How to determine duration of a common stock whose dividends grow in perpetuity?
"The Dividend Discount Model is based on the concept that the present value of a stock is the sum of all future dividends, discounted back to the present. Since you said: dividends are expected to grow at a constant rate in perpetuity ... the Gordon Growth Model is a simple variant of the DDM, tailored for a firm in ""steady state"" mode, with dividends growing at a rate that can be sustained forever. Consider McCormick (MKC), who's last dividend was 31 cents, or $1.24 annualized. The dividend has been growing just a little over 7% annually. Let's use a discount, or hurdle rate of 10%. MKC closed today at $50.32, for what it's worth. The model is extremely sensitive to inputs. As g approaches r, the stock price rises to infinity. If g > r, stock goes negative. Be conservative with 'g' -- it must be sustainable forever.  The next step up in complexity is the two-stage DDM, where the company is expected to grow at a higher, unsustainable rate in the early years (stage 1), and then settling down to the terminal rate for stage 2. Stage 1 is the present value of dividends during the high growth period. Stage 2 is the Gordon Model, starting at the end of stage 1, and discounting back to the present. Consider Abbott Labs (ABT). The current annual dividend is $1.92, the current dividend growth rate is 12%, and let's say that continues for ten years (n), after which point the growth rate is 5% in perpetuity. Again, the discount rate is 10%. Stage 1 is calculated as follows: Stage 2 is GGM, using not today's dividend, but the 11th year's dividend, since stage 1 covered the first ten years. 'gn' is the terminal growth, 5% in our case. then... The value of the stock today is 21.22 + 51.50 = 72.72 ABT closed today at $56.72, for what it's worth." "The fact that dividends grow in perpetuity does not prevent one from calculating duration.  In fact, many academic papers look at exactly this problem, such as Lewin and Satchell.  This Wilmott thread discusses some of the pros and cons of the concept in some detail. PS: Although I was already broadly familiar with the literature and I use the duration of equities in some of my every-day work as a professional working in finance, I found the links above doing a simple google search for ""equity duration."""
"Dividends telegraph that management has a longer term focus than just the end of quarter share price.  There is a committment to at least maintain (if not periodically increase) the dividend payout year over year.  Management understands that cutting or pausing dividends will cause dividend investors in market to dump shares driving down the stock price.   Dividends can have preferential tax treatment in some jurisdictions, either for an individual compared to capital gains or compared to the corporation paying taxes themselves.  For example, REITs (real estate investment trusts) are a type of corporation that in order to not pay corporate income tax are required to pay out 95% of income as dividends each year.  These are not the only type, MLP (master limited partnerships) and other ""Partnership"" structures will always have high dividend rates by design. Dividends provide cash flow and trade market volatility for actual cash.  Not every investor needs cash flow, but for certain investors, it reduces the risks of a liquidity crisis, such as in retirement.  The alternative for an investor who seeks to use the sale of shares would be to maintain a sufficient cash reserve for typical market recessions." "If by saying you wish to invest ""for the long term 5-10 years"" I take it you mean to hold a stock for between 5-10 years. If this is the case, this is the fundamental flaw in your screening algorithm. No company stock price continues to go up without end for 5-10 years. The price of every company's stock goes down at some point. You have to decide on a company by company basis whether you want to ride out the downturn or sell and get out. This is a personal decision based on your own research. The list of screening criteria you list indicates you are looking for solid earnings companies. Try not to apply these rules rigidly because every company runs through a rough patch. At times past, GE (for example) met all of your criteria. However, in 2017, it would not and therefore would not meet your screening criteria. Would you sell GE if you owned it? Maybe, or maybe you would hold through the downturn. The same be said for MSFT in 2010 or AAPL pre-Jobs return. A rule you may want to add to your list: know the company business well; that is, don't invest in companies you have no understanding of their business model." "Hello! I am working on a project where I am trying to determine the profit made by a vendor if they hold our funds for 5 days in order to collect the interest on those funds during that period before paying a third party.  Currently I am doing ""Amount x(Fed Funds Rate/365)x5"" but my output seems too low. Any advice?" The real value of a share of stock is the current cash value of all dividends the owner will receive, plus the current cash value of the final liquidation if any.  Since people with different needs may judge the current cash value of an income stream differently, there would be a market basis for people to buy and sell stocks even if everyone could predict all future payouts perfectly. If shareholders knew that a company wouldn't pay any dividends until it was liquidated in the year 2066, whereupon it would pay $2000/share, then each share would in 2016 effectively be a fifty-year zero-coupon bond with a $2000 maturity value.  