Stanley W. Black
Vice President
Dimensional Fund Advisors
Samuel Wang
Researcher
Dimensional Fund Advisors
PURELY ACADEMIC
September 2015
The Cost of a Perfect Market
Timing Strategy
What if we could construct a strategy with a correlation
close to 1 with a perfect market timing strategy? Can
we estimate what its returns would have been? Can we
use these estimates to understand what the market would
have “charged” for a perfect market timing strategy?
To answer these questions, first let’s define what a
perfect market timing strategy needs to know and what
it is. A perfect market timing strategy needs to know,
with certainty, the future returns of the assets that are
eligible for investment. Armed with this information,
the perfect market timing strategy always chooses the
highest returning asset to invest in. For example, imagine
a strategy that can invest in US stocks [for simplicity, the
S&P 500 Total Return (TR) Index] and One-Month US
Treasury Bills. It rebalances once per month and on that
day knows with certainty the return of US stocks and
US Treasury bills over the following month. This perfect
market timing strategy would then invest all in equities
or all in the One-Month US Treasury Bill, depending on
which one will have the higher return over the following
month. The challenge here is obvious. How can we know
with certainty what the future returns on any asset class
will be? Without that knowledge, there is no way to create
a perfect market timing strategy. Or is there?
Merton1 (1981) shows how to create a strategy that has an
almost perfect correlation with the perfect market timing
strategy. In particular, this equity strategy buys one share
of stock and a one-month put option with a strike price
equal to the current stock price adjusted for the yield
on the one-month T-bill. We refer to it as a protected
equity strategy. In the absence of costs, the payoffs from
the protected equity strategy are identical to the perfect
market timing strategy. As shown in the Appendix, the
options used in the protected equity strategy are “in-themoney” by the amount of the yield on the T-bill when
written at the beginning of each month.
In the absence of costs, the hypothetical returns and
wealth generated by the equivalent perfect market timing
and protected equity investment strategies are astonishing.
For the 1962–2014 sample period, a $100 initial investment
grows to over $146 million, much larger than the $18,273
from investing in the S&P 500 Index (see Exhibit 1). But
what about the costs associated with the protected equity
1. Robert Merton provides consulting services to Dimensional Fund Advisors LP.
DIMENSIONAL FUND ADVISORS
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strategy? Merton argues that the equilibrium market
price of the perfect market timing strategy is equal to
the cost of the put option required to guarantee at least
the T-bill return each month.
Since the put option price can be estimated using the
Black-Scholes formula, we could estimate the returns
of the protected equity strategy net of the cost of the
required put option. The Black-Scholes formula to price
stock options requires these inputs: current price of the
stock, strike price of the option, riskless interest rate and
expected volatility of the stock’s returns over the life of the
option. The expected volatility of stock returns is the only
input to the Black-Scholes model that must be estimated.
In the simulation, we estimate this as the annualized daily
volatility of the 60 previous daily returns.2 Exhibit 1
summarizes the results for the 1962–2014 period.
The monthly returns of the perfect market timing and
downside protection strategies are highly correlated at 0.94.
The minimum monthly return of the perfect market timing
strategy was 0.00%. However, the minimum monthly return
of the downside protection strategy is −7.34%. This occurred
in January 2009 and reflects the cost of purchasing the put
option that month. In fact, the estimated average cost of
the one-month put option is 1.63% over the sample period.
