_{1}

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The paper compares three portfolio optimization models. Modern portfolio theory (MPT) is a short-horizon volatility model. The relevant time horizon is the sampling interval. MPT is myopic and implies that investors are not concerned with long-term variance or mean-reversion. Intertemporal portfolio choice is a multiple period model that revises portfolios continuously in response to relevant signals to reduce variance of terminal wealth over the holding period. Digital portfolio theory (DPT) is a non-myopic, discrete time, long-horizon variance model that does not include volatility. DPT controls mean-reversion variances in single period solutions based on holding period and hedging and speculative demand.

Advances in signal processing in recent years have resulted in a “golden age of digital signal processing” characterized by stability and reliability. In addition to revolutionizing communication systems and electronic devices, this technology is now taking on a new role of understanding human-centric signals and information. In this same period, the field of finance has enjoyed an age of volatility characterized by high leverage and uncertainty. This paper looks at the potential benefit of the digital revolution when applied to portfolio theory and compares it with existing models. Markowitz [

Classical Markowitz portfolio selection finds portfolio weights based on first moments and a parametric representation of second moments assuming no predictable time variation. Mean variance MPT assumes investment opportunities are constant, or returns are independently distributed over time (IID). The static mean variance solution is myopic and does not consider events beyond the present period. The nonlinear mean variance optimization is ill-conditioned because solution methodologies introduce problems such as; estimation error amplification, extreme positions, unstable solutions, excessive sensitivity to mean and covariance estimates, a tendency to find solutions that diverge from pricing model recommendations, and no solutions for large universes do to a singular covariance matrix.

Portfolio theory evolved from the myopic mean variance paradigm to the multi-period or continuous-time models of Samuelson [

DPT is a long horizon paradigm based on a nonparametric representation of variances, covariances, and autocovariances and is solved with a linear programming (LP) solution. The DPT mean-variance-autocovariance problem allows multiple horizon hedging, or speculative demands based on holding period and has a static single period solution. Horizon effects result when returns have mean-reversions. Mean-reversions over different horizons influence the single period decision. DPT uses contributions to single period risk of multiple mean- reversion lengths but uses no information about of the pattern of mean-rever- sions. Instead, it uses long-term forecasts of the level of mean-reversion risks. Unconditional hedging demand for mean-reversion risk occurs in single period DPT when the holding period is greater than or equal to the mean-reversion length. For example, an investor with a 2-year holding period will have unconditional hedging demand for 6-month mean-reversion risk and 1-year mean-re- version risk but will have zero hedging demand for 4-year mean-reversion risk. Conditional speculative demand for mean-reversion risk occurs in DPT when an investor has a forecast of the cyclical period of a mean-reversion. For example, if we have a forecast that 4-year mean-reversion is currently at a 4-year peak we could speculate by reducing 4-year mean-reversion risk. If we have a forecast that 6-month mean-reversion is at a bottom we can speculate by increasing 6- month mean-reversion risk.

DPT defines the non-myopic asset allocation problem based on autocovariance and variance of mean reverting returns. In DPT hedging demand for risky assets is larger the longer the holding period because more mean-reversion lengths are included in the holding period. As the holding period gets shorter fewer of the mean-reversion lengths can be used to hedge. Hedging demand normally decreases and speculative demand increases as an investors holding period shortens. In the Jones [

To understand the mechanism used to optimize a portfolio over multiple periods in the presence of mean-reversion we need to examine prior research. The analysis of hedging demands originated with Merton and was followed by a large number of studies utilizing multiple period problems. These multiple period models often include both the portfolio selection and the consumption decision. The studies examine the multi-period portfolio choice problem in the presence of return predictability, frequently for a two-asset universe. Campbell and Viceira [

Many models focus on optimal dynamic multi-period policies based on conditional predictability of first moments. Ait-Sahalia and Brandt [

