are four names for one number: the market's price for capital, which the market sets, not the company.

IIPricing Two Stages

The price is stage one's discounted dividends plus stage two's perpetuity, valued at year $T$ and discounted to today.

Stage-1 growth (g) 6% Growth years (T) 7 Rate (r) 20% Stage-2 growth (g₂, ours) 0% Inspect year 7

💡 Every stage-two dollar rests on forecasts more than seven years out, and the video's benchmark for the forecasting job is sobering: a superb analyst is right about 52% of the time.

IIIThe General Formula in Blocks

Three level blocks, each valued as an annuity and discounted to year 0, reproduce the term-by-term sum exactly.

C₁ 1.00 C₂ 2.00 C₃ 3.00 r₁ 10% r₂ 9% r₃ 8%

💡 The cash flows, the rates and the discount-factor rule are ours: the video leaves $d_m$ unspecified, and here each year is discounted at its own block's rate, so $d_{10} = 1/(1+r_1)^{10}$ and $d_{30} = d_{10}/(1+r_2)^{20}$ .

IVThe Price of Credit Risk

A basis point is 0.01% of the amount a year, so 700 bp of insurance costs 7% of the debt annually.

Measure Amount ($ million) 100

VQuiz Questions

Problem 1 · Valuing One Block

Given: a block of dividends pays $2 a share each year for 20 years, and the block's rate is $r = 10\%$ . Find the block's value one period before its first payment.

✅ Correct! The annuity is a perpetuity worth

C/r = 20

minus the part of it that starts after year 20.

❌ Not quite. $2 × 20 adds the payments without discounting any of them.

❌ Close, but…

C/r = 20

is a perpetuity; this block stops after 20 years, so subtract the value of the payments it does not make.

❌ Not quite.

\frac{C}{r}\cdot\frac{1}{(1+r)^n}

is the value of the payments after year 20, the part the bracket removes, not the block itself.

❌ Not quite. Use

V = \frac{C}{r}\left[1 - \frac{1}{(1+r)^n}\right]

with

C = 2

r = 0.10

n = 20

Show solution

With $C = 2$ , $r = 0.10$ and $n = 20$ :

V = \frac{2}{0.10}\left[1 - \frac{1}{(1.10)^{20}}\right] = 20\,[1 - 0.14864] \approx 17.03

since $(1.10)^{20} \approx 6.7275$ and $1/6.7275 \approx 0.14864$ . The block is worth about $17.03 at the date one year before its first payment. Placed later in a multi-stage valuation, this value is then multiplied by the discount factor for that date.

Problem 2 · Growth, Then Flat

Given: a company has just paid a dividend of $2 a share. Dividends grow at $5\%$ a year for three years and then stay at the year-3 level forever; the discount rate is $r = 10\%$ . Find the price today.

✅ Correct! The flat perpetuity is worth 23.15 at year 3, but only about 17.39 today.

❌ Not quite.

D_1/(r-g) = 2.10/0.05

assumes 5% growth forever; here growth stops after year 3, so the constant-growth shortcut no longer describes the firm.

❌ Close, but…

D_3/r = 23.15

is the value of stage two at year 3. It has to be discounted three years before it is added to stage one.

❌ Close, but…

D_3/r

values the level payments from year 4 on as of one year before the first of them, which is year 3, so discount by

(1.10)^3

, not

(1.10)^4

❌ Not quite. Discount the three growing dividends, then add the year-3 perpetuity value discounted three years.

Show solution

Stage 1: $D_1 = 2.10$ , $D_2 = 2.205$ , $D_3 = 2.31525$ .

\frac{2.10}{1.10} + \frac{2.205}{1.21} + \frac{2.31525}{1.331} \approx 1.9091 + 1.8223 + 1.7395 = 5.4709

Stage 2: the dividend stays at $D_3$ from year 4 on, a perpetuity worth, at year 3,

\frac{D_3}{r} = \frac{2.31525}{0.10} = 23.1525, \qquad \frac{23.1525}{(1.10)^3} = \frac{23.1525}{1.331} \approx 17.3948

Price: $5.4709 + 17.3948 = 22.8657$ , so the price is about $22.87. About three quarters of it (17.39 of 22.87) comes from dividends after year 3.

