Discounted cash flow (DCF) valuation prices assets by finding the present value of the asset's future cash flows. The single cash flow, annuity, and perpetuity equations are very useful for this.
Which of the following equations is the 'perpetuity with growth' equation?
The perpetuity with growth equation is:
###V_0=\dfrac{C_1}{r-g} = \sum\limits_{t=1}^\infty \left( \dfrac{C_t.(1+g)^t}{(1+r)^t} \right) ###See the dividend discount model Wikipedia page for a derivation of how the infinite sum has a closed-form solution.
The following equation is called the Dividend Discount Model (DDM), Gordon Growth Model or the perpetuity with growth formula: ### P_0 = \frac{ C_1 }{ r - g } ###
What is ##g##? The value ##g## is the long term expected:
The pronumeral ##g## is the expected growth rate of the dividend which must also be the expected growth rate of the stock price, which is the expected capital return of the stock.
If the stock price grew by a rate less than the dividend in perpetuity, then the dividend would eventually grow bigger than the stock price which is impossible.
If the stock price grew by a rate more than the dividend in perpetuity, then the stock price would eventually grow so big compared to the dividend that the dividend yield (expected dividend in one year divided by stock price now) would be close to zero. To maintain the same total return, the capital return of the stock price must increase until it is very close (or equal) to the total return. If the total return is more than the country's GDP growth rate, then the capital return of the stock will be more than the average firm in perpetuity (forever), so the firm must take over the country. This is very unlikely.
Mathematically:
### P_0 = \frac{ C_1 }{ r - g } ### ### r - g = \frac{ C_1 }{ P_0 } ### ###g = r - \frac{ C_1 }{ P_0 } ###Substitute for the total return, ##r = \dfrac{P_1 - P_0 + C_1}{P_0}##
###\begin{aligned} g &= \frac{ P_1 - P_0 + C_1 }{ P_0 } - \frac{ C_1 }{ P_0 } \\ &= \frac{ P_1 - P_0 }{ P_0 } \\ &= r_\text{capital} \\ \end{aligned}###For a price of $13, Carla will sell you a share paying a dividend of $1 in one year and every year after that forever. The required return of the stock is 10% pa.
The first payment of a constant perpetual annual cash flow is received at time 5. Let this cash flow be ##C_5## and the required return be ##r##.
So there will be equal annual cash flows at time 5, 6, 7 and so on forever, and all of the cash flows will be equal so ##C_5 = C_6 = C_7 = ...##
When the perpetuity formula is used to value this stream of cash flows, it will give a value (V) at time:
The perpetuity equation gives a value one period before the first cash flow.
###V_{t-1} = \dfrac{C_t}{r} ###In this question the first cash flow is at time 5, and there's a one year period between each cash flow. So the value of the perpetuity will be at time 4, one year before time 5.
###V_4 = \dfrac{C_5}{r} ###For a price of $1040, Camille will sell you a share which just paid a dividend of $100, and is expected to pay dividends every year forever, growing at a rate of 5% pa.
So the next dividend will be ##100(1+0.05)^1=$105.00##, and the year after it will be ##100(1+0.05)^2=110.25## and so on.
The required return of the stock is 15% pa.
The following equation is the Dividend Discount Model, also known as the 'Gordon Growth Model' or the 'Perpetuity with growth' equation.
### P_{0} = \frac{C_1}{r_{\text{eff}} - g_{\text{eff}}} ###
What would you call the expression ## C_1/P_0 ##?
For a stock, the expected income return is the ratio of the dividend expected to be paid in one year to the stock price now. Another name for it is the expected dividend yield.
Rearranging this equation:
###P_{0} = \frac{C_1}{r_{\text{eff}} - g_{\text{eff}}} ### ###r_{\text{eff}} - g_{\text{eff}} = \frac{C_1}{P_{0}} ### ###r_{\text{eff}} = g_{\text{eff}} + \frac{C_1}{P_{0}} ###This is equivalent to:
###\begin{aligned} r_{\text{eff, total}} =& r_{\text{eff, capital}} + r_{\text{eff, income}} \\ \end{aligned} ###
Where ##r_{\text{eff, capital}} = g## and ##r_{\text{eff, income}} = d_1/p_0##.
The following is the Dividend Discount Model (DDM) used to price stocks:
###P_0=\dfrac{C_1}{r-g}###
If the assumptions of the DDM hold and the stock is fairly priced, which one of the following statements is NOT correct? The long term expected:
All statements are true except for answer c. The expected dividend yield is defined as ##c_1/p_0## whereas the expected growth rate in the dividend is ##g##.
