Portfolios: Rebalancing


Kevin Crotty
BUSI 448: Investments

Where are we?

Last time:

  • Short-sales constraints

Today:

  • Simulation
  • Investing over multiple periods
  • Rebalancing

Simulation

General steps

  1. Identify one or more random inputs
  2. Set up a function that generates a random draw of the inputs and does some calculations to produce output(s).
  3. Run the function in step 2 many times to collect the simulated distribution of the output(s).
  4. Summarize the output distribution in some way (average value, percentiles, etc.)

Simulating a single return series

from scipy.stats import norm
N_SIMS=1000
sims = pd.DataFrame(dtype=float,columns=['avg_ret'],index=np.arange(N_SIMS))
for s in sims.index:
    rets = norm.rvs(loc=MN, scale=SD, size=T)
    sims.loc[s] = np.mean(rets)

Using a function for a single realization

# Function to run a single realization
def sim_calc(mean, sd, n_time):
    rets = norm.rvs(loc=mean, scale = sd, size=n_time)
    return np.mean(rets)  

# Collect N_SIMS runs of the simulation function
sims = pd.DataFrame(dtype=float,columns=['avg_ret'],index=np.arange(N_SIMS))
for s in sims.index:
    sims.loc[s] = sim_calc(MN,SD,T)

Simulating multiple return series

# Simulate a single realization and do calculation(s)
def sim_calc(means, cov, n_time):
    n = len(means)
    rets = pd.DataFrame(data=mvn.rvs(means, cov, size=T), 
            columns=['ret' + str(i+1) for i in np.arange(n)])
    x = rets.corr()
    corr12 = x.loc['ret1', 'ret2']
    corr13 = x.loc['ret1', 'ret3']
    return corr12, corr13

# Collect N_SIMS runs of the simulation function
sims = pd.DataFrame(dtype=float,columns=['corr12', 'corr13'],
        index=np.arange(N_SIMS))
for s in sims.index:
    sims.loc[s,['corr12','corr13']] = sim_calc(MNS,COV,T)

Let’s look at a notebook that simulates some of our input statistics for portfolio optimization.

Rebalancing

60-40 Stock-Bond Portfolio

What if we leave this portfolio alone?

  • Suppose returns for the stock fund are 12.3% over the first year and the bond fund returns 5.9% over the same period.

  • Weights become: 61.4% and 38.6%

  • The new weight for asset \(i\) is:\[ w_{i,t+1} = w_{i,t}\frac{1+r_{i,t}}{1+r_{p,t}} \] where \(r_{p,t}=\sum_{j=1}^N w_{j,t} r_{j,t}\) is the time \(t\) realized portfolio return for an \(N\) asset portfolio

Five years of possible returns

30 years of possible returns

Ex Ante Sharpe Ratios

Realized Sharpe Ratios

The realized Sharpe ratio of a strategy is its realized average excess return scaled by its realized standard deviation:

\[ SR = \frac{\overline{r_{p} - r_{f}}}{\text{sd}[r_{p} - r_{f}]}\]

For the particular returns above, the realized Sharpe ratio of rebalancing to 60-40 is -0.0277 versus -0.09 for the non-rebalanced portfolio.

Realized Sharpe Ratios

What if we ran 1000 versions of the 30 year investment period?

Rebalancing: 3 assets

Assumptions

We have been assuming:

  • returns are independently and identically distributed each period

  • the risk-free rate is constant each period.

This means the tangency portfolio is optimal each period.

If expected returns are mean-reverting, then it is also advantageous to rebalance.

Practical Issues: Taxes

  • Rebalancing is a contrarian strategy: sell winners and buy losers.

  • Selling winners may result in capital gains taxes

  • One must weigh the potential benefit of improved portfolio allocation vs. the potential tax exposure of selling overweighted assets

Practical Issues: Transactions Costs

  • Similarly, trading may result in fixed or variable transactions costs

  • One must weight the potential benefit of improved portfolio allocation vs. the expected costs of the rebalancing transactions

For next time: Input Sensitivity





BUSI 448