Problem 3 · A Growing Second Stage

Given: the video's company ($1 just paid, $r = 20\%$ , $g = 6\%$ for years 1 to 7), whose stage one is worth about 4.3942. Suppose instead that from year 8 on the dividend grows at $g_2 = 2\%$ a year forever. Find the price today.

✅ Correct! Two percent of stage-two growth lifts the price from about $6.49 to about $6.77, all of it from the farthest forecasts.

❌ Close, but… A growing perpetuity valued at year 7 uses the next dividend,

D_8 = D_7(1.02)

, over

r - g_2

, not

D_7

❌ Not quite. 8.52 is stage two's value at year 7; it must be divided by

(1.2)^7 \approx 3.58

before it is added to stage one.

❌ Close, but…

D_8/(r - g_2)

is a value one period before the first payment, year 7, so the discount is

(1.2)^7

, not

(1.2)^8

❌ Not quite. Value stage two at year 7 with

D_8/(r - g_2)

, discount it seven years, then add stage one.

Show solution

$D_7 = (1.06)^7 \approx 1.50363$ , so $D_8 = 1.50363 \times 1.02 \approx 1.5337$ . At year 7 stage two is a growing perpetuity (the Gordon form):

\frac{D_8}{r - g_2} = \frac{1.5337}{0.20 - 0.02} = \frac{1.5337}{0.18} \approx 8.5206

\frac{8.5206}{(1.2)^7} = \frac{8.5206}{3.5832} \approx 2.3779

Price: $4.3942 + 2.3779 = 6.7721$ , about $6.77. The video's flat stage two gives $4.3942 + 2.0982 = 6.4924$ ; the extra 0.28 comes entirely from dividends after year 7.

Problem 4 · One-Time or Forever

Given: a firm pays a regular dividend of $2 a share and its rate is $r = 10\%$ . After selling a division it declares an extraordinary dividend of $5 a share, paid one year from now, after which the regular policy resumes.

How much does the extraordinary dividend add to the dividend discount sum?

How much would a permanent $0.50 raise in the regular dividend, starting next year, add instead?

✅ Correct! A $0.50 raise forever adds more than a one-time $5: a regular level enters every future term of the sum, which is why a board raises it only when it believes the new level can be sustained.

❌ Not quite.

5/0.10

values $5 paid every year forever. The extraordinary dividend is explicitly one-time: it is a single term of the sum.

❌ Close, but… The $5 arrives in one year, so it enters the sum as

5/1.10

❌ Close, but…

5/(1.10)^2

discounts the payment for two years; it arrives one year from now, so it is discounted once:

5/1.10

❌ Not quite. $0.50 is only next year's extra payment, and undiscounted; a permanent raise adds $0.50 to every future dividend, a perpetuity worth

0.50/0.10

today.

❌ Close, but…

0.50/1.10

counts only next year's extra $0.50; a permanent raise adds $0.50 to every future dividend.

❌ Not quite.

2.50/0.10 = 25

is the value of the whole new dividend stream; the question asks what the raise adds to the old $2 stream.

❌ Not quite. One payment of $5 in one year is one term:

5/(1.10)^1

❌ Not quite. An extra $0.50 every year from next year on is a perpetuity:

0.50/0.10

Show solution

The general dividend discount model prices any path of dividends, including a one-time spike. The extraordinary dividend is a single extra term:

\frac{5}{1.10} \approx 4.55

A permanent raise adds $0.50 to every term from year 1 on, a level perpetuity valued one period before its first payment, which is today:

\frac{0.50}{0.10} = 5.00

The one-time $5 adds about $4.55; the $0.50 raise adds $5.00. A change in the regular level moves the whole forecast path, which is why dividends change infrequently and a cut is read as bad news about the firm.

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