MPT provides a useful diversification paradigm but ignores risks beyond one period ahead. The Markowitz solution gives the myopic demand or the investment policy of an investor who optimizes over the next period without regard for changes in the investment opportunity set. Myopic mean variance demand does not include intertemporal hedging. In a myopic strategy the optimal portfolio in the current revision period does not depend on the investment opportunities in future revision periods. The MPT solution is optimal when the investment opportunity set is constant, when returns are IID over time, or when investor’s utility is logarithmic. With constant investor opportunities, investors will behave as single period optimizers. If the asset returns are IID, risky assets cannot hedge changes in investment returns, hedging demand is zero, and all risky asset demand is myopic. The dynamic problem reduces to the single period problem when returns are IID. The traditional approach of applying myopic mean variance optimization each period ignores mean-reversions and intertemporal hedging demands. Over a multiple period horizon, static buy-and-hold single period Markowitz solutions using multiple period returns are not optimal when portfolio revisions are possible each period, and return distributions have predictable time variation. Mean variance solutions exclude Merton’s hedging components driven by predictive state variables. With predictable returns, intertemporal portfolio theory with multiple revisions will result in a lower variance of terminal wealth compared to the one period MPT solution.

The MPT database approach uses return time series to estimate means and covariance. It does not consider the composition of risk in terms of systematic (passive) risk, or unsystematic (active) risk, nor does it consider investment horizon risk. Capital market theory suggests that with a risk-free asset, two-fund separation applies and the tangent efficient portfolio will contain only systematic risk. Merton and Samuelson [

One difficulty with MPT is that correlations between securities are unreliable. To improve correlations and describe active and passive risk, model-based approaches find parameters by applying an asset pricing relation such as the CAPM, or APT and use benchmark returns to describe optimal portfolios. To give greater control over systematic sources of risk the covariance matrix is represented with multiple factors or scenarios that describe common sources of covariance and assumes that remaining variability is security specific and uncorrelated (Perold [

The quadratic function that describes the weights in MPT magnifies estimation errors. Michaud [

MPT solution weights tend to be extreme. Green and Hollifield [

Another problem is that MPT solution weights tend to be unstable. Instability is the result of sensitivity of the weights to small variations in the sample mean, variance, and correlation resulting from parameter uncertainty. The solution weights are subject to high levels of estimation error with correlations between securities estimated at low significance.

Merton [

In the Merton [

The Merton [

In the Merton model, mean-reversion is the only source of hedging demand in each revision period. Hedging demands arise in the long horizon problem with intermediate revision since investors can partially hedge against future changes in returns by deviating from their single period optimal portfolio. Serial covariance of returns induces hedging demands. When returns are predictable using a set of lagged state variables, multi-period optimizers’ hedging demands will affect the composition of their equity portfolios. When there is predictability in the return process, a sequence of myopic solutions that ignore autocorrelation is not an optimal trading strategy. With predictability, the non-myopic investor will try to reduce risk by holding securities with mean-reversions that are out of phase, negatively correlated, or shorter than the holding period. Alternatively, the non-myopic investor may speculate by moving to lower risk assets in anticipation that the predictable mean-reversion will allow the high-risk asset to be purchased at a lower price in a subsequent revision. The non-myopic investor may also speculate based on expected return forecasts by holding high-risk assets with mean-reversion length longer than the holding period. Hedging and speculative demands depend on; forecasting variables, mean-reversion lengths (mean-reversion speeds), holding period, and revision frequency. Using a finite life investor, Lynch and Balduzzi [

In a dynamic setting with mean reverting returns, the myopic demand for the risky asset decreases with an increase in risk aversion, while the intertemporal hedging demand for risky assets increases. As horizon shortens, the myopic demand generally increases and the hedging demand decreases. In the Merton model, hedging demand goes to zero in the last period of the investment horizon. Shorter-term mean-reversion or faster mean-reversion speed may result in lower hedging demand but may increase myopic demand. Hedging demand will be larger for longer-term mean-reversions, or slower mean-reversion speeds if the holding period is longer than the mean-reversion length.

Kim and Omberg [

One problem with the Merton multi-period solution is the difficulty of modeling the conditional return distributions (mean, variance, and covariance) when they are time varying and predictable. State variables forecast conditional moments of return. Forecasting asset returns and state variables over long horizons is problematic. Intertemporal models frequently make the unrealistic assumption that investors know the predictive model and its parameters for the dynamics of asset returns and state variables. Predicting future returns or state variables at long horizons using term structures, or dividend yields is subject to large forecasting errors. When a statistical model such as the autoregressive is used, it imposes a structure on the dynamics of returns. Many studies allow the risk premium to follow Ornstein-Uhlenbeck process in order to capture stochastic mean-rever- sion effects (Kim and Omberg [

