May 18, 2011
Return On Equity: A Quiz
Return On Equity: A Quiz
Assuming you want to earn a return of 10% and ignoring taxes, what multiple of earnings should you pay for the following business:
- return on equity equals 15%
- the business pays out 60% of its earnings as dividends, and reinvests the other 40%. You expect it to also earn 15% on the reinvested earnings?
I’ll even let you have multiple choice:
- 14.5
- 15.0
- 15.9
Tomorrow, I’ll give the answer and explain why this little question is important.
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Comments
This question is not important. Who wants a 10% return :P
Retirees perhaps. I've averaged over 18% compounding for the last 11 years but now that I'm about to retire, a low risk 10% would suit me fine.
Colin maybe you should apply for a job at the intelligent investor!share your method to obtain an 18% return
Is it 15 times earnings?
Earnings Multiple 14.5 15 15.9
Buy in $1,450 $1,500 $1,590
If Earnings $100
Divident $60 $60 $60 $60
Capital Reinvested $40
Expected Earnings on reinvested amount $6
Capital Growth attributed to reinvested earnings $87 $90 $95
Total Return $147 $150 $155
10% Return on buy in would be $145 $150 $159
Variance $2.00 $- $3.60
15.9
Price(P) = Next Coupon (C1) / (Required Return(R) - Growth in Dividends (G))
P = C1/(R-G)
G = (1-Payout Ratio (PR)) * Return on Capital
G = (1-60%)*15%
G = 6%
So assuming C0 = 10
C1 = C0 *(1+G)
C1 = 10 * (1.06)
C1 = 10.6
P = C1/(R-G)
P = 10.6 /(10% - 6%)
P = $265
P/E = 265/(10/60%) (Earnings * 60% = C0(Dividend))
P/E = 265/16.67)
P/E = 15.9
15 if the first dividend you receive is tomorrow, d0, or 15.9 if the first dividend is a year away but you are measuring the pe against e0 (I.e. d1 is the first div you get, per Erik's calculation.
Cheers
Justin S
Div (cf) multiple = 1 / r - g
= 1 / .1 - .06 = 25
Price = 25 x 9 = 225
Pe = 225/15 = 15
Before delving into the calculations, the forecast return on retained earnings would suggest earnings and dividend growth of 6%. The Gordon growth model, which provides a multiple of 15.9, assumes this rate continues in perpetuity when the reality is that such growth is impossible (barring inflationary phenomenon) and this company will eventually become the economy.
The alternative is that 15% ROE is sustainable and assuming no growth, an investor would be happy to pay up to 1.5x book value or a PE of 15.
In any case, the maximum/theoretically sound PE to pay would be 15.9 but realistically, one would want to pay lower than that.
my apologies, ignore that first post of mine.
if you are buying the business today (t0) and you do not receive your first dividend (d1) until one year out, paid from that years earnings (e1), then the pe today (p0/e1) is 15.0
if you are comparing the price today with trailing earnings, then the pe (p0/e0) is 15.9
the key formulae are:
growth = roe x reinvestment rate = .15 x .4 = .06
dividend (fcf) multiple = 1 / r-g = 1/(.1-.06) = 25
earnings multiple = fcf multiple x payout ratio = 25 x 60% = 15
trailing multiple = earnings multiple x (1+g) = 15 x 1.06 = 15.9
cheers
Justin S
Assumptions:
1. No franking credits.
2. ROE is based on return on en-of-year equity (not return on average annual equity).
The growth rate in earnings (and dividends since payout is assumed constant) is:
growth, g = ROE x r = 6%, where r=retained earnings ratio (40%)
The value of the dividend stream is:
V = d x (1+g)/(R-g) = e x (1-r) x (1+g)/(R-g)
where:
d = current (year zero) dividend = e x (1-r)
e = current (year zero) earnings
R = required return = 10%
Hence: PE = V/e = (1-r) x (1+g)/(R-g) = (1-0.4) x (1.06)/(0.1-0.06)
ie PE = 15.9
Ok I'm getting this wrong. I'm not familiar with the standard formulae others have used but I approached it this way. My assumptions are the first dividend arrives tomorrow and the return is on end-of year equity.
Say E(n) = earnings in year n. R = ROE, Q = fraction of retained earnings.
Then E(n) = E(n-1) x (1 + RQ)
Therefore E(n) = E(0) x (1 + RQ)^n (to the power n that is)
So value of earnings at discount rate d is
E(0) + E(1)(1 + RQ)d + E(2)(1 + RQ)^2 x d^2 + E(3)(1 + RQ)^3 x d3 + ...
Geometric series formula gives this as
E(0) x (1/(1 - d(1 + RQ))
Plugging in R = 0.15, Q = 0.4, and d = 0.9 (10% discount rate), then multiplying by 0.6 to value the dividend stream gives 13.0 times earnings. If you set d = 1/1.1 (it depends what standard convention for discount rate is?) and everything else the same gives 16.5 times earnings.
You could subtract one off these numbers if you're not getting the first dividend tomorrow. Either way, I'm wrong - anyone able to correct me?
You are summing the present values of the earnings, not the dividends. What you, the investor, recieved is the dividend - not the earnings.
Doesn't my multiplying it by 0.6 for the final step mean I'm summing the dividends?
Ok having read Steve's answer, I now see what I did wrong. In my original post I had the *earnings* value at E(0) x (1/1 - d(1 + RQ)), and multiplying this by the payout ratio (call it P) to get the value of the dividend stream. I said subtract one if the first dividend doesn't arrive for a year, but this was a mistake: you need to subtract the earnings payout ratio (P) times earnings, because this is the amount you'd receive "this afternoon" if the dividend came straight away (and following the algebra shows this).
So to summarise in general: If the first dividend arrives in a year, you're expressing your price as a function of trailing earnings, the equity in the ROE is year-end equity, and the discount rate is expressed as d = 1/(1+rate), we end up with
value = P/(1 - d(1 + RQ)) - P (times trailing earnings)
In this case P = 0.6, d = 1/1.1, R = 0.15, Q = 1 - P = 0.4, so
value = 15.9 x trailing earnings.
Quite a handful. Glad that the real-world factors make the difference unimportant :)
-- Maz
15
This business sounds pretty good and lovely consistency.
My 10 year super return nett is 7%.
Happy to pay 15.9 P/E ratio. It will improve my average.
What we know:
Return on Equity = Profit/Equity = 15%
We want a return of 10% i.e.:
Dividend/Amount Paid = 10%
Assuming we own the whole business and we have $1 (1 share of $1) of equity
PER = Share Price/(Profit per share - Dividends Per Share)
Profits per share = $0.15
Dividends per share = 0.15 x 60% = $0.09
Dividends per share is only 0.09. We want 0.09 to be 10% of the share price.
Therefore Share Price x 0.1 = 0.09
Share Price = 0.09/0.1
= 0.9
Therefore:
PER = 0.9 / (0.15 - 0.09)
PER = 0.9 / 0.06
PER = 15
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