While some investors would be willing to trade in such an instrument, the amount of money a company could charge for such an instrument would be far lower than the money it could charge for one with payouts that were more evenly distributed through time. Since the founders of most companies want their companies to be around for a long time, that would mean that shareholders would have no expectation of their shares ever yielding anything of value within any foreseeable timeframe.  Even those who would be more interested in share-price appreciation than dividends wouldn't be able to see share prices rise if there wasn't any likelihood of the stock being bought by someone who wanted the dividends. The owner of a long futures contract does not receive dividends, hence this is a disadvantage compared to owning the underlying stock. If the dividend is increased, and the future price would not change, there is an arbitrage possibility. For the sake of simplicity, assume that the stock suddenly starts paying a dividend, and that the risk free rate is zero (so interest does not play a role). One can expect that the future price is (rougly) equal to the stock price before the dividend announcment. If the future price would not change, an investor could buy the stock, and short a futures contract on the stock. At expiration he has to deliver the stock for the price set in the contract, which is under the assumptions here equal to the price he bought the stock for. But because he owned the stock, he receives the announced dividend. Hence he can make a risk-free profit consisting of the divivends. If interest do play a role, the argument is similar. "Your company actually will most likely use some sort of options pricing model, either a binomial tree or black-scholes to determine the value for their accounting and, subsequently, for their issuance and realization.  First, market value of equity will be determined. Given you're private (although ""pre-IPO could mean public tomorrow,""), this will likely revolve around a DCF and/or market approaches.  Equity value will then be compared to a cap table to create an equity waterfall, where the different classes of stock and the different options will be valued along tranches. Keep in mind there might be liquidation preferences that would make options essentially further out of the money.  As such, your formulae above do not quite work. However, as an employee, it might be difficult to determine the necessary inputs to determine value.  To estimate it, however, look for three key pieces of information:  1. Current equity value  2. Option strike price  3. Maturity for Options  If the strike is close to the current equity value, and the maturity is long enough, and you expect the company to grow, then it would look like the options have more value than not.   Equity value can be derived from enterprise value, or by directly determining it via a DCF or guideline multiples.   Reliable forecasts should come from looking at the industry, listening to what management is saying, and then your own information as an insider." You simply add the dividend to the stock price when calculating its annual return. So for year one, instead of  it would be EDIT: After reading one of the comments on the original question, I realized that there is a much more intuitive way to think about this. If you look at it as a standard PV calculation and hold each of the cashflows constant. Really what's happening is that because of inflation the discount rate isn't the full value of the interest rate. Really the discount rate is only the portion of the interest rate above the inflation rate. Hence in the standard perpetuity PV equation PV = A / r r becomes the interest rate less the inflation rate which gives you PV = A / (i - g). That seems like a much better way to get to the answer than all the machinations I was originally trying. Original Answer: I think I finally figured this out. The general term for this type of system in which the payments increase over time is a gradient series annuity. In this specific example since the payment is increasing by a percentage each period (not a constant rate) this would be considered a geometric gradient series. According to this link the formula for the present value of a geometric gradient series of payments is: Where P is the present value of this series of cashflows.   A_1 is the initial payment for period 1 (i.e. the amount you want to withdraw adjusted for inflation).   g is the gradient or growth rate of the periodic payment (in this case this is the inflation rate)   i is the interest rate   n is the number of payments This is almost exactly what I was looking for in my original question. The only problem is this is for a fixed amount of time (i.e. n periods). In order to figure out the formula for a perpetuity we need to find the limit of the right side of this equation as the number of periods (n) approaches infinity. Luckily in this equation n is already well isolated to a single term: (1 + g)^n/(1 + i)^-n}. And since we know that the interest rate, i, has to be greater than the inflation rate, g, the limit of that factor is 0. So after replacing that term with 0 our equation simplifies to the following: Note: I don't do this stuff for a living and honestly don't have a fantastic finance IQ. It's been a while since I've done any calculus or even this much algebra so I may have made an error in the math. I use the following method. For each stock I hold long term, I have an individual table which records dates, purchases, sales, returns of cash, dividends, and way at the bottom, current value of the holding. Since I am not taking the income, and reinvesting across the portfolio, and XIRR won't take that into account, I build an additional column where I 'gross up' the future value up to today() of that dividend by the portfolio average yield at the date the dividend is received. The grossing up formula is divi*(1+portfolio average return%)^((today-dividend date-suitable delay to reinvest)/365.25) This is equivalent to a complex XMIRR computation but much simpler, and produces very accurate views of return.  