Rearranging this equation:
###P_{0} = \dfrac{C_1}{r - g} ### ###r - g = \dfrac{C_1}{P_{0}} ### ###r = g + \dfrac{C_1}{P_{0}} ###This is equivalent to:
###\begin{aligned} r_{\text{total}} =& r_{\text{capital}} + r_{\text{income}} \\ \end{aligned} ###
Where ##r_{\text{capital}} = g## and ##r_{\text{income}} = C_1/P_0##. Note that the expected dividend yield can also be called the share's expected income return ##r_{\text{income}}##.
A stock just paid its annual dividend of $9. The share price is $60. The required return of the stock is 10% pa as an effective annual rate.
What is the implied growth rate of the dividend per year?
The $9 dollar dividend just paid will not be included in the price since buying the stock now will not give the owner the right to receive that dividend since it's already paid, it's too late. The next dividend will be in one year, and it will have grown by g, so ## c_1 = 9 \times (1+g)^1## .
###p_0=\dfrac{c_1}{r-g}### ###60 = \dfrac{9 \times (1+g)^1}{0.1-g}### ###60 \times (0.1-g) = 9 \times (1+g)### ###6 - 60g = 9 + 9g### ###69g = -3### ###g = -3/69 = -0.043478261 ###Error, '49 7' is not a number.
A stock is expected to pay a dividend of $15 in one year (t=1), then $25 for 9 years after that (payments at t=2 ,3,...10), and on the 11th year (t=11) the dividend will be 2% less than at t=10, and will continue to shrink at the same rate every year after that forever. The required return of the stock is 10%. All rates are effective annual rates.
What is the price of the stock now?
The following equation is the Dividend Discount Model, also known as the 'Gordon Growth Model' or the 'Perpetuity with growth' equation.
###P_0=\frac{d_1}{r-g}###
A stock pays dividends annually. It just paid a dividend, but the next dividend (##d_1##) will be paid in one year.
According to the DDM, what is the correct formula for the expected price of the stock in 2.5 years?
The growth rate in the dividend (##g##) must equal the growth rate in the stock price measured between a whole number of dividend periods. But the growth rate in stock price, also known as the capital return, is actually equal to the total return ##r## in between dividend payments. This is best seen in a saw-tooth graph, where the 'dividend drop-off' price fall can be seen every time a stock pays a dividend. But here it's explained in words:
- The expected capital return measured just after a dividend is paid to just after the next dividend is paid is ##g##.
- The expected capital return measured just before a dividend is paid to just before the next dividend is paid is also ##g##.
- But, the expected capital return measured just after a dividend is paid to just before the next dividend is paid is actually ##r##. The price growth must be higher than ##g## since the stock price must accumulate the next dividend payment as well as the usual price gain over a whole dividend period. Thus the price growth between dividend payments must be ##r = d_1/P_0 + g##.
Using this logic, the growth rate in the share price from just after the current (t=0) dividend was paid to just after the next dividend is paid in one year will be ##g##.
###P_\text{1, just after div} = P_\text{0, just after div}(1+g)^1### Similarly for the next year, just after that dividend is paid (at t=2).###\begin{aligned} P_\text{2, just after div} &= P_\text{1, just after div}(1+g)^1 \\ &= P_\text{0, just after div}(1+g)^2 \\ \end{aligned}###
But from just after the second dividend is paid at t=2 to t=2.5, that period is in between dividend payments, so the share price growth will be the total return ##r##.
###\begin{aligned} P_\text{2.5} &= P_\text{2, just after div}(1+r)^{0.5} \\ &= P_\text{0, just after div}(1+g)^2(1+r)^{0.5} \\ \end{aligned}###
In the dividend discount model:
###P_0 = \dfrac{C_1}{r-g}###
The return ##r## is supposed to be the:
Future cash flows and returns are important.
Owners of assets such as shares are entitled to the future cash flows only, not the past cash flows which have already been paid. This is why asset prices are the present value of future cash flows.
To calculate the present value of future dividends, the dividend discount model must use the future expected return ##r## and growth rate ##g## of the market price of equity.
Of course the future is impossible to predict. Often the best guide to the future is the past, so in practice the actual historical return and growth rate are used as a proxy for what's expected in the future.
Market prices are important.
In finance, current market prices are always more important and relevant than old historical cost book prices. The market price of a share is the price that it trades for every day on the stock exchange. It's the price that a buyer will actually pay to buy the share.
When the share was first bought, the market price and book price were the same. But after that, the book price never changes while the market price goes up and down every day. Therefore the book price is old and out of date. Generally it is not the same as the current market price, unless by coincidence.
Owners equity recorded by an accountant in the firm's balance sheet is the sum of the shareholders' equity (also called contributed equity), retained profits and reserves such as asset revaluation reserve. This is often very different to the market price of equity. If the firm has been successful in the past, usually the market price of equity will be much higher than the book price.
