Dynamic Portfolio Strategies

1
Dynamic Portfolio Strategies

• Part 3
• Dynamic Properties of Returns and Implications
  for Long Horizon Forecasts

2
Forecasting Long-Horizon Prices IID Returns
(review)

• Keep in mind the effect of horizon on return
  distributions in the IID normal case

3
Forecasting Returns AR(1) Returns

• An alternative model for returns, with dependence
  among the returns, is an AR(1) model
• AR autoregressive
• 1 lag length.
• In this case, the model for returns is

4
AR(1) Returns

• Note
• Returns are not independent since future returns
  depend on past returns.
• ? must be in (-1,1) in order for returns to have
  a long-run mean.
• The current level of r affects the short-run mean
  returns.
• ? will have an impact on long-horizon variances.

5
Conditional Expectations

• Now that information is useful for forecasting
  future returns (here that information is past
  returns) we have to work with conditional
  expectations.
• In this case, we wish to estimate the
  distribution of two period ahead returns
  conditional on the current return.

6
Forecasting Two-Period Returns AR(1)

• Suppose weve just observed the log of prices
  change from pt-1 to pt (i.e. we know rt)

7
Forecasting Two-Period Returns AR(1)

• Note
• The impact of rt on the mean declines with time.
• If ?lt1 the ?2 is small.
• Old shocks to rt1 also show up in rt2 (the
  returns are correlated).

8
Forecasting Two-Period Returns AR(1)

• The two-period return can be represented by

9
Forecasting Two-Period Returns AR(1)

• Note
• The shocks are independent and normal, so the sum
  is normal. Thus, the two-period return is
  normally distributed (prices two periods from now
  are lognormal).
• The mean return is about twice the unconditional
  one-period expected return, ?.
• The variance of returns can be larger or smaller
  than twice the one-period variance, depending on
  the sign of ?.

10
The Two-Period Return Variance

• More details on the two-period return variance

11
Notes on Variance

• Again, notice that the variance can be larger or
  smaller than in the IID return case.
• Although it is possible to derive a formula for
  the conditional means and variances at all
  horizons, its instructive (and probably more
  useful) to see how to derive these forecasts
  using simulation.

12
Simulating Long-Horizon AR(1) Returns with excel

• There are many commercial excel-based simulation
  tools
• Crystal Ball http//www.decisioneering.com/crysta
  l_ball/
• _at_Risk http//www.palisade.com/
• Well be using a freeware addin called SIMTOOLS
  http//home.uchicago.edu/rmyerson/addins.htm

13
Simulating Returns

• Two excel functions are needed to generate a
  random normally distributed number
• rand() generates a random number, uniformly
  distributed on 0,1
• normsinv() inverts the standard normal
  distribution
• So to generate a N(0,1) random draw you call
  normsinv(rand())

14
Simulating Returns

• With a sequence of N(0,1) variables in hand, we
  can recursively apply the return model to
  generate a sequence of one-period returns
• To determine the distribution of the sum
• Add the returns.
• Simulate the sum many times, keeping track of
  each result (this is where SIMTOOLS is handy).

15
Simulating Returns An Example

• Suppose youve determined that an index has the
  following properties
• Long-run mean .96/month.
• Standard deviation of shock 4.33/month.
• Autocorrelation 4/month.
• If the return last month was 1, calculate the
  conditional mean and variance of returns one year
  from now?
• How do the mean and variance change if the
  autocorrelation is 4/month?

16
Estimating a model of AR(1) Returns

• In order to simulate returns from the AR(1)
  model, we need to know what the model parameters
  are
• In order to do this, we use regression techniques.

17
Determining AR(1) Return Parameters

• To estimate the model
• Generate a series of log prices over some time
  horizon and form continuous returns.
• Regress the returns on their own lags and
  estimate coefficients in the model
• Calculate the standard deviation of the error
  term, e.

18
Estimating an AR(1) Model Example

• Using the returns from a broad-based index,
  estimate the coefficients in the following AR(1)
  model

19
Predictive Models of Returns

• Returns, it seems, can be predicted.
• We will
• Examine evidence on dividend yields and
  predictability.
• Estimate a model in which dividend yields predict
  returns.
• Use the model to forecast returns over long
  horizons.