The 'weighted combined' XIRR calculated across all holdings then agrees very nearly with the overall portfolio XIRR. I have done this for very along time.  TR1933 Yes, 1933 is my year of birth and still re investing divis! "In the USA there are two ways this situation can be treated. First, if your short position was held less than 45 days. You have to (when preparing the taxes) add the amount of dividend back to the purchase price of the stock. That's called adjusting the basis. Example: short at $10, covered at $8, but during this time stock paid a $1 dividend. It is beneficial for you to add that $1 back to $8 so your stock purchase basis is $9 and your profit is also $1. Inside software (depending what you use) there are options to click on ""adjust the basis"" or if not, than do it manually specifically for those shares and add a note for tax reviewer. Second option is to have that ""dividednd payment in lieu paid"" deducted as investment expence. But that option is only available if you hold the shorts for more than 45 days and itemize your deductions. Hope that helps!" A futures contract is based upon a particular delivery date.  In the case of a stock index futures contract is a cash settled futures contract based upon the stock index value at a particular point in time (i.e. this is when the final settlement is determined). In your example, the S&P 500 (SPX) is a price return index - that is, it is not affected by dividends and therefore dividends are not incorporated into the index value.  Dividends will affect the price of the constituent stocks (not necessarily by the same amount as the dividend) so they do have influence on the stock index value.  Since the dividends are known ahead of time (or at least can be estimated), this has already been factored into the futures price by the market. In terms of the impact of a dividend by AAPL, AAPL is approximaetely 3.6% of the index.  Apple pays out dividends 4 times a year (currently paying out $0.52 dividends).  Assuming the market is otherwise steady and AAPL drops by $0.52 due to the dividend and Apple is priced at around $105, this would result in a drop in the index of 0.0178% or around 0.35 points. Interesting fact:  There are some futures contracts that are based upon Total Return indexes, such as the German DAX and the above logic would need to be reversed. No - there are additional factors involved. Note that the shares on issue of a company can change for various reasons (such as conversion/redemption of convertible securities, vesting of restricted employee shares, conversion of employee options, employee stock purchase programs, share placements, buybacks, mergers, rights issues etc.) so it is always worthwhile checking SEC announcements for the company if you want an exact figure.  There may also be multiple classes of shares and preferred securities that have different levels of dividends present. For PFG, they filed a 10Q on 22 April 2015 and noted they had 294,385,885 shares outstanding of their common stock.  They also noted for the three months ended March 31 2014 that dividends were paid to both common stockholders and preferred stockholders and that there were Series A preferred stock (3 million) and Series B preferred stock (10 million), plus a statement: In February 2015, our Board of Directors authorized a share repurchase program of up to $150.0 million of our outstanding common stock. Shares repurchased under these programs are accounted for as treasury stock, carried at cost and reflected as a reduction to stockholders’ equity. Therefore the exact amount of dividend paid out will not be known until the next quarterly report which will state the exact amount of dividend paid out to common and preferred shareholders for the quarter. "Below I will try to explain two most common Binomial Option Pricing Models (BOPM) used. First of all, BOPM splits time to expiry into N equal sub-periods and assumes that in each period the underlying security price may rise or fall by a known proportion, so the value of an option in any sub-period is a function of its possible values in the following sub period. Therefore the current value of an option is found by working backwards from expiry date through sub-periods to current time. There is not enough information in the question from your textbook so we may assume that what you are asked to do is to find a value of a call option using just a Single Period BOPM. Here are two ways of doing this: First of all let's summarize your information: Current Share Price (Vs) = $70 Strike or exercise price (X) = $60 Risk-free rate (r) = 5.5% or 0.055 Time to maturity (t) = 12 months Downward movement in share price for the period (d) = $65 / $70 = 0.928571429 Upward movement in share price for the period (u) = 1/d = 1/0.928571429 = 1.076923077 ""u"" can be translated to $ multiplying by Vs => 1.076923077 * $70 = $75.38 which is the maximum probable share price in 12 months time. If you need more clarification here - the minimum and maximum future share prices are calculated from stocks past volatility which is a measure of risk. But because your textbook question does not seem to be asking this - you probably don't have to bother too much about it yet. Intrinsic Value: Just in case someone reading this is unclear - the Value of an option on maturity is the difference between the exercise (strike) price and the value of a share at the time of the option maturity. This is also called an intrinsic value.  Note that American Option can be exercised prior to it's maturity in this case the intrinsic value it simply the diference between strike price and the underlying share price at the time of an exercise. But the Value of an option at period 0 (also called option price) is a price you would normally pay in order to buy it. So, say, with a strike of $60 and Share Price of $70 the intrinsic value is $10, whereas if Share Price was $50 the intrinsic value would be $0. The option price or the value of a call option in both cases would be fixed.  