Equity returns calculated from book prices are also therefore not very useful to determine value. They reflect the past, not the future. Therefore accounting ratios such as ROE (Net Income/Owners Equity) and ROA (Net Income/Total Assets) are not very useful for pricing stocks. But they are a reasonable guide to past performance.
Two years ago Fred bought a house for $300,000.
Now it's worth $500,000, based on recent similar sales in the area.
Fred's residential property has an expected total return of 8% pa.
He rents his house out for $2,000 per month, paid in advance. Every 12 months he plans to increase the rental payments.
The present value of 12 months of rental payments is $23,173.86.
The future value of 12 months of rental payments one year ahead is $25,027.77.
What is the expected annual growth rate of the rental payments? In other words, by what percentage increase will Fred have to raise the monthly rent by each year to sustain the expected annual total return of 8%?
The perpetuity with growth equation is suitable for valuing real estate since rental payments from land last forever. Re-arranging the perpetuity with growth equation into the 'total return' equation gives:
###P_0 = \frac{C_1}{r-g} ### ###r = \frac{C_1}{P_0} + g ###The growth rate in the rental payments g must equal the capital return which is the growth rate in the house price. See question 3 for an explanation of why.
Values can now be substituted into the equation to find g which is the growth rate in income cash flows (and the capital return). The price is supposed to be the market value now ($500,000) not the historical cost book value ($300,000) years ago. The income yield is the future value of income cash flows divided by the current market price:
###r = \frac{C_1}{P_0} + g ### ###0.08 = \frac{25,027.77}{500,000} + g ### ###\begin{aligned} g &= 0.08 - \frac{25,027.77}{500,000} \\ &= 0.08 - 0.050055546 \\ &= 0.029944454 \\ \end{aligned}###Question 31 DDM, perpetuity with growth, effective rate conversion
What is the NPV of the following series of cash flows when the discount rate is 5% given as an effective annual rate?
The first payment of $10 is in 4 years, followed by payments every 6 months forever after that which shrink by 2% every 6 months. That is, the growth rate every 6 months is actually negative 2%, given as an effective 6 month rate. So the payment at ## t=4.5 ## years will be ## 10(1-0.02)^1=9.80 ##, and so on.
The cash flows occur every 6 months, so the discount rate and growth rate in the perpetuity formula need to be effective 6 month rates. The growth rate already is, but the discount rate needs converting from an effective annual rate to an effective 6 month rate.
###\begin{aligned} r_\text{eff,6mth} =& (1+r_\text{eff,annual})^{1/2}-1 \\ =& (1+0.05)^{1/2}-1 \\ =& 0.024695077 \\ \end{aligned}###
Now to value the perpetuity. The subscripts denote semi-annual periods.
###\begin{aligned} V_{0} =& \frac{\left(\dfrac{V_{8}}{r_\text{eff,6mth} - g_\text{eff,6mth}}\right)}{(1+r_\text{eff,6mth})^{7}} \\ =& \frac{\left(\dfrac{10}{0.024695077 - -0.02}\right)}{(1+0.024695077)^{7}} \\ =& \frac{223.7382878}{(1+0.024695077)^{7}} \\ =& 188.62 \\ \end{aligned}###
A share just paid its semi-annual dividend of $10. The dividend is expected to grow at 2% every 6 months forever. This 2% growth rate is an effective 6 month rate. Therefore the next dividend will be $10.20 in six months. The required return of the stock 10% pa, given as an effective annual rate.
What is the price of the share now?
The inputs into the perpetuity with growth formula must be consistent with the frequency of the cash flow, which is every 6 months. Therefore the discount rate and growth rate must be effective 6 month rates. The growth rate of 2% is already given as an effective 6 month rate.
###g_\text{eff 6mth} = r_\text{capital, eff 6mth} = 0.02### The total required return of 10% is given as an effective annual rate, so first we convert this to an effective 6 month rate by compounding down: ###(1+r_\text{total, eff 6mth})^2 = (1+r_\text{total, eff annual})^1### ###(1+r_\text{total, eff 6mth})^2 = (1+0.1)^1### ###1+r_\text{total, eff 6mth} = (1+0.1)^{1/2}### ###r_\text{total, eff 6mth} = (1+0.1)^{1/2} - 1 = 0.048808848 ###Since the $10 dividend was just paid, in the past, it will not be paid to anyone who buys the share now, so we ignore that dividend. The next dividend will be in 6 months and it will have grown to be ##c_1 = 10 \times (1+g)^1##. Putting this into the perpetuity with growth equation:
###\begin{aligned} p_0 &= \dfrac{c_1}{r-g} \\ &= \dfrac{c_\text{1, every 6 months}}{r_\text{total, eff 6mth} - r_\text{capital, eff 6mth}} \\ &= \dfrac{10 \times (1+0.02)^1}{0.048808848 - 0.02} \\ &= \dfrac{10.20}{0.028808848} \\ &= 354.0578901 \\ \end{aligned}###A stock pays annual dividends which are expected to continue forever. It just paid a dividend of $10. The growth rate in the dividend is 2% pa. You estimate that the stock's required return is 10% pa. Both the discount rate and growth rate are given as effective annual rates. Using the dividend discount model, what will be the share price?
