• Discounting Cash Flows

How to Discount Future Cash Flows

The Problem

Consider an asset that trades at price \(P_t\) and pays dividends \(D_{t+1},D_{t+2},\dots\).
For example, the asset might be the S&P 500 portfolio, and the dividends the cash flows that this portfolio pays in month \(t\).
How does the asset's price relate to its future dividends?
Intuitively, if the economy performs well in the future, companies will be able to increase their dividend payments and this should be good news for the value of the asset, but we are also uncertain about future dividend payments and in general we prefer cash rather sooner than later and one bird in the hand is better than two in the bush so when we calculate the value of this asset we should form some sort of expectation about how fast the payouts will grow and apply some sort of discount rate that takes the time value of money and uncertainty into account.
How do we do all of this?
As it turns out, this problem is more complicated than one might think and there are several different approaches, each one with its own advantages and disadvantages, none of them clearly dominating the others.
In the following, I explain the different discounting models and discuss how they relate to each other.

Pricing Risk-free Bonds

To start with a simple case, assume for the moment we are interested in a risk-free bond.
Our bond costs \(P_t\) and it pays coupons \(C\) from time \(t+1\) to maturity at some future date \(t+\tau\) at which the bond also repays its face value \(F\) (also called principal).
In this case we can define \(R\) as the number that solves the following equation: $$ P_t^{\text{bond}} = \frac{C}{R} + \frac{C}{R^2} + \cdots + \frac{C}{R^{t+\tau}} + \frac{F}{R^{t+\tau}} $$ The number \(R\) (or \(R-1\)) is called the yield to maturity of the bond.
This yield is the return per period we would achieve if we hold this bond until maturity and reinvest all coupons at the same rate \(R\).
The set of all such yields for various maturities is called the yield curve or term structure of interest rates and we can draw beautiful pictures of this curve such as this one.
Alternatively, we can also think of our bond as a portfolio of zero-coupon bonds (such as Treasury strips).
A zero-coupon bonds makes a single payment at the end of its life, but it does not pay any coupons in between. For example, if the zero-coupon bond pays $1000 at time \(t+\tau\), we can calculate its yield as $$ P_t^{\text{zero}} = \frac{1000}{(R_{ft,t+\tau})^\tau} $$ where \(R_{ft,t+\tau}\) is the risk-free rate at time \(t\) for payments that occur at \(t+\tau\).
We can replicate each single cash flow of the coupon-paying bond with a zero-coupon bond with corresponding maturity (for example, to replicate the first coupon payment we purchase \(C/1000\) zero-coupon bonds that mature at \(t+1\) where we conveniently assume that these bonds are perfectly divisible which is a good enough approximation if we talk about large portfolios).
The price of this replicating portfolio is given by $$ P_t^{\text{replicating portfolio}} = \frac{C}{(R_{ft,t+1})^1} + \frac{C}{(R_{ft,t+2})^2} + \cdots + \frac{C}{(R_{ft,t+\tau})^{t+\tau}} + \frac{F}{(R_{ft,t+\tau})^{t+\tau}} $$ Since the replicating portfolio generates the same cash flows as the original bond we have $$ P_t^{\text{bond}} = P_t^{\text{replicating portfolio}} $$ If these two prices where not equal, investors could make a risk-free profit by buying the cheaper securities and selling the more expensive securities (such a profit opportunity is called an arbitrage opportunity).
Hence, if there is no arbitrage, we can find the price of any bond by discounting its cash flows with the risk-free rates for the corresponding maturities.

