FINM36700 - Homework 3 - Solved

The Risk of Stocks in the Long-Run: The Barnstable College Endowment [HBS 9-296073].


1.    Barnstable’s Philosophy

(a)     What has Barnstable’s investment strategy been in the past?

(b)    Explain the logic behind their view that stocks are safer in the long run.

(c)     What assumptions underly Barnstable’s belief in the long-run safety of stocks?

2.    Two Proposals

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

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

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

(d)    Do the two proposals differ in their risk?

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

2           Estimating Underperformance
Use the returns on the S&P 500 (rM) and 1-month T-bills, (rf) provided in “barnstable analysis data.xlsx”.

Barnstable’s estimates of mean and volatility are based on the subsample of 1965 to 1999. We consider this subsample, as well as 2000-2021, as well as the full sample of 1926-2021. We only have data through August of 2021, but no adjustment is needed for the fact that you have only the partial year–just use what you have.

1.    Summary Statistics

(a) Report the following (annualized) statistics.

 

1965-1999 2000-2021 1926-2021 mean vol mean vol mean vol

 
rM
 
 
 
 
 
levels
r˜M rf
 
 
 
 
 
 
rM
 
 
 
 
 
logs
˜rM rf
 
 
 
 
 
(b) 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.    Recall the following...

•    If , then

Pr[x < `] = ΦN (L)

 

where ΦN denotes the standard normal cdf.

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

h

r 

i=1 • 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,

 

Calculate the probability that the cumulative market return will fall short of the cumulative risk-free return.[1]

                                                                                                    h M                           f             i

                                                                                                 Pr Rt,t+h < Rt,t+h                                                                                                                         (1)

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

(a)     Calculate (1) using the subsample 1965-1999.

(b)    Report the precise probability for h = 10 and h = 30 years.

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

3.    Suppose that at the end of 2021, an investor is considering the issues Barnstable analyzed in 1999. Re-do the probability calculations for the sample 1965-2021. How different are the long-run probabilities now considering this extra data?

As of the end of 2021, calculate the probability of underperforming the risk-free rate over the next 30 years. That is, Rt,tM+h underperforming Rt,tf +h for 0 < h ≤ 30.

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

(a)     What was the probability (based on the 1999 estimate of µ,) that the 21-year market return,

 , would be smaller than that realized in 2000-2021? Note that we are asking about the market return, not the excess market return. Continue using the 1965-1999 sample standard deviation for σ.

(b)    Suppose Barnstable had implemented the put-option strategy with the growing strike of 6%? Based on the 2000-2021 performance, what is the probability that the 2000-2029 cumulative market return will be low enough to make the puts in the money? For the calculation, update your estimates of µ,σ to be the estimates based on 1965-2021.

3           Modeling Volatility and VaR
For the full sample of market returns, 1926-2021, calculate the LEVEL, (not log,) excess market returns. We use this level excess return data throughout this section.

1.    Historic VaR.

Starting at t = 61, calculate the historic-based VaR, based on the expanding sample from period 1 (Jan 1926) to t − 1. By historic VaR, we mean simply taking the 5th quantile for the historic sample up to time t − 1. Of course, a sample size that is not a multiple of 100 will require some interpolation to get a 5th quantile. Your statistical package should handle this fine.

Denote this as ˜rt,VaRhistoric,.05 , which is the best estimate of the time-t VaR based on data through t − 1.

(a)     Plot ˜               over time.

(b)    Calculate the frequency of periods in which ˜   .

(c)     What drawbacks do you see in this historic VaR?

2.    Volatility

We will calculate a time-series of volatility estimates using a few different methods. For each, we use σt to denote our estimate of the time-t return volatility, as based on data over periods 1 (Jan 1926) through t − 1, but not including t itself. • Expanding Series[2]

σt,2expanding  

Begin the calculation at t = 61, so that the first estimate is based on 60 data points.

•    Rolling Window

σt,2rolling  

Use m = 60, and begin the calculation at the t = 61, (so that the calculation has a full 60 data points.)