The $10 dividend that was just paid (at t=0, a moment ago) in the past will not be paid to whoever buys the stock now, therefore it should be left out of the valuation. Every investment is priced based on its future cash flows, not past cash flows. Therefore, we're more interested in the next annual dividend ##C_1##, which will equal the old one ##C_0## grown by the growth rate: ##C_1 = C_0 (1+g)^1 = 10 \times (1+0.02)^1 = 10.2##
###\begin{aligned} P_{0} &= \frac{C_1}{r - g} \\ &= \frac{C_0(1+g)^1}{r - g} \\ &= \frac{10(1+0.02)^1}{0.1 - 0.02} = \frac{10.20}{0.08} = 127.5 \\ \end{aligned} ###
A stock is expected to pay the following dividends:
| Cash Flows of a Stock | ||||||
| Time (yrs) | 0 | 1 | 2 | 3 | 4 | ... |
| Dividend ($) | 0.00 | 1.00 | 1.05 | 1.10 | 1.15 | ... |
After year 4, the annual dividend will grow in perpetuity at 5% pa, so;
- the dividend at t=5 will be $1.15(1+0.05),
- the dividend at t=6 will be $1.15(1+0.05)^2, and so on.
The required return on the stock is 10% pa. Both the growth rate and required return are given as effective annual rates. What is the current price of the stock?
There is a little trick in this question, which is that the growth rate of 5% only occurs from t=4 onwards. It is tempting to take a short cut and grow the dividend from t=1 by 5% since the dividend increases by $0.05 each year. But this will give the wrong answer $20 (c) rather than the correct answer $19.88 (d). This is because the growth rate of 5% compounds, whereas the addition of $0.05 does not compound. Growing $1 by 5% each year will lead to a bigger number than growing by $0.05 each year due to the power of compound interest.
###\begin{aligned} P_0 &= \dfrac{C_1}{(1+r)^1} + \dfrac{C_2}{(1+r)^2} + \dfrac{C_3}{(1+r)^3} + \dfrac{\left( \dfrac{C_4}{r-g} \right)}{(1+r)^3} \\ &= \dfrac{1}{(1+0.1)^1} + \dfrac{1.05}{(1+0.1)^2} + \dfrac{1.1}{(1+0.1)^3} + \dfrac{\left( \dfrac{1.15}{0.1- 0.05} \right)}{(1+0.1)^3} \\ &= 0.909090909 + 0.867768595 + 0.826446281 + \frac{23}{(1+0.1)^3} \\ &= 2.603305785 + 17.28024042 \\ &= 19.88354621 \\ \end{aligned} ###
An alternative method that gives the same answer is to grow the perpetuity from year 5 rather than from year 4.
###\begin{aligned} P_0 &= \dfrac{C_1}{(1+r)^1} + \dfrac{C_2}{(1+r)^2} + \dfrac{C_3}{(1+r)^3} + \dfrac{C_4}{(1+r)^4} + \dfrac{\left( \dfrac{C_5}{r-g} \right)}{(1+r)^4} \\ &= \dfrac{1}{(1+0.1)^1} + \dfrac{1.05}{(1+0.1)^2} + \dfrac{1.1}{(1+0.1)^3} + \dfrac{1.15}{(1+0.1)^4} + \dfrac{\left( \dfrac{1.15(1+0.05)^1}{0.1- 0.05} \right)}{(1+0.1)^4} \\ &= 0.909090909 + 0.867768595 + 0.826446281 + 0.785465474 + \frac{24.15}{(1+0.1)^4} \\ &= 3.388771259 + 16.49477495 \\ &= 19.88354621 \\ \end{aligned} ###
A stock is expected to pay the following dividends:
| Cash Flows of a Stock | ||||||
| Time (yrs) | 0 | 1 | 2 | 3 | 4 | ... |
| Dividend ($) | 0.00 | 1.00 | 1.05 | 1.10 | 1.15 | ... |
After year 4, the annual dividend will grow in perpetuity at 5% pa, so;
- the dividend at t=5 will be $1.15(1+0.05),
- the dividend at t=6 will be $1.15(1+0.05)^2, and so on.
The required return on the stock is 10% pa. Both the growth rate and required return are given as effective annual rates.
What will be the price of the stock in three and a half years (t = 3.5)?