The optimal portfolio weights for Merton’s dynamic portfolio choice problem are solutions to nonlinear partial differential equations (PDE). The solution is a trading strategy given by weights on each asset at each period and is frequently restricted to the choice between a risk-free and a risky asset. A state variable’s stochastic process conditions returns and variances. The optimal weights are found by defining a dynamic programming problem assuming conditional expectation of moments in each period based on state variables (interest rates, or dividend yield, etc.) in the prior period. Backward recursion, assuming the investor’s utility or value function, is used to solve the multi-period problem. Optimal dynamic policies for multi-period problems are found numerically, by stochastic dynamic programming, or by approximation using restrictive statistical models. Dynamic intertemporal models introduced additional estimation errors and instability to the Markowitz static approach. In the Merton model, long-term investors should revise their portfolios daily in response to state variables. Frequent portfolio adjustment adds a greater cost to this long-term model.

DPT is a theoretical and practical enhancement to the implementation of estimating efficient portfolios by using mean-reversion risks. Like MPT, DPT gives a single period solution. Like intertemporal portfolio choice, the DPT solution is non-myopic because it depends on holding period and estimates of mean-rever- sion risk levels. The discrete time signal processing of a discrete return process measures levels of mean-reversion risks. Unlike MPT, DPT does not assume any return distribution. DPT uses a nonparametric description of long-term risk. Unlike intertemporal portfolio choice, DPT does not forecast state variables or returns but uses forecasts of the long-term variance and its mean-reversion risk components. DPT uses low-frequency signal processing to describe mean-re- version risk levels of the unconditional variance. The discrete time Fourier transform gives a nonparametric description of the total variance and autocovariance by making a one-to-one transform of a finite length discrete return signal to a discrete risk spectrum. The digital return signal, r ˜ [ n ] consists of a sequence of returns of length T,

r ˜ j [ n ] , n = 1 , 2 , 3 , ⋯ , T (1)

The square brackets indicate a digital process. The integer n indicating the place in the sequence and r ˜ j [ n ] is the stochastic return at time nδt for asset j. The observation horizon is δt. T gives the finite return signal length (bandwidth) but is not the holding period. DPT uses a finite duration signal, or finite bandwidth. Digital return signals are constructed from discontinuous prices and dividends. The finite discrete time, discrete frequency Fourier transform (DFT) of the low-frequency return signal gives the mean-reversion risk makeup of the original signal,

R k j = R j [ k ω ] = ∑ n = 1 T r ˜ j [ n ] exp ( − i [ k ω ] n ) , k = 1 , 2 , ⋯ , K = T / 2 δ t (2)

R_{kj}, or R_{j}[kω] represent the mean-reversion risk, or standard deviation of return of the kth mean reverting return component contained in the return signal. The K mean-reversion risks describe the total variance of the original signal. The number of harmonic periods, K = T/2δt depends on δt, the sampling interval, or observation horizon and the signal length, T. The angular frequency, ω = 2π/T depends on the signal length, T. R_{kj} is the mean-reversion risk with mean-rever- sion length p_{k},

p k = T δ t / k , k = 1 , 2 , , K = T / 2 δ t (3)

The period, p_{k} is the mean-reversion length of the kth mean-reversion. The lengths of the mean-reversions decrease harmonically. The K mean-reversion variance components are uncorrelated, non-overlapping, and form a complete orthogonal set. The power spectrum or mean-reversion variance spectrum of the original signal consists of mean-reversion risks for K horizon lengths,

σ j 2 = var ( r j ˜ [ n ] ) = 1 2 ∑ k = 1 K R k j 2 (4)

The total single period variance of the return signal (1) is ½ the sum of K longer horizon variance components, R k 2 . Jones [

σ p 2 = var ( r ˜ p [ n ] ) = 1 2 ( ∑ k = 1 K ( ∑ j = 1 N w j R k j cos θ k m j ) 2 + ( ∑ j = 1 N w j R k j sin θ k m j ) 2 ) (5)

where

σ p 2 = var ( r ˜ p [ n ] ) the total single period variance of a portfolio of N assets,

r ˜ p [ n ] = stochastic portfolio return in period n, n = 1 , 2 , ⋯ , T ,

w_{j} = weight of asset j in the portfolio,

R_{kj} = the mean-reversion risk of period p_{k}-length returns of asset j,

θ_{kmj} = relative mean-reversion phase between asset m and asset j, for the kth mean-reversion length,

cosθ_{kmj} = correlation between asset m and asset j, for the kth mean-reversion length,

R_{kj}cosθ_{kmj} = systematic or beta risk of the kth mean-reversion of asset j relative to asset m,

R_{kj}sinθ_{kmj} = unsystematic or alpha risk of the kth mean-reversion of asset j relative to asset m,

K = the number of mean-reversion lengths measured, k = 1, 2,..., K; K = ½T,

T = the signal length (for example for T = 48 months, K = 24),

p_{k} = kth mean-reversion length; p k = T δ t / k , k = 1 , 2 , ⋯ , K = T / 2 δ t ,

δt = sampling interval or revision length (1-month).

