Cash Flow News, Discount Rate News, and Return Variation

# Cashflow Risk vs Discount Risk


# Cashflow Risk vs Discount Risk












Studying asset pricing, I often hear the terms cashflow risk and discount risk but I'm not sure what they mean? The Campbell/Shiller (1988) decomposition includes cashflows (future dividends) and discount rates (expected returns) and hence identifies both risks?

Apparently, the long run risk model from Bansal and Yaron (2004) and the duration model from Lettau and Wachter (2007) discuss cashflow risk whereas the external habit model from Campbell and Cochrane (1999) is about discount risk? The investment decision model from Berk Green and Naik (1999) apparently includes both? What about the simple CAPM and CCAPM?

Campbell and Vuolteenaho (2004) use an ICAPM set-up to decompose market beta in cashflow and discount component and show that value stocks have higher CF betas.

## Answer by phdstudent (score 3, accepted)

https://quant.stackexchange.com/a/55876

The answer to your question could fill an entire asset pricing text book. Your question mixes theory and empirics.

A different way of looking at it is to look at the identity:

$$ 1 = E[M_t R_t]$$

To generate a sufficient risk premium either you need to have the covariance of the SDF with the the return to be sufficiently high.

Campbell and Cochrane basically change $M_t$ to generate a sufficiently volatile SDF.

Bansal and Yaron, use Epstein-Zin utility and change the standard cash-flow component of dividends. Lettau and Wachter similarly.

Empirically I think this blog post explains it super well: https://johnhcochrane.blogspot.com/2015/04/the-sources-of-stock-market-fluctuations.html

## Answer by fes (score 3)

https://quant.stackexchange.com/a/55867

The cash flow news / discount rate news decomposition is given by

$$r_{t+1}-\mathbb{E}_t[r_{t+1}]=(\mathbb{E}_{t+1}-\mathbb{E}_t)\sum_{j=0}^{\infty}\rho^j\Delta d_{t+1+j}-(\mathbb{E}_{t+1}-\mathbb{E}_t)\sum_{j=1}^{\infty}\rho^j\Delta r_{t+1+j},$$

where $r_{t}$ is log-return $d_{t}$ is log-dividend and $\rho$ is a constant. This follows directly from the Campbell-Shiller decomposition.

Here the second term is discount rate news that determines shocks to the path of expected log-returns. This will be zero if expected stock returns are constant as in older finance theories. On the other hand, it is generally non-zero if returns are predictable. To see this assume we find $\beta\neq 0$ for some predictor $x_t$ so that

$$r_{t+1}=\alpha +\beta x_t+\epsilon_{t+1}.$$

Then the discount rate news component is

$$(\mathbb{E}_{t+1}-\mathbb{E}_t)\sum_{j=1}^{\infty}\beta\rho^j\Delta x_{t+1+j}$$

For simplicity assume the predictor is AR(1) with persistence $\lambda$.

$$(\mathbb{E}_{t+1}-\mathbb{E}_t)\sum_{j=1}^{\infty}\beta\rho^j\Delta x_{t+1+j}=(x_{t+1}-\lambda x_t)\frac{\beta\lambda\rho}{1-\rho\lambda}.$$

Hence return predictability implies that return variation is partly driven by discount rate news. Modern asset pricing theories try to explain why certain variables $x_t$ can forecast returns. In the habit model the key predictor is consumption growth so higher consumption means lower expected returns. This can also explain why price-dividend ratios forecast returns. In the long run risk model there are two predictors: expected consumption growth and consumption volatility.

The cash flow news component does not create return predictability but creates variance in returns as a positive shock to future cash flows leads to higher returns.