###\begin{aligned} P_{3.5} =& \frac{C_4}{r-g} \times (1+r)^{0.5} \\ =& \frac{1.15}{0.1- 0.05} \times (1+0.1)^{0.5} \\ =& 23 \times (1+0.1)^{0.5} \\ =& 24.12260351 \\ \end{aligned} ###
Note that ## \frac{C_4}{r-g} ## is ## P_3 ##, and this amount is grown forward half a period by the total return (r), not the capital return (g). This may seem counter-intuitive since you would normally grow the dividend or the price forward by g. But in this case the stock price needs to be grown by the total return (r) because we are growing the price in between dividend payments. The stock price needs to grow by the higher total return so it is big enough to pay the dividend and fall in price, but still have realised a capital return (g).
Another way of thinking about it is that the growth rate in the stock price between t=3 and 3.5 needs to include not just the capital growth (g), but also the accrued dividend which will be paid at t=4 and which is part of the stock price until it is paid.
This concept is best illustrated by the 'saw-tooth' graph of expected share price vs time.
The following is the Dividend Discount Model (DDM) used to price stocks:
### P_0 = \frac{d_1}{r-g} ###Assume that the assumptions of the DDM hold and that the time period is measured in years.
Which of the following is equal to the expected dividend in 3 years, ## d_3 ##?
Re-arranging the dividend discount model we can break up total return into its dividend yield and capital yield components:
###P_0 = \frac{d_1}{r-g} ### ###r-g = \frac{d_1}{P_0} ### ###r = \frac{d_1}{P_0} + g ### ###r_\text{total} = r_\text{dividend} + r_\text{capital} ###So the following expressions are all equal to the dividend yield:
###r_\text{dividend} = \frac{d_1}{P_0} = r_\text{total} - r_\text{capital} = r - g###
Therefore, starting from answer (e),
###\begin{aligned} &P_0(1+g)^2(r-g) \\ &= P_2 \times (r-g) \\ &= P_2 \times \frac{d_1}{P_0} \\ &= P_2 \times \frac{d_1 \times (1+g)^2}{P_0 \times (1+g)^2} \\ &= P_2 \times \frac{d_3}{P_2} \\ &= d_3 \\ \end{aligned} ###
Note that all of the other answers give ## d_4 ##, the dividend in year 4.
The following equation is the Dividend Discount Model, also known as the 'Gordon Growth Model' or the 'Perpetuity with growth' equation.
### p_0 = \frac{d_1}{r - g} ###
Which expression is NOT equal to the expected dividend yield?
Answer (e) is incorrect because it grows the stock price by the total return (r) instead of the capital return (g). While it is true that in between dividend payments the stock price grows by the total return (r), the dividend payments reduce the stock price. This means that the stock price grows by the total return (r) less the dividend yield (##d_{t+1}/p_t##) which equals the capital return (g) over a whole period.
This concept is best illustrated in a 'saw-tooth' graph of expected share price versus time.
The following equation is the Dividend Discount Model, also known as the 'Gordon Growth Model' or the 'Perpetuity with growth' equation.
###p_0=\frac{d_1}{r_\text{eff}-g_\text{eff}}###
Which expression is NOT equal to the expected capital return?
Answer (d) is the dividend yield minus one (which is not very useful!). All of the other expressions will be equal to the firm's capital yield which is the same as the growth rate of the stock price and also the growth rate of the dividend, provided that the assumptions of the DDM hold.
A fairly valued share's current price is $4 and it has a total required return of 30%. Dividends are paid annually and next year's dividend is expected to be $1. After that, dividends are expected to grow by 5% pa in perpetuity. All rates are effective annual returns.
What is the expected dividend income paid at the end of the second year (t=2) and what is the expected capital gain from just after the first dividend (t=1) to just after the second dividend (t=2)? The answers are given in the same order, the dividend and then the capital gain.
Since dividends are expected to grow in perpetuity and the share is fairly priced, the dividend discount model (DDM) applies. The dividends and share price will grow at the same rate of 5% pa. For an explanation of why see question 3.
Let the dividend cash flow be ##c##, the price be ##p##, the capital return be ##r_\text{capital}## and the total return be ##r_\text{capital}##.
###\begin{aligned} c_2 &= c_1(1+r_\text{capital})^1 \\ &= 1(1+0.05)^1 \\ &= 1.05 \\ \end{aligned}###To find the capital gain from just after the first dividend (t=1) to just after the second dividend (t=2), find the price increase between time 1 and 2.
###\begin{aligned} p_1 &= p_0(1+r_\text{capital})^1 \\ &= 4(1+0.05)^1 \\ &= 4.2 \\ \end{aligned}### ###\begin{aligned} p_2 &= p_0(1+r_\text{capital})^2 \\ &= 4(1+0.05)^2 \\ &= 4.41 \\ \end{aligned}###The capital gain is the price increase which is the difference between ##p_2## and ##p_1##. ###\begin{aligned} \text{Capital gain over second year} &= p_2 - p_1 \\ &= 4.41 - 4.2 \\ &= 0.21 \\ \end{aligned}###
Another method which gives the same expected share prices is to grow by the total return and subtract the dividends at the appropriate time.