Discounting Dividend Strips

Is there a way to apply the discounting method for bonds to assets that pay risky dividends?
Assume for the moment that we could purchase a claim to each dividend individually and lets call these claims dividend strips. These dividend strips are the risky equivalent to the zero-coupon bonds of the previous section.
Let \(P_{t,t+\tau}\) be the time \(t\) price for the time \(t+\tau\) dividend and define the geometric average return of this asset as $$ (R_{t,t+\tau})^\tau = \frac{D_{t+\tau}}{P_{t,t+\tau}} $$ This equation is just a definition and holds therefore for any realization of prices and dividends. If today is time \(t\) and we don't know the future dividend, we can take the mathematical expectation on both sides of this equation, and we get $$ E_t\left[(R_{t,t+\tau})^\tau\right] = \frac{E_t[D_{t+\tau}]}{P_{t,t+\tau}} \hspace{1cm}\implies\hspace{1cm} P_{t,t+\tau} = \frac{E_t[D_{t+\tau}]}{E_t\left[(R_{t,t+\tau})^\tau\right]} $$ where the subscript \(t\) of \(E_t\) indicates that we calculate this expectation conditional on our information at time \(t\), and \(P_{t,t+\tau}\) is not inside the expectation since this price is known at time \(t\).
Now we can apply the same replicating idea as in the case of the risk-free bond and we can treat the asset as a portfolio of an infinite number of dividend strips and assume that investors will exploit any arbitrage opportunity and therefore the price of the asset is given by $$ P_t = \sum_{\tau=1}^\infty P_{t,t+\tau} = \sum_{\tau=1}^\infty\frac{E_t[D_{t+\tau}]}{E_t\left[(R_{t,t+\tau})^\tau\right]} $$ where we of course assume that the series of dividend strip prices converges (which means that dividends that are paid far in the future are discounted so much that their present values have no significant effect on the overall value of the portfolio so that for practical purposes we don't really need an infinite number of dividend strips and we can approximate this equation with a sufficiently large finite number).
To compare this last result with the risk-free case we can define \(\pi_{t,t+\tau}\) as the numbers that solve $$ E_t\left[(R_{t,t+\tau})^\tau\right] = (R_{ft,t+\tau}+\pi_{t,t+\tau})^\tau $$ If the dividend \(D_{t+\tau}\) would be risk-free than investors would discount it at the risk-free rate \(R_{ft,t+\tau}\). But since the dividend is uncertain, investors will in general demand a higher return (or pay a lower price) for this dividend which means they expect to earn a premium in excess of the risk-free rate for exposing themselves to risk, and therefore we can interpret \(\pi_{t,t+\tau}\) as the (time \(t\)) risk premium for the dividend strip with maturity \(\tau\).
Plugging this definition into our equation for the asset price we get $$ P_t = \sum_{\tau=1}^\infty\frac{E_t[D_{t+\tau}]}{(R_{ft,t+\tau}+\pi_{t,t+\tau})^\tau} $$ Since it is often more convenient to work with growth rates than levels, we define the one-period grow rate of dividends as $$ G_{Dt+1}=\frac{D_{t+1}}{D_t} $$ and the corresponding compound growth as $$ G_{Dt}^{t+\tau}=G_{Dt+1}\times G_{Dt+2}\times\cdots\times G_{Dt+\tau} = \frac{D_{t+1}}{D_t}\times \frac{D_{t+2}}{D_{t+1}}\times\cdots\times \frac{D_{t+\tau}}{D_{\tau-1}} = \frac{D_{t+\tau}}{D_t} $$ and now we can divide the pricing equation by the current dividend \(D_t\) and get our final result $$ \frac{P_t}{D_t} = \sum_{\tau=1}^\infty\frac{E_t\left[G_{Dt}^{t+\tau}\right]}{(R_{ft,t+\tau}+\pi_{t,t+\tau})^\tau}. $$ This ratio is called the price-dividend ratio (the inverse of the dividend yield ).
The equation says that the price-dividend ratio of the asset is a function of the expected future growth rates of the dividends and of the rates at which investors discount these dividends. For example, the price-dividend ratio increases if investors become more optimistic about the growth rate of future dividends or if the risk-free decreases or if the amount of risk and therefore the risk premium decreases.
The price-dividend equation is great because it provides us with a way to relate economic growth rates and uncertainty and the term structure of interest rates to the fundamental value of the asset.
However, there is one problem with our approach.
The risk premiums \(\pi\) are not equal to the risk premium that investors demand for purchasing this asset, but instead they are risk premiums for hypothetical claims to individual dividends.
Since the dividend strips do not exist in the market, we cannot observe their prices and we therefore have a hard time even answering basic questions such as how do these premiums depend on the maturity of the dividends?