•    Exponentially Weighted Moving Average (EWMA)

σt,2EWMA  ,EWMA  

Rather than estimating θ, simply use θ = 0.97, and initialize with σ1 = 0.15. • GARCH(1,1) model.

To estimate GARCH(1,1), try using the ARCH package in Python. The default estimation implementation is fine.[3]

(a)     For each of these three methods, plot σt. (Plot the vol, not the variance.)

(b)    For each method, calculate the 5th percentile, 1-month-ahead VaR. We use a slight simplification of the normal VaR formula, by dropping µ from that formula, and rounding the normal distribution z-score to -1.65.

 

(c)     For each of these three vol-based VaR estimates, calculate the frequency of periods in which

 .

(d)    Compare and contrast your results among each other and relative to the historic method in the previous problem.

4           Extensions
This section is not graded, and you do not need to submit your answers. We may discuss some of these extensions.

1.    Let’s consider how Barnstable would have done if they had positioned the problem in excess returns rather than beating a fixed benchmark based on historic risk-free rates.

(a)     Based on the end-of-1999 parameters, ˜µ and ˜σ, what was the probability that the log market excess return, ˜rM would have a smaller 22-year average than what happened in 2000-2021, based on ˜µ estimated from the 1965-1999 average??

 

(b)    Suppose they struck the puts to the realized risk-free rate instead of a fixed 6%. Calculate the probability that the puts will be in the money, given what has already occurred in 2000-2021. As above, calculate this using full-sample parameter estimates of 1965-2021.

(c)     Explain the risk Barnstable took on the future movements in the risk-free rate.

2.    The probability of shortfall calculations for Barnstable presume that we can simply use sample estimates for the probability calculations that involve population parameters for the means and variances of log returns. Let’s consider how this estimation uncertainty leads to uncertainty in the probability of shortfall calculation.

(a)     Assuming log returns are normally distributed, (or just using the CLT,) calculate the 5th and 95th percentiles for our confidence interval of the mean of excess log returns: ˜µ = µ−µf.

(b)    Denote these ˜µ.05,µ˜.95, and recalculate the probability of shortfall from (1). Plot the confidence bounds as a function of the investment horizon, 0 < h ≤ 30.

(c)     Has ˜µ from 2000-2021 been outside the 5th and 95th percentile estimates derived from the data in 1965-1999?

(d)    Does uncertainty around ˜µ materially impact the estimates? Does it change your recommendation to Barnstable regarding the trust versus the puts?

3.    The case assumes stock returns are iid. and lognormally distributed.

 

The following table lists return stats given by the case,

Notice that Barnstable’s mean estimate is for the level-return, rtM, while Barnstable’s volatility estimate corresponds to the log-return, rtM. Here and elsewhere it is at times useful to translate mean and volatility from level to logs—at least for the lognormal case.

Barnstable case estimates

 
mean
variance
level return of stocks
E[rtM] = 0.13
var[rtM] =?
log return of stocks
E[rtM] =?
var[rMt ] = (0.16)2
log riskless benchmark
E[rf] = 0.06
 
Use the following formulas to translate log-return means and variances to level-return means and varianaces.

 

(a) Use Barnstable’s estimates in the table above, along with these equations to calculate the mean log return, µ.

Do you get the same answer as calculated by the case, 0.117? (In footnote 3.) (b) Also calculate var[rtM].

(c) Are the mean and volatility bigger for level or log returns?

4. Does the analysis change much if we use the pre-1965 data in our estimates of the market return and risk-free rate?


 
[1] Note that this is essentially what Barnstable is estimating, but with the 1965-1999 sample estimate that the risk-free rate was 6% during that period.
[2] Note that here we use a slightly biased version of the usual variance estimator by ignoring µ and dividing by the number of data points.
[3] It is okay if you have not used GARCH much before; at this point I do not expect you to have a deep understanding of GARCH, but I would like you to explore using it to estimate volatility.