###\begin{aligned} p_1 &= p_0(1+r_\text{total})^1 - c_1 \\ &= 4(1+0.3)^1 - 1 \\ &= 5.2 - 1 \\ &= 4.2 \\ \end{aligned}### ###\begin{aligned} p_2 &= \left( p_1 - c_1 \right)(1+r_\text{total})^1 - c_2 \\ &= \left( p_0(1+r_\text{total})^1 - c_1 \right)(1+r_\text{total})^1 - c_1(1+r_\text{capital})^1 \\ &= \left( 4(1+0.3)^1 - 1 \right)(1+0.3)^1 - 1(1+0.05)^1 \\ &= (5.2 - 1) \times 1.30 - 1.05 \\ &= 4.41 \\ \end{aligned}###A stock pays semi-annual dividends. It just paid a dividend of $10. The growth rate in the dividend is 1% every 6 months, given as an effective 6 month rate. You estimate that the stock's required return is 21% pa, as an effective annual rate.
Using the dividend discount model, what will be the share price?
To find the present value of the semi-annual dividends, the effective annual required return on equity needs to be transformed into an effective semi-annual rate:
###\begin{aligned} r_\text{eff 6 mth} &= (1 + r_\text{eff annual})^{0.5} - 1 \\ &= (1 + 0.21)^{0.5} - 1 \\ &= 0.10 \\ \end{aligned}###
Applying the Dividend Discount Model (DDM),###\begin{aligned} P_0 &= \frac{d_1}{r - g} \\ &= \frac{d_0(1+g)}{r - g} \\ &= \frac{10(1+0.01)}{0.1 - 0.01} \\ &= \frac{10.1}{0.09} \\ &= 112.2222222 \\ \end{aligned}###
Question 50 DDM, stock pricing, inflation, real and nominal returns and cash flows
Most listed Australian companies pay dividends twice per year, the 'interim' and 'final' dividends, which are roughly 6 months apart.
You are an equities analyst trying to value the company BHP. You decide to use the Dividend Discount Model (DDM) as a starting point, so you study BHP's dividend history and you find that BHP tends to pay the same interim and final dividend each year, and that both grow by the same rate.
You expect BHP will pay a $0.55 interim dividend in six months and a $0.55 final dividend in one year. You expect each to grow by 4% next year and forever, so the interim and final dividends next year will be $0.572 each, and so on in perpetuity.
Assume BHP's cost of equity is 8% pa. All rates are quoted as nominal effective rates. The dividends are nominal cash flows and the inflation rate is 2.5% pa.
What is the current price of a BHP share?
Method 1: Make two perpetuities.
Find the present value of the annual interim dividends and the annual final dividends as two separate perpetuities and add them together to get the share price.
###\begin{aligned} P_0 &= V_\text{0, perp of interim dividends} + V_\text{0, perp of final dividends} \\ &= \frac{C_\text{0.5, interim}}{r_\text{eff yrly} - g_\text{eff yrly}}(1+r_\text{eff yrly})^{0.5} + \frac{C_\text{1, final}}{r_\text{eff yrly} - g_\text{eff yrly}} \\ &= \frac{0.55}{0.08 - 0.04}(1+0.08)^{0.5} + \frac{0.55}{0.08 - 0.04} \\ &= 13.75(1+0.08)^{0.5} + 13.75 \\ &= 14.28941916 + 13.75 \\ &= 28.03941916 \\ \end{aligned}###
Note that nominal cash flows can be discounted by nominal discount rates to give a correct valuation without any problem. It is only important to discount nominal cash flows by nominal discount rates and real cash flows by real discount rates. Both methods should give the same answer.
Method 2: Single perpetuity of a year's payments.
Make a perpetuity out of the future value of the first interim dividend in 6 months plus the first annual dividend in one year. Use this dividend value inside the perpetuity with growth equation.
###\begin{aligned} P_0 &= \frac{C_{1\text{, dividends over first year}}}{r-g} \\ &= \frac{C_\text{0.5, interim dividend}(1+r)^{0.5} + C_\text{1, final dividend}}{r-g} \\ &= \frac{0.55(1+0.08)^{0.5} + 0.55}{0.08-0.04} \\ &= \frac{1.121576766}{0.08-0.04} \\ &= 28.03941916 \\ \end{aligned}###
You own an apartment which you rent out as an investment property.
What is the price of the apartment using discounted cash flow (DCF, same as NPV) valuation?
Assume that:
- You just signed a contract to rent the apartment out to a tenant for the next 12 months at $2,000 per month, payable in advance (at the start of the month, t=0). The tenant is just about to pay you the first $2,000 payment.