Discounting Dividends with Expected Asset Returns

In the previous section, we learned how to discount dividends with the returns of hypothetical claims to individual dividends.
Is there a way to replace these discount rates for individual dividends with the discount rate (or return) of the entire asset?

Define the (gross) return as $$ R_{t+1} = \frac{P_{t+1}+D_{t+1}}{P_t}. $$ Solving this equation for the current price: $$ P_t = \frac{P_{t+1}+D_{t+1}}{R_{t+1}}. $$ Hence the price of the asset equals its payoff in the next period, discounted at its return \(R_{t+1}\).
Since this equation holds for any time \(t\), we can replace \(P_{t+1}\) with its discounted payoff for \(t+2\) and get: $$ P_t = \frac{\frac{P_{t+2}+D_{t+2}}{R_{t+2}}+D_{t+1}}{R_{t+1}} = \frac{D_{t+1}}{R_{t+1}}+\frac{D_{t+2}}{R_{t+1}R_{t+2}}+\frac{P_{t+2}}{R_{t+1}R_{t+2}}. $$ Defining the compound return from time \(t\) to some future time \(t+\tau\) as $$ R_t^{t+\tau}=R_{t+1}\times R_{t+2}\times\cdots\times R_{t+\tau}, $$ we can write our pricing equation more succinctly as $$ P_t = \frac{D_{t+1}}{R_{t+1}}+\frac{D_{t+2}}{R_t^{t+2}}+\frac{P_{t+2}}{R_t^{t+2}}. $$ If we continue to solve this equation forward until some future time \(\tau\), we get $$ P_t = \sum_{j=1}^\tau\frac{D_{t+j}}{R_t^{t+j}}+\frac{P_{t+\tau}}{R_t^{t+\tau}}. $$ Hence the price equals the discounted value of the next \(\tau\) future dividends plus the discounted value of the future price at time \(\tau\).
If we divide both sides of the pricing equation by the current dividend \(D_t\) and plug in the definition for the compound dividend growth, we get an expression for the price relative to the current dividend: $$ \frac{P_t}{D_t} = \sum_{j=1}^\tau\frac{G_{Dt}^{t+j}}{R_t^{t+j}} + \frac{1}{D_t}\frac{P_{t+\tau}}{R_t^{t+\tau}}. $$ Since all we did so far was rearraging definitions, our expression for the price-dividend ratio must hold for any arbitrary realization of prices and dividends.
But what happens if we are today at time \(t\) and we don't know what future dividends and returns will be? In this case, we can apply the mathematical expectation on both sides of the equation: $$ E_t\left[\frac{P_t}{D_t}\right] = E_t\left[\sum_{j=1}^\tau\frac{G_{Dt}^{t+j}}{R_t^{t+j}}+\frac{1}{D_t}\frac{P_{t+\tau}}{R_t^{t+\tau}}\right], $$ where the subscript \(t\) indicates that we calculate the expectation conditional on our information at time \(t\). Using the fact that the current price-dividend ratio is known at time \(t\) and that the expectation is a linear operator, we can rewrite this equation as $$ \frac{P_t}{D_t} = \sum_{j=1}^\tau E_t\left[\frac{G_{Dt}^{t+j}}{R_t^{t+j}}\right] + \frac{1}{D_t}E_t\left[\frac{P_{t+\tau}}{R_t^{t+\tau}}\right]. $$ Now we have the price-dividend ratio as a function of expected future dividend growth rates and returns.
One slightly annoying component of our expression is the term on the right involving the final price \(P_{t+\tau}\).
What happens if we continue to solve this equation forward until some very distant time \(\tau\), say 200 years from now?
Intuitively, if the asset pays dividends, \(P_{t+\tau}\) \(/\) \(R_t^{t+\tau}\) should tend to zero in the long run since the return including dividends (in the denominator) exceeds the price appreciation (in the nominator). To understand this better we can go back to our definition of returns and rearranage the equation for the compound return as $$ R_t^{t+\tau} = \frac{P_{t+\tau}}{P_t} \left(1+\frac{D_{t+1}}{P_{t+1}}\right) \left(1+\frac{D_{t+2}}{P_{t+2}}\right)\cdots\left(1+\frac{D_{t+\tau}}{P_{t+\tau}}\right) $$ Hence the compound return is the product of the price appreciation and the "compound dividend yield".
As we can see from the last equation, we can expect returns to grow faster than prices if the dividend yield exceeds some lower bound (say 0.00001) "often enough with high enough probability" (we could make this statement mathematically precise but those technicalities just distract us from the big picture here).
So taking the limit and assuming convergence we arrive at: $$ \frac{P_t}{D_t} = \sum_{j=1}^\infty E_t\left[\frac{G_{Dt}^{t+j}}{R_t^{t+j}}\right] $$ This equation says that the value of the asset depends on how fast we expect dividends to grow relative to the return we require for investing in the asset.
There is only one problem with our result: we have no way to disentangle growth rates and discount rates since $$ E_t\left[\frac{G_{Dt}^{t+j}}{R_t^{t+j}}\right] \ne \frac{E_t\left[G_{Dt}^{t+j}\right]}{E_t\left[R_t^{t+j}\right]}. $$ This really sucks because we would like to estimate growth rates and discount rates to arrive at a fundamental value for our asset but instead now we also have to know the joint probability distribution of these processes which is a huge mess and very difficult to implement.
So what now?
One possible way out of this malaise is to make some assumptions on the nature of expected growth and discount rates, which brings us to the next section.

