# Present-Value Relations Kevin C.H. Chiang. Fundamental Value Cash flow (dividend) pricing models say that security price/value is the sum of discounted.

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Present-Value Relations Kevin C.H. Chiang

Fundamental Value Cash flow (dividend) pricing models say that security price/value is the sum of discounted values of expected cash flows (dividends). With the assumptions of (1) constant dividend growth rate, g, and (2) constant discount rate/expected return rate, R, the present security price/value, P(t), is given by: P(t) = D(t+1)/[R – g], where D(t+1) is the expected dividends in the next period.

Intuitions One can rewrite P(t) = D(t+1)/[R – g] as follows: D(t)/P(t) = [R – g] / [1 + g], where D(t)/P(t) is called dividend yield. Now suppose that, holding other factors constant, stock price, P(t), went down sharply today. What happened to dividend yield? Dividend yield went up sharply. The above equation says that (1) either expected return/discount rate, R, was revised upward, (2) dividend growth rate, g, was revised downward, (3) or a combination of both.

D(t)/P(t) = [R – g] / [1 + g] Alternatively, (1) If for some reasons investors do not like to take risk and demand a higher expected return/discount rate for investing in stocks, then R goes up; i.e., the RHS of the equation goes up. To balance the equation, P(t) needs to be lower. So, stock price drops. (2) If for some reasons investors believe that the growth rate of dividends would be lower than expected, then the RHS of the equation becomes higher. P(t) thus needs to be lower. (3) If scenarios (1) and (2) happen simultaneously, you can expect stock price drops “a lot.” Q: Between scenarios (1) and (2), which is the main driver of stock prices? Alternatively, should we pay more attention to discount rate/expected return or dividend growth rate?

Different views at the market level Traditionally, researchers looked at this question/issue at the market level; e.g., S&P 500. Before 1970s, researchers tended to think scenario/story (2) is more of the case. That is, S&P 500 drops mainly because dividend growth is revised downward by the investors. After 1970s, most researchers changed their minds and thought story (1) is more close to the truth. That is, stock price drops because investors demand higher expected returns.

Why researchers changed their minds? They did so because they found a positive relation between S&P dividend yields and subsequent (particularly 3-5 years) S&P 500 returns. That is, when S&P 500 drops (i.e., S&P 500 dividend yield goes up), S&P 500 tend to have good performance during the next 3-5 years. This effect is weak and close to none in a short horizon, say less than 1-2 years. In contrast, there is no relationship between S&P 500 dividend yields and subsequent S&P 500 dividend growth rates.

If this view is correct If you think the view is correct and you are a top- down investor, then you should focus most of your research on expected return/discount rate. If the market is believed to be willing to take on more risk in the near future (i.e., investors will require lower risk premiums/expected returns on S&P 500), you can infer from it that S&P 500 dividend yield would be lower. Strategy: You should take a long position on S&P 500 because its price will go up to achieve a lower dividend yield.

A naïve example D(t)/P(t) = [R – g] / [1 + g]: Suppose that the current dividend yield is 4%; that is, the price/dividend ratio is 25. Then, R – g is roughly 4%. If for some reasons the market is experiencing a 1% permanent change in expected return/discount rate, this would translate into a 25% change in price! Intuition: a small change in expected return can leads to a big change in price. Note that this is an extreme, overstated example. In the real life, we rarely see a permanent change in anything.

Alternatively, If you believe that the view is correct and the S&P 500 dividend yield is now unusually high, then you should take a long position on S&P 500. The positive association between market dividend yields and the subsequent market returns implies that current high S&P 500 dividend yield will tend to lead to higher S&P 500 returns in the long run; e.g., next 3-5 years. Q: suppose that you are a short-term (e.g., 1-6 months) market timer and S&P 500 dividend yield is now relatively high. What should you do?

Can we really be sure about the dominant view? Although the variation in expected return/discount rate seems to be the main driver of market prices, we should be carefully about its robustness. Recent evidence suggests that the positive relationship between dividend yield and subsequent market returns is not robust after the 1990s. Chiang (2009) documents a positive relationship between REIT dividend yield and subsequent REIT dividend growth. It could be that the dominant view and the alternative view are both partially responsible for setting prices.

At the individual stock level The above discussions focused present- value relations at the market level. For bottom-up investors, present-value relations at the individual stock level are more relevant. It turns out that the present-value relations at the individual stock level are quite different from those documented at the market level.

Vuolteenaho (2002, J. of Finance) He found that firm-level stock returns are mainly driven by cash flow (i.e., dividends/earnings growth) news. Expected return news are highly correlated across firms. So, their impacts are observable at the market level. In contrast, cash flow news can largely diversified away at the market level. So, their impacts are not that observable at the market level.

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