- The contract states that monthly rental payments are fixed for 12 months. After the contract ends, you plan to sign another contract but with rental payment increases of 3%. You intend to do this every year.
So rental payments will increase at the start of the 13th month (t=12) to be $2,060 (=2,000(1+0.03)), and then they will be constant for the next 12 months.
Rental payments will increase again at the start of the 25th month (t=24) to be $2,121.80 (=2,000(1+0.03)2), and then they will be constant for the next 12 months until the next year, and so on. - The required return of the apartment is 8.732% pa, given as an effective annual rate.
- Ignore all taxes, maintenance, real estate agent, council and strata fees, periods of vacancy and other costs. Assume that the apartment will last forever and so will the rental payments.
This question can be calculated in three steps.
First find the total required return as an effective monthly rate since the rental payments are monthly.
###r_\text{eff mthly} = (1+r_\text{eff yrly})^{1/12} - 1 = (1+0.08732)^{1/12} - 1 = 0.007000721###
Second, find the present value of a year's worth of rental payments which can be done with an annuity of the 12 equal monthly payments. Note that the annuity's first payment is just about to occur right now (at t=0). The annuity equation gives a value one period before (at t=-1 month), so we grow it forward by one period to get the present value (at t=0), which makes what some call the 'annuity due' formula.
###\begin{aligned} V_\text{0, 12 months rent} &= \dfrac{C_\text{0, mthly}}{r_\text{mthly}} \left(1- \dfrac{1}{(1+r_\text{mthly})^{12}} \right) (1+r_\text{mthly})^1 \\ &= \dfrac{2,000}{0.007000721} \left(1- \dfrac{1}{(1+0.007000721)^{12}} \right) (1+0.007000721)^1 \\ &= 23,103.2659 \\ \end{aligned}###
Third, this present value of annual rental payments will grow by 3% pa forever, so we can now discount this using the perpetuity formula. Again we have to take care since the annual rental value is a present value (at t=0) and the perpetuity gives a value one period before (at t=-1 year) so we grow it forward by one year to get the present value of the rental payments in perpetuity.
###\begin{aligned} P_\text{0} &= \left( \frac{ V_\text{0, 12 months rent} }{r_\text{yrly}-g_\text{yrly}} \right) (1+r_\text{yrly})^1\\ &= \left( \frac{ 23,103.2659 }{0.08732-0.03} \right) (1+0.08732)^1\\ &= 438,252.6707 \\ \end{aligned}###
Combining all steps in one big formula,
###\begin{aligned} P_\text{0} &= \left( \vcenter{ \frac{ \dfrac{C_\text{0, mthly}}{r_\text{mthly}} \left(1- \dfrac{1}{(1+r_\text{mthly})^{12}} \right) (1+r_\text{mthly})^1 }{r_\text{yrly}-g_\text{yrly}} } \right) (1+r_\text{yrly})^1\\ &= \left( \vcenter{ \frac{ \dfrac{2,000}{0.007000721} \left(1- \dfrac{1}{(1+0.007000721)^{12}} \right) (1+0.007000721)^1 }{0.08732-0.03} } \right) (1+0.08732)^1\\ &= \left( \frac{ 23,103.2659 }{0.08732-0.03} \right) (1+0.08732)^1\\ &= 438,252.6707 \\ \end{aligned}###
This formulaic model can be made a little more elegant by not requiring the monthly effective total return as follows:
###\begin{aligned} P_\text{0} &= \vcenter{ \frac{ \left( \dfrac{C_\text{0, mthly}}{(1+r_\text{yrly})^{1/12}-1} \right) \left(1- \dfrac{1}{(1+r_\text{yrly})^{1}} \right) (1+r_\text{yrly})^{13/12} }{r_\text{yrly}-g_\text{yrly}} } \\ \end{aligned}###
Another way to approach this question is to make an annuity of perpetuities. Think of the monthly payments as 12 separate perpetuities, one for each month, that are received each year and increase by the annual growth rate. These 12 perpetuities will all be equal so they can be present valued using the annuity formula.
###\begin{aligned} P_\text{0} &= \left(\dfrac{C_\text{0, yrly}}{r_\text{yrly}-g_\text{yrly}}\right)(1+r_\text{yrly})^1 { \dfrac{ 1 }{r_\text{mthly}} \left(1- \dfrac{1}{(1+r_\text{mthly})^{12}} \right) (1+r_\text{mthly})^1 } \\ &= \left(\dfrac{2,000}{0.08732-0.03}\right)(1+0.08732)^1 { \dfrac{ 1 }{0.007000721} \left(1- \dfrac{1}{(1+0.007000721)^{12}} \right) (1+0.007000721)^1 } \\ &= 34,891.83531 \times (1+0.08732)^1 \times 11.47132541 \times (1+0.007000721)^1 \\ &= 438,252.6707 \\ \end{aligned}###
Comments
This question is interesting because it values property using discounted cash flows, whereas most property valuers and real estate agents use a comparable sales or multiples approach to valuing property.