Constant Expected Growth Rates and Returns

Going back to the definition of returns, we have: $$ P_tR_{t+1}=P_{t+1}+D_{t+1} \hspace{1cm}\implies\hspace{1cm} P_tE_t[R_{t+1}] = E_t[P_{t+1}+D_{t+1}]. $$ Hence we can write the price as the expected payoff discounted at the expected return: $$ P_t = \frac{E_t[P_{t+1}+D_{t+1}]}{E_t[R_{t+1}]}. $$ At this point we make a big assumption: all expected returns are constant (and therefore know at time \(t\)): $$ E[R] = E_t[R_{t+1}] = E_{t+1}[R_{t+2}] = E_{t+2}[R_{t+3}] = \cdots $$ Since the equation for the price holds for every time \(t\) we can solve it forward (same trick as above) as $$ P_t = \frac{E_t\left[\frac{E_{t+1}[P_{t+2}+D_{t+2}]}{E[R]}+D_{t+1}\right]} {E[R]} = \frac{E_t[D_{t+1}]}{E[R]} + \frac{E_t[E_{t+1}[D_{t+2}]]}{E[R]^2} + \frac{E_t[E_{t+1}[P_{t+2}]]}{E[R]^2} $$ Using the rule of iterated expectation (my expectation today about my expectation tomorrow about \(X\) equals my expectation today about \(X\)), we can simplify the expression to $$ P_t = \frac{E_t[D_{t+1}]}{E[R]} + \frac{E_t[D_{t+2}]}{E[R]^2} + \frac{E_t[P_{t+2}]}{E[R]^2} $$ Continuing to solve this equation forward we get $$ P_t = \sum_{j=1}^\tau\frac{E_t[D_{t+j}]}{E[R]^j} + \frac{E_t[P_{t+\tau}]}{E[R]^\tau} $$ Dividing by the current dividend and taking the limit (see discussion about convergence above): $$ \frac{P_t}{D_t} = \sum_{j=1}^\infty\frac{E_t[G_{Dt}^{t+j}]}{E[R]^j} $$ So now we got what we wanted. The price-dividend ratio depends on expected dividend growth rates and expected discount rates and we can form our own beliefs about these quantities and then calculate the fundamental value of the asset.
Unfortunately this result comes at a price: we assumed expected returns are constant.
Is this assumption a big deal? Yes.
There is plenty of empirical evidence that expected returns vary quite a bit over time and some people who spent a considerable fraction of their days staring at these data even believe that most of the variation of the price-dividend ratio originates from movements in the discount rates and in any case we know for sure that risk-free rates are not constant which implies that expected returns can only be constant if the risk premium moves excactly opposite to the risk-free rate and that doesn't make much sense at all.
So we established that this is a dead end, but since we already spent all this effort to get here and since this is so much fun lets go one step further and assume that expected growth rates are constant as well: $$ E[G_D] = E_t[G_{Dt+1}] = E_{t+1}[G_{Dt+2}] = E_{t+2}[G_{Dt+3}] = \cdots $$ Plugging this growth rate into our pricing equation we get $$ \frac{P_t}{D_t} = \sum_{j=1}^\infty\frac{E[G_D]^j}{E[R]^j} = \frac{E[G_D]}{E[R]-E[G_D]} $$ (provided that \(E[R]>E[G_D]\)). This result is also know as the Gordon growth model (even though Myron Gordon was not the first person to come up with this formula, and by the way there is also a much simpler way to derive this model using the formula for a growing perpetuity but our derivation fits nicely into the framework we developed so far).
The constant-growth model is nice because its simple, for example the current price-dividend ratio of the S&P 500 is somewhere around 50, so if we assume that dividends will continue to grow at their historical rate of somewhere around 5% per year, we can conclude that the expected return of the index is about 7% per year.
Unfortunately this model is not just simple but really much too simple since all expectations on the right-hand side of the equation are constant which implies that the price-dividend ratio is constant and that is clearly not the case (historically, the S&P 500 price-dividend ratio varied somewhere between 10 and 90).
Of course one might wonder what happens if investors today believe dividends grow at 5% and tomorrow they become more optimistic and adjust their forecast to 6%, wouldn't that change the price-dividend ratio and in that way explain why the market moves up or down, but this idea would be misguided since we derived this model under the assumption that expectations never change and therefore if we allow investors to change their mind we cannot apply this model anymore.