The single period total portfolio variance; σ p 2 = var ( r ˜ p [ n ] ) , is composed of K mean-reversion variance contributions that describe the variance, covariance, and autocorrelation of the portfolio return process. These K portfolio mean-re- version risks are uncorrelated and non-overlapping. DPT portfolio selection is more robust than time domain models because of the orthogonal nature of mean-reversion risks. For example, 1-year mean-reversion risk is independent of 3-month mean-reversion risk. In Equation (5) the portfolio variance uses the relative mean-reversion phases, θ_{kmj}. The correlation between assets is measure by the phase relative to the market index, or benchmark index. DPT uses index relative correlations rather than individual inter-asset correlations. The K mean- reversion risks are further composed of passive and active risk contributions. The cosine terms describe the systematic or beta risk and the sine terms the unsystematic or alpha risk. The cosine of the relative phase (cosθ_{kmj}) is the correlation between the security and a benchmark for the kth mean-reversion length. The objective of DPT is the same as MPT, to maximize expected portfolio return. Jones [

Maximize

E ( r ˜ p ( t ) ) = ∑ j = 1 N w j E ( r ˜ j ( t ) ) = ∑ j = 1 N w j μ j (6)

subject to: 4K constraints (K = 24) k = 1 , 2 , 3 , ⋯ , K

∑ j = 1 N w j R k j cos θ k m j ≤ c β k (7)

∑ j = 1 N w j R k j cos θ k m j ≥ − c β k (8)

∑ j = 1 N w j R k j s i n θ k m j ≤ c α k (9)

∑ j = 1 N w j R k j s i n θ k m j ≥ − c α k (10)

∑ j = 1 N w j = 1 (11)

w j ≥ 0 , j = 1 , 2 , 3 , ⋯ , N (12)

where w_{j} is the weight of asset j in the portfolio, R_{kj} is the mean-reversion risk for the kth horizon length, or standard deviation of kth period returns of asset j. The kth phase-shift between index m and security j’s returns is, θ_{kmj}. The k cross-cor- relations, θ_{kmj}, define the multiple horizon correlation structure. K is the number of horizon lengths measured ( k = 1 , 2 , ⋯ , K ) and is 1/2 the signal length T (for example if T = 48 months, K = 24,). cβ_{k} is the right hand side (RHS) constant that limits systematic or β-risk of mean-reversion length p_{k}. cα_{k} is a RHS constant that limits unsystematic or α-risk of the portfolio’s mean-reversion risk with length p_{k}. DPT is a linear model and is stable for very large universes and long time periods. It allows control over the risk from long-term mean reverting components generating single period portfolio variance.

The relaxed constraint set, (7) to (10), describes the long-term portfolio variance (5) that results in single period risk. These risk constraints are uncondi-

k | Period p_{k } (Months) | Calendar Based Risk | Calendar Effect | k | Period p_{k } (Months) | Calendar Based Risk | Calendar Effect |
---|---|---|---|---|---|---|---|

1 | 48.0 | 4-yr | Presidential | 13 | 3.7 | ||

2 | 24.0 | 2-yr | Election | 14 | 3.4 | 8-yr | Business Cycle |

3 | 16.0 | 15 | 3.2 | 16-yr | Long Term | ||

4 | 12.0 | 1-yr | Annual/Summer | 16 | 3.0 | 1/4-yr | Quarter |

5 | 9.6 | 17 | 2.8 | ||||

6 | 8.0 | 8-yr | Business Cycle | 18 | 2.7 | ||

7 | 6.9 | 19 | 2.5 | ||||

8 | 6.0 | 1/2-yr | Sell in May | 20 | 2.4 | 1-mo | Jan/Nov |

9 | 5.3 | 21 | 2.3 | ||||

10 | 4.8 | 8-yr | Business Cycle | 22 | 2.2 | 8-yr | Business Cycle |

11 | 4.4 | 23 | 2.1 | ||||

12 | 4.0 | 1-mo | Jan/Nov | 24 | 2.0 | 1-mo | Jan/Nov |

^{a}There are calendar periods corresponding to long, intermediate, and short-term institutional event intervals with a 48-month signal length and monthly returns using low-frequency digital signal processing. The remaining periods are non-calendar. In the discrete Fourier transform (DFT) framework, the 24 mean-re- version lengths generate total 1-month risk.