This can be thought of as the market assigned value for
perfect market timing. So while the perfect market timing
strategy has an average monthly return of 2.32%, the net
of cost monthly return of the downside protected equity
investment is 0.69% per month, which is smaller than
the average monthly return of the S&P 500 Index’s 0.93%
per month. Hypothetical growth of $100 turns out to be
just $6,086 after costs, much less than the value of $100
invested in the S&P 500 Index ($18,273).3
We can now answer our original question: What if we could
construct a strategy with a correlation close to 1 with the
perfect market timing strategy? It turns out that while this
is possible, the cost of implementing would have resulted
in inferior performance (in terms of ending wealth) to just
holding the S&P 500 Index. If volatility and maximum
drawdowns are something an investor is particularly
concerned with, a balanced portfolio may be a more costeffective way to achieve this goal. For example, 60% S&P
500 and 40% T-Bills would have achieved a similar level of
volatility to the downside protected strategy with a higher
average return—so your $100 would have grown to $7,215.3
Exhibit 1: Summary Stats for S&P 500 with Downside Protection Simulation
Oct 1962–Dec 2014 4
Monthly Returns
Average Return
Downside Protected
S&P 500
(Perfect Market Timing)
Downside
Protected S&P 500
Net of Cost of Put
S&P 500 TR Index
T-Bills
60% S&P 500/
40% T-Bills
2.32%
0.69%
0.93%
0.40%
0.72%
Standard Deviation
2.55%
2.51%
4.28%
0.26%
2.57%
Highest Return
16.57%
13.21%
16.57%
1.35%
10.14%
Lowest Return
0.00%
−7.34%
−21.52%
0.00%
−12.67%
Average Annual
Compound Return
31.22%
8.18%
10.48%
4.96%
8.53%
$146,495,809
$6,086
$18,273
$1,257
$7,215
N/A
0.94
0.83
0.06
0.84
Growth of $100
Correlation to Perfect
Market Timing
2. By using actual trailing volatility of 60 previous daily returns, we are following a standard methodology similar to Merton (1981), which uses the
volatility of 12 previous monthly returns. However, we note that the market prices of options on average tend to reflect a higher implied level
of volatility than the actual realized equity volatility. This would cause our cost estimates to be conservative.
3. Past performance is no guarantee of future results. It is not possible to invest directly in an index.
4. Data Source: S&P 500 Total Return (TR) Index from CRSP Value-Weighted Universe (1962–1988) and S&P 500 TR Index from Bloomberg (1988–2014);
One-Month US Treasury Bills from Morningstar. Past performance is not a guarantee of future results. It is not possible to invest directly in an index.
Simulated strategy returns based on a model/back-tested simulation. This is not a strategy managed by Dimensional. The performance was
achieved with the retroactive application of a model designed with the benefit of hindsight; it does not represent actual investment performance.
Back-tested model performance is hypothetical (it does not reflect trading in actual accounts) and is provided for informational purposes only.
The securities held in the model may differ significantly from those held in client accounts. Model performance may not reflect the impact that
economic and market factors might have had on the advisor’s decision making if the advisor were actually managing client money. This strategy
was not available for investment in the time periods depicted. Actual management of this type of simulated strategy may result in lower returns
than the back-tested results achieved with the benefit of hindsight. Past performance (including hypothetical past performance) does not guarantee
future or actual results. The simulated performance shown is “gross performance,” which includes the reinvestment of dividends but does not
reflect the deduction of investment advisory fees and other expenses.
DIMENSIONAL FUND ADVISORS
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REFERENCE
[1] Merton, R. C. 1981. “On Market Timing and Investment
Performance. I. An Equilibrium Theory of Value for Market
Forecasts.” Journal of Business, 1981, Vol. 54, No. 3.
APPENDIX
Assume that the monthly return of one-month T-bill is
r f (t) and the monthly return of the equity index is rM (t).
Denote S(t) as the index level at month t.
The monthly return of the perfect market timing (PMT)
strategy is:
R PMT (t)=max(r f (t), rM (t))
The protective put strategy is the following:
1.At the end of month t−1, the investor buys a share of
index at price S(t−1), and a one-month put option with
strike price equal to S(t−1)×(1+r f (t)). Notice that r f (t),
the T-bill return at month t is known at month t−1, as
it could be derived from the yield at month t−1.
2.Hold the portfolio until the end of month t when the put
option expires; exercise the option if in-the-money.
3.Rebalance at the end of month t: Buy a share of stock
S(t) and a new one-month put option with strike price
S(t−1)×(1+r f (t+1)).
The monthly return of the protective put strategy is:
R PUT (t)=
S(t)+max[0, S(t−1)(1+r f (t))−S(t)]−(S(t−1)+P(t−1))
S(t−1)+P(t−1)
Here P(t−1) is the put option price derived from BlackScholes formula 5 with underlying index level S(t−1), strike
price S(t−1)(1+r f (t)) and the (annualized) continuously
compounded risk-free rate r(t)=12 ln(1+r f (t)).
Note that, if P(t−1) is zero, R PUT (t)=R PMT (t). In other words,
if the cost of the put option is zero, the perfect market
timing strategy and the protective put strategy are identical.
5. No early exercise, “European” stock options that are relevant for this problem.
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investors, and is not intended for public use.
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