Log-Linear Approximation

We are still looking for sensible way to express the price-dividend ratio as a function of expected growth rates and expected returns.
Since we had no luck so far with trying to separate growth rates from returns we will try a different approach and look for an approximate solution instead of an exact one.
Lets rearrange our original definition of the return one more time: $$ R_{t+1} = \frac{P_{t+1}+D_{t+1}}{P_t} = \frac{\left(\frac{P_{t+1}}{D_{t+1}}+1\right)D_{t+1}}{P_t} = \frac{\left(\frac{P_{t+1}}{D_{t+1}}+1\right)\frac{D_{t+1}}{D_t}}{\frac{P_t}{D_t}} $$ Taking logs on both sides of this equation (I write \(\hat{X}\) for log\((X)\)): $$ \hat{R}_{t+1} = \hat{G}_{Dt+1} + \log\left(\frac{P_{t+1}}{D_{t+1}}+1\right) - \log\left(\frac{P_t}{D_t}\right) $$ We can write log\((P_{t+1}/D_{t+1}+1)\) as $$ \log\left(\frac{P_{t+1}}{D_{t+1}}+1\right) = \log\left[e^{\log\frac{P_{t+1}}{D_{t+1}}}+1\right] = f\left(\log\frac{P_{t+1}}{D_{t+1}}\right) $$ and apply a first-order Taylor approximation $$ f(x) \approx f(x_0) + f'(x_0)\times(x-x_0) $$ around $$ x_0 = 1 + e^{E\left[\log\frac{P_{t+1}}{D_{t+1}}\right]} $$ and if we plug this into the Taylor approximation we get $$ \log\left(\frac{P_{t+1}}{D_{t+1}}+1\right) \approx k_0+k_1 \log\left(\frac{P_{t+1}}{D_{t+1}}\right). $$ where $$ k_0 = \log\left[1+Y\right] - \frac{Y}{1+Y}\log Y, \hspace{1cm} k_1 = \frac{Y}{1+Y}, \hspace{1cm} Y = e^{E\left[\log\frac{P_{t+1}}{D_{t+1}}\right]} $$ Hence we have for the log return: $$ \hat{R}_{t+1} \approx \hat{G}_{Dt+1} + k_0 + k_1\log\frac{P_{t+1}}{D_{t+1}} - \log\frac{P_t}{D_t} $$ Now we have a linear relation between the return, the dividend growth rate and the price-dividend ratio, which is great since this linear form allows to achieve our desired separation of growth rates and discount rates.
Solving the return equation for the current price-dividend ratio: $$ \log\frac{P_t}{D_t} \approx \hat{G}_{Dt+1} - \hat{R}_{t+1} + k_0 + k_1\log\frac{P_{t+1}}{D_{t+1}} $$ Since this equation holds for every time \(t\) we can replace the future price-dividend ratio as $$ \log\frac{P_t}{D_t} \approx \hat{G}_{Dt+1} - \hat{R}_{t+1} + k_0 + k_1\left(\hat{G}_{Dt+2} - \hat{R}_{t+2} + k_0 + k_1\log\frac{P_{t+2}}{D_{t+2}}\right) $$ and if we continue to solve this equation forward we get $$ \log\frac{P_t}{D_t} \approx \sum_{j=1}^\infty k_1^{j-1}\big(\hat{G}_{Dt+j}-\hat{R}_{t+j}\big) + \frac{k_0}{1-k_1}. $$ Taking expectations conditional on information at time \(t\): $$ \log\frac{P_t}{D_t} \approx \sum_{j=1}^\infty k_1^{j-1}\big(E_t[\log G_{Dt+j}]-E_t[\log R_{t+j}]\big) + \frac{k_0}{1-k_1}. $$ Now we have a nice result: we can express the fundamental value of the asset as a function of expected dividend growth rates and expected returns on the asset.