So we also need to find intrinsic option values when price falls to the lowest probable and rises to the maximum probable (Vcd and Vcu respectively) (Vcd) = $65-$60 = $5 (remember if Strike was $70 then Vcd would be $0 because nobody would exercise an option that is out of the money) (Vcu) = $75.38-$60 = $15.38 1. Setting up a hedge ratio:  h = Vs*(u-d)/(Vcu-Vcd) h = 70*(1.076923077-0.928571429)/(15.38-5) = 1 That means we have to write (sell) 1 option for each share purchased in order to hedge the risks. You can make a simple calculation to check this, but I'm not going to go into too much detail here as the equestion is not about hedging. Because this position is risk-free in equilibrium it should pay a risk-free rate (5.5%). Then, the formula to price an option (Vc) using the hedging approach is: (Vs-hVc)(e^(rt))=(Vsu-hVcu) Where (Vc) is the value of the call option, (h) is the hedge ratio, (Vs) - Current Share Price, (Vsu) - highest probable share price, (r) - risk-free rate, (t) - time in years, (Vcu) - value of a call option on maturity at the highest probable share price. Therefore solving for (Vc): (70-1*Vc)(e^(0.055*(12/12))) = (75.38-1*15.38) => (70-Vc)*1.056540615 = 60 => 70-Vc = 60/1.056540615 => Vc = 70 - (60/1.056540615)  Which is similar to the formula given in your textbook, so I must assume that using 1+r would be simply a very close approximation of the formula above. Then it is easy to find that Vc = 13.2108911402 ~ $13.21 2. Risk-neutral valuation: Another way to calculate (Vc) is using a risk-neutral approach. We first introduce a variable (p) which is a risk-neutral probability of an increase in share price. p = (e^(r*t)-d)/(u-d) so in your case: p = (1.056540615-0.928571429)/(1.076923077-0.928571429) = 0.862607107 Therefore using (p) the (Vc) would be equal: Vc = [pVcu+(1-p)Vcd]/(e^(rt)) => Vc = [(0.862607107*15.38)+(0.137392893*5)]/1.056540615 => Vc = 13.2071229185 ~ $13.21 As you can see it is very close to the hedging approach. I hope this answers your questions. Also bear in mind that there is much more to the option pricing than this. The most important topics to cover are: Multi-period BOPM Accounting for Dividends Black-Scholes-Merton Option Pricing Model" "A bit strange but okay.  The way I would think about this is again that you need to determine for what purpose you're computing this, in much the same way you would if you were to build out the model. The IPO valuation is not going to be relevant to the accretion/dilution analysis unless you're trying to determine whether the transaction was net accretive at exit. But that's a weird analysis to do. For longer holding periods like that you're more likely to look at IRR, not EPS. EPS is something investors look at over the short to medium term to get a sense of whether the company is making good acquisition decisions. And to do that short-to-medium term analysis, they look at earnings.   Damodaran would say this is a shitty way of looking at things and that you should probably be looking at some measure of ROIC instead, and I tend to agree, but I don't get paid to think like an investor, I get paid to sell shit to them (if only in indirect fashion). The short answer to your question is that no, you should not incorporate what you are calling liquidation value when determining accretion/dilution, but only because the market typically computes accretion/dilution on a 3-year basis tops. I've never put together a book or seen a press release in my admittedly short time in finance that says ""the transaction is estimated to be X% accretive within 4 years"" - that just seems like an absurd timeline.  Final point is just that from an accounting perspective, a gain on a sale of an asset is not going to get booked in either EBITDA or OCF, so just mechanically there's no way for the IPO value to flow into your accretion/dilution analysis there, even if you are looking at EBITDA/shares. You could figure the gain on sale into some kind of adjusted EBITDA/shares version of EPS, but this is neither something I've ever seen nor something that really makes sense in the context of using EPS as a standardized metric across the market. Typically we take OUT non-recurring shit in EPS, we don't add it in. Adding something like this in would be much more appropriate to measuring the success of an acquisition/investing vehicle like a private equity fund, not a standalone operating company that reports operational earnings in addition to cash flow from investing. And as I suggest above, that's an analysis for which the IRR metric is more ideally situated.  And just a semantic thing - we typically wouldn't call the exit value a ""liquidation value"". That term is usually reserved for dissolution of a corporate entity and selling off its physical or intangible assets in piecemeal fashion (i.e. not accounting for operational synergies across the business). IPO value is actually just going to be a measure of market value of equity." "The other two folks here are right with the math and such, so I'll just throw some intuition out there for you.  The basis for this valuation model is really just tacking the Gordon growth model (which is really just a form of valuing a perpetuity) onto a couple of finite discounted cash flows. So that ending part is the Gordon growth model *at the future point* discounted back to the present.  The Gordon growth model uses a ""next period"" dividend for the very simple reason that it's the next one you'd get if you bought the stock.  Is that explanation clear enough, or were some of these points not adequately explained in your class? I'll help a bit more, if I can."