Empirical Methods in Finance

Homework 4

Problem 1: Return forecasting regressions

1. Using CRSP, get monthly market returns ex and cum dividends, as well as the monthly

t-bill rate, from 1963 to 2018. Create the market dividend yield by summing the dividends over the last 12 months and divide by current price (you can do this using

information extracted using the ex- and cum-dividend returns). Construct excess returns by subtracting the log of the 1-month gross t-bill rate from the 1-month gross

cum-dividends returns. Note: to get to gross returns you may have to add 1 to the

original data series.

From the St. Louis Fed data page (FRED; https://fred.stlouisfed.org/), get monthly

data on the term and default spreads for the same sample. For the former, use the “10-

Year Treasury Constant Maturity Minus Federal Funds Rate,” for the latter subtract

“Moodyís Seasoned Aaa Corporate Bond Minus Federal Funds Rate ” from “Moodyís

Seasoned Baa Corporate Bond Minus Federal Funds Rate.”

2. Plot your data.

3. Using your three predictive variables (the lagged dividend yield, term spread, and

default spread), forecast excess equity returns at the 1-month, 3-month, 12-month,

24-month, and 60-month horizons. Report your results from each of these regressions

(regression coe¢ cients, standard errors, and R2

s). The underlying data is monthly, so

make sure to explain your choice of standard errors.

4. Plot the estimated expected 12-month excess return that obtains from the forecasting regressions of the 12-month excess return regression. What type of periods are

1

associated with high expected returns and what type of periods are associated with

low expected returns. Does the patterns make sense to you? What are the economic

stories you would tell to explain these patterns?

5. In the last class, we used our AR toolkit to infer expected returns at long horizons.

Why do you think using, say, an AR(1)-type setting could be useful instead of simply

regressing returns at di§erent horizons on lagged predictive variables as we have done

in this problem?

Problem 2: M/B ratios and the present value formula

In class, we derived a log-linear present value formula using the Campbell-Shiller return

decomposition (rt+1 0 + pdt+1 pdt + dt+1). While this decomposition is very accurate, it does require a stationary pd-ratio and positive dividends. Many Örms do not pay

dividends, which means this formula is not always useful at the Örm level.

However, Ohlson (Contemporary Accounting Research, 1995) and Vuolteenaho (Journal

of Finance, 2002) derive a return decomposition that uses market-to-book ratios and returnon-equity instead of the pd-ratio and dividends (See also Lochstoer and Tetlock (forthcoming

Journal of Finance)). In particular, the alternative return decomposition is:

rt+1 mbt+1 mbt + roet+1; (1)

where = 0:97 with annual data, mbt = ln Mt=Bt and roet = ln (1 + ROEt). Here, Mt

is

the market value of equity, Bt

is the book value of equity, ROEt =

Et

Bt1

is return-on-equity

where Et

is earnings. See the above papers for details. With this decomposition we have:

mbt Et

P1

j=1

j1

roet+j Et

P1

j=1

j1

rt+j

: (2)

Thus, if the market-to-book ratio is high today, either future expected return on equity is

high or expected future returns are low. Vice versa if the market-to-book ratio is low today.

1. You estimate a Örmís annual roe to follow the ARMA(1,1):

roet+1 = 0:05 + 0:9 (roet 0:05) 0:6″t + “t+1

where (“t) = 0:1. Assume the current value of mbt = ln Mt=Bt = 0:7, roet =

ln(1 + ROEt) where ROEt = 0:2, and “t = 0:1. From this information, derive the

current values of the below:

CFt = Et

P1

j=1

j1

roet+j

;

DRt = Et

P1

j=1

j1

rt+j

:

2. Assume the unconditional average of mbt

is 0:2. What then is the unconditional average

of CF and DR? Now, you can, as a value investor, assess if the long-run discount rate

of this Örm is currently higher or lower than normal.