How does this approximation compare to the exact value?
The function \(f(x)=\log(1+e^x)\) is convex: $$ f'(x)=\frac{e^x}{1+e^x}>0, \hspace{0.5cm}\implies\hspace{0.5cm} f''(x) = \frac{e^x(1+e^x-e^xe^x)}{(1+e^x)^2} = \frac{e^x}{(1+e^x)^2} > 0 $$ Hence $$ k_0+k_1\log\frac{P_t}{D_t}\le\log\left(1+\frac{P_t}{D_t}\right) $$ (with strict inequality at \(P/D\ne E[P/D]\)) and therefore the approximation underestimates the exact price-dividend ratio. But, in the data, this approximation works very well, and the approximation errors are small enough that we can neglect them.

One important difference to the previous discount models is that this model uses the current expectations of future short-term returns \(E_t[\log R_{t+j}]\) instead long-term returns compounded from today to the future (\(E_t[R_t^{t+j}]\)).
Therefore, if we would like to use interest rates to estimate expected returns, we need to estimate future short-term interest rates. Estimating these rates is difficult since the the difference between short-term and long-term rates does not only depend on expected future rates but also on a risk premium associated with long-term bonds.

Comparing the Three Models

Here are the main three models for the price-dividend ratio:

1. Discount rates = returns on dividend strips: $$ \frac{P_t}{D_t} = \sum_{\tau=1}^\infty \frac{E_t\left[G_{Dt}^{t+\tau}\right]}{(R_{ft,t+\tau}+\pi_{t,t+\tau})^\tau} $$ Good: separation of growth rates and discount rates.
   Bad: the risk premiums apply to claims on individual dividends which do not exist in the market and that makes it difficult to estimate or interpret these premiums.
2. Discount rates = returns on the asset: $$ \frac{P_t}{D_t} = \sum_{j=1}^\infty E_t\left[\frac{G_{Dt}^{t+j}}{R_t^{t+j}}\right] $$ Good: discount rates are returns on an asset that exists.
   Bad: we cannot separate growth rates from discount rates.
3. Log-linear approximation of (2): $$ \log\frac{P_t}{D_t} \approx \sum_{j=1}^\infty k_1^{j-1}\Big(E_t\big[\log G_{Dt+j}\big]-E_t\big[\log R_{t+j}\big]\Big) + \frac{k_0}{1-k_1} $$ Good: discount rates are returns on an asset that exists and we can separate them from growth rates.
   Bad: We need to know expected future interest rates which are difficult to estimate.