Barnstable and Long-Run Risk

Barnstable and Long-Run Risk#

HBS Case#

The Risk of Stocks in the Long-Run: The Barnstable College Endowment#

1. READING - Barnstable#

1 Barnstable’s Philosophy#

  • What has Barnstable’s investment strategy been in the past?

  • Explain the logic behind their view that stocks are safer in the long run.

  • What assumptions underlie Barnstable’s belief in the long-run safety of stocks?

2. Two proposals#

Describe the two proposals Barnstable is considering for how to take advantage of their view regarding the long-run safety of stocks.

3. The trust#

How is the trust different from simply shorting the risk-free rate to take a levered position in stocks?

4. Payoff differences#

You may not have had a course in options. It’s okay if you are only vaguely familiar with the mechanics below and the option payoffs.

Do these proposals take the same bet on long-run stock performance? In what outcomes will they have different returns?

The payoff at maturity of the common share is:

\[\Gamma_{30}^{\text{common}} = \max\left(\texttt{r}^m_{t,t+30} - \exp\left\{.06(30)\right\},~ 0\right)\]

The payoff at maturity of selling puts is:

\[\Gamma_{30}^{\text{selling puts}} = - \max\left(\exp\left\{.06(30)\right\} - \texttt{r}^m_{t,t+30}, 0\right)\]

5. Risk differences#

Do the two proposals differ in their risk?

6. Recommendation#

Do you recommend a direct investment in the S&P, the trust or the puts?

2. Estimating Underperformance#

Data#

Use the returns on the S&P 500 (\(r^m\)) and 1-month T-bills, (\(r^f\)) provided in barnstable_analysis_data.xlsx.

  • Data goes through END_YR=2024.

Barnstable’s estimates of mean and volatility are based on the subsample of 1965 to 1999.

  • We consider this subsample, as well as 2000-{END_YR}, as well as the full sample of 1926-{END_YR}.

Notation#

  • \(r\) = level return rates

  • \(R\) = cumulative return factor

  • \(\texttt{r}\) = log return rates

\[ R \equiv 1+r\]
\[ \texttt{r} \equiv \ln(1+r) = \ln(R)\]

1. Summary Statistics#

Report the following (annualized) statistics.

1965-1999

2000-{END_YR}

1926-{END_YR}

mean

vol

mean

vol

mean

vol

levels

\(r^m\)

\(\tilde{r}^m\)

\(r^f\)

logs

\(\texttt{r}^m\)

\(\tilde{\texttt{r}}^m\)

\(\texttt{r}_f\)

  • Comment on how the full-sample return stats compare to the sub-sample stats.

  • Comment on how the level stats compare to the log stats.

2. Probability of Underperformance#

Recall the following:

  • If \(x\sim\mathcal{N}\left(\mu_x,\sigma_x^2\right)\), then

    \[\Pr\left[x<\ell\right] = \Phi_\mathcal{N}\left(L\right)\]

    where \(L = \frac{\ell-\mu_x}{\sigma_x}\) and \(\Phi_\mathcal{N}\) denotes the standard normal cdf.

  • Remember that cumulative log returns are simply the sum of the single-period log returns:

    \[\texttt{r}^m_{t,t+h} \equiv \sum_{i=1}^h \texttt{r}^m_{t+i}\]

  • It will be convenient to use and denote sample averages. We use the following notation for an \(h\)-period average ending at time \(t+h\):

    \[\bar{\texttt{r}}^m_{t,t+h} = \frac{1}{h}\sum_{i=1}^h \texttt{r}^m_{t+i}\]

Calculate the probability that the cumulative market return will fall short of the cumulative risk-free return:

\[\Pr\left[R^m_{t,t+h} < R^f_{t,t+h}\right]\]

To analyze this analytically, convert the probability statement above to a probability statement about mean log returns.

2.1#

Calculate the probability using the subsample 1965-1999.

2.2#

Report the precise probability for \(h=15\) and \(h=30\) years.

2.3#

Plot the probability as a function of the investment horizon, \(h\), for \(0<h\le 30\) years.

Hint: The probability can be expressed as:

\[p(h) = \Phi_{\mathcal{N}}\left(-\sqrt{h}\;\text{SR}\right)\]

where \(\text{SR}\) denotes the sample Sharpe ratio of log market returns.

3. Full Sample Analysis#

Use the sample 1965-{END_YR} to reconsider the 30-year probability. As of the end of {END_YR}, calculate the probability of the stock return underperforming the risk-free rate over the next 30 years. That is, \(R^m_{t,t+h}\) underperforming \(R^f_{t,t+h}\) for \(0<h\le 30\).

4. In-Sample Estimate of Out-of-Sample Likelihood#

Let’s consider how things turned out relative to Barnstable’s 1999 expectations.

What was the probability (based on the 1999 estimate of \(\mu\)) that the h-year market return, \(R^m_{t,t+h}\), would be smaller than that realized in 2000-{END_YR}?

Hint: You can calculate this as:

\[p = \Phi_{\mathcal{N}}\left(\sqrt{h}\; \frac{\bar{\texttt{r}}_{out-of-sample} - \bar{\texttt{r}}_{\text{in-sample}}}{\sigma_{\text{in-sample}}}\right)\]

where “in-sample” denotes 1965-1999 and “out-of-sample” denotes 2000-{END_YR}.

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