- Comparable sales: find properties that have sold recently with similar characteristics and take an average of their sale prices.
- Multiples valuation: a rough and simple measure of how much a property is worth. An example that is often used in Sydney Australia is to multiply the weekly rent of the property by 1,000 to get the price. So a property that rents for $2,000/month equates to about $500 per week, so this property would be worth around $500,000. Of course this is just a rule of thumb used to make quick estimates.
The discounted cash flow valuation used here depends heavily on the inputs. A small decrease in annual forecast rental growth, or a small increase in the total required return will lower the price considerably. These variables have a non-linear effect on price.
The ongoing costs of renting such as agent fees and maintenance have a linear effect on the value. If half of the gross monthly rent was spent on ongoing costs, then the property price would be half as much.
Question 488 income and capital returns, payout policy, payout ratio, DDM
Two companies BigDiv and ZeroDiv are exactly the same except for their dividend payouts.
BigDiv pays large dividends and ZeroDiv doesn't pay any dividends.
Currently the two firms have the same earnings, assets, number of shares, share price, expected total return and risk.
Assume a perfect world with no taxes, no transaction costs, no asymmetric information and that all assets including business projects are fairly priced and therefore zero-NPV.
All things remaining equal, which of the following statements is NOT correct?
All statements are true except for statement b. BigDiv and ZeroDiv will have the same required total return into the future since returns are a proportional measure of performance. They are scaled by the starting price. Despite the fact that ZeroDiv's market value of assets, equity, share price, dividends and profit are expected to grow faster than BigDiv's, both firms' total returns will be equal to each other and unchanged through time.
Another point to note is that assets with equal risk should have equal expected total returns. Since the firms' risks are equal, their required total returns should also be equal.
Due to its higher dividends, BigDiv will have a lower capital return than ZeroDiv (statement a), and ZeroDiv's asset value and share price will grow faster than BigDiv's (statements c and d). This is because ZeroDiv will have more money available to re-invest in more assets to generate more money, whereas BigDiv won't since it pays out high dividends.
BigDiv has a higher payout ratio than ZeroDiv (statement e) because it pays higher dividends. ##\text{Payout ratio} = (\text{total dividends})/(\text{net income})##
The boss of WorkingForTheManCorp has a wicked (and unethical) idea. He plans to pay his poor workers one week late so that he can get more interest on his cash in the bank.
Every week he is supposed to pay his 1,000 employees $1,000 each. So $1 million is paid to employees every week.
The boss was just about to pay his employees today, until he thought of this idea so he will actually pay them one week (7 days) later for the work they did last week and every week in the future, forever.
Bank interest rates are 10% pa, given as a real effective annual rate. So ##r_\text{eff annual, real} = 0.1## and the real effective weekly rate is therefore ##r_\text{eff weekly, real} = (1+0.1)^{1/52}-1 = 0.001834569##
All rates and cash flows are real, the inflation rate is 3% pa and there are 52 weeks per year. The boss will always pay wages one week late. The business will operate forever with constant real wages and the same number of employees.
What is the net present value (NPV) of the boss's decision to pay later?
This question is very interesting because there are two methods to solve it and they both give the same answer.
Common sense method
The boss avoids paying $1 million right now. Therefore the NPV of his decision is positive $1m.
Mathematical method
The boss will receive the interest on the extra $1m in the bank forever. The question then becomes, what's the present value of earning the interest on $1m forever? The answer is $1m if we make the reasonable assumption that the bank interest rate is the required return. The calculation is hardly necessary, but here it is anyway as a perpetuity of weekly interest payments. Note that the returns and cash flows are real so there is no need to convert either to nominal.
###\begin{aligned} V_0 &= \dfrac{C_\text{1, weekly interest, real}}{r_\text{real}} \\ &= \dfrac{1m \times r_\text{eff weekly, real}}{r_\text{eff weekly, real}} \\ &= 1m \\ \end{aligned}###So the NPV is $1m, as found above.
For those who prefer actual numbers rather than algebraic symbols, ##r_\text{eff weekly, real} = (1+0.1)^{1/52}-1 = 0.001834569##, so ##C_\text{1, weekly interest, real} = 1m \times 0.001834569 = 1,834.569 ## which is the interest income at the end of every week due to the extra $1m in the bank. The present value of this is:
###\begin{aligned} V_0 &= \dfrac{C_\text{1, weekly interest, real}}{r_\text{real}} \\ &= \dfrac{1,834.569}{0.001834569} \\ &= 1m \\ \end{aligned}###