20
Dividend Yields

• The dividend yield of an index is defined as the
  dividend paid by that index over some period
  divided by the current price of the stock.
• In US data there is strong evidence that
  variation in dividend yields reflects
  time-varying expected returns.
• The predictive ability of dividend yields
  improves as the forecasting horizon expands. For
  example, even though dividend yields are poor
  predictors of one-month returns, they are good
  predictors of four year returns.

21
Present Value Identity

• Stock returns must come from two sources
  dividends and price appreciation.
• This simple observation has some less obvious
  implications.
• Begin with the identity

22
Present Value Identity

• Rearrange and divide through by current
  dividends
• Applying this recursive relation forward

23
Present Value Identity

• Note that this is an identity. It must be true
  given any sequence of future returns and
  dividends.
• The relationship must, therefore, hold when the
  conditional expectation is taken

24
Present Value Identity

• This identity says nothing more than todays
  price is the discounted value (using future, as
  yet unspecified, discount rates) of future (again
  unspecified) dividends.
• It is difficult to work with this relation
  statistically. Taking logs yields an equivalent
  approximating relation.

25
An Approximate Present Value Relation

• The continuous compounding version of the
  identity
• Again, this must hold taking expectations

26
Interpreting the Present Value Identity

• The identity makes a straightforward point
• If the current price/dividend ratio is high, then
  either
• Future dividend growth must be high, or
• Future returns must be low.
• Another way of saying the same thing
• If the price/dividend ratio varies (and it does)
  then it must be forecasting changing future
  dividends or changing future returns.

27
The Source of Variation in Dividend Yields

• We can decompose dividend yield variation into
  two components

28
What We Learn About Returns

• Think back to some simple descriptions that have
  been used for returns and dividends
• IID returns (unpredictable).
• Growing unpredictable dividends.
• In this case, according to the formula above, the
  price/dividend ratio should be a constant. Again,
  this is not true.

29
Empirical Evidence

• Historical dividend growth has been very smooth.
• Dividend yields are negatively correlated with
  future dividend growth.
• Dividend yields are highly correlated with future
  returns.
• Another interpretation if prices are currently
  low (relative to dividends) then returns are
  expected to be high (historically, low dividends
  do not accompany low prices).

30
Conclusions to Draw from the Present Value
Identity

• In light of the fact that dividend growth seems
  to be largely unpredictable, current dividend
  yields must predict future expected returns.
• This has important implications for long-horizon
  investors. There are potentially large benefits
  to
• Quantifying the nature of return predictability.
• Applying this knowledge to the portfolio choice
  problem.

31
Dividend Yield and Return Predictability VAR
Models

• The ability of dividend yields to predict
  expected returns can be incorporated into a
  formal statistical model.
• This model is referred to as a Vector
  Auto-Regression (VAR)
• The model is a multiple variable version of the
  AR(1) description of returns discussed previously.

32
A Dividend Yield VAR

• The following model relates expected returns to
  dividend yields and future dividend yields to
  current

33
Estimating the VAR

• In order to estimate this model we
• Create a time-series of returns and log dividend
  yields using prices and dividends.
• Regress returns on dividend yields.
• Regress dividend yields on lagged dividend
  yields.
• Use the regression residuals to estimate the
  covariance structure of the errors.

34
Estimating a VAR Example

• Using the returns from a broad-based index,
  estimate the following VAR model

35
Forecasting Two-Period Returns with a VAR

• The VAR model can be applied recursively to
  generate forecasts of long-horizon returns.
• The two-period forecast gives some intuition
  about the distribution of these returns.

36
Two-Period Return Forecasts

• The forecasts of the next two returns
  (conditional on the current level of dividend
  yields)

37
Two-Period Return Forecasts

• The sum of the returns is

38
Two-Period Return Forecasts The Effect on
Variance

• Note
• Returns are normally distributed.
• The variance of the two-period returns may be
  larger or smaller than the variance of the
  one-period returns

39
Forecasting Long-Horizon Returns

• The distribution of long-horizon returns can be
  simulated, again by applying the VAR model
  recursively.
• One complication that arises in this setting is
  to generate correlated random variables.
• The SIMTOOLS function corand() is designed to do
  this.

40
Forecasting Long-Horizon Returns Example

• Using the following estimates for the VAR,
  determine the properties of 5-year returns