Period | Associated Calendar Effect |
---|---|

1-month | January/November |

1-quarter | Quarter |

6-month | Sell in May |

1-year | Annual/Summer |

2-year | Election |

4-year | Presidential |

8-year | Business Cycle |

16-year | Long Term |

Noise | Random Walk |

^{a}Using digital signal processing (DSP) with a 4-year signal and a monthly observation horizon there are eight calendar mean-reversion lengths. These calendar mean-reversion lengths are associated with calendar effects. Multiple seasonal and cyclical effects as well as non-calendar effects such as white noise influence one-month variance.

tional; they do not change over time since mean-reversion lengths in all return process are assumed stationary. They describe the distribution of portfolio risk by mean-reversion lengths. As with MPT, the expected returns in the objective function of DPT may be estimated using historical means, or using multi-period forecasts based on state variables or past returns. With mean reverting returns the expected geometric mean can be used when the holding period is longer than one month. The DPT optimal portfolios depend on multiple horizon risks. DPT allows hedging, or speculative demands for multiple length mean-reversion risks. Holding mean-reversion risks with lengths equal to or shorter than the holding period will hedge unexpected changes in returns while reducing the variance of terminal wealth. We can hedge the risk of remaining unexposed to in-holding-period mean-reversions. For mean-reversion lengths longer than the holding period speculative demands will change the risk exposure of the optimal solution. Speculative demand may also apply to mean-reversion risks with lengths shorter than the holding period when the investor has a forecast of the pattern of the mean-reversion. The DPT formulation does not predict mean reverting turning points, or their cyclical phase, only the risk given the mean-re- version length. Cyclical or periodic speculation must be based on signals, or forecasts of expected returns or risk premiums that predict mean-reversion patterns.

DPT finds optimal single period policies that depend on intertemporal hedging and speculative demands for multiple length, unconditional, non-overlap- ping, mean-reversion risks. Covariance and autocovariance are estimated using low-frequency DSP. DPT solutions are non-myopic unless all return processes are white noise. In DPT, single period variance results from contributions to variance from longer horizon returns. Risk is constrained based on mean-reversion length, holding period, and desired exposure to active and passive risk. DPT does not utilize predictable expected return patterns because the phase is lost in taking the Fourier transform. The DPT paradigm assumes that the levels of risk that long-term mean reverting patterns generate are stable. DPT assumes no information about what patterns other than their length. By choosing appropriate values of mean-reversion risk RHS constants; cβ_{k} and cα_{k} in constraints (7) to (10), the linear DPT portfolio selection problem independently controls K horizon systematic and unsystematic mean-reversion risks. The kth risk constraint set (7) to (10) is statistically independent from the k + 1 constraint set. The risk contribution of 4-year returns to one-month risk is independent of the risk contribution of 2-year returns, or 1-year returns, etc. to one-month risk. The non- overlapping variance contributions of multiple mean-reversion lengths are an essential feature of the frequency domain. There are four risk constraints for each mean-reversion length. The first two constraints (7) and (8) control the upper and lower bounds on systematic (beta) risk (cosine terms), and the second two risk constraints (9) and (10) control upper and lower bounds on unsystematic (alpha) risk (sine terms).

The relative phases, θ_{kmj}, are estimated with a cross-spectral density between the security and the market index or benchmark returns. The phase-shifts, θ_{kmj} in constraint equations (7) to (10), give the relative lead or lag of the kth mean- reversion length component between security j and index m. The cosine and sine terms relate to the time domain. The cosine of the phase shift, θ_{kmj} is the correlation between p_{k}-length returns of the two processes,

− 1 ≤ ρ k m j = cos θ k m j ≤ 1 (13)

where ρ_{kmj} is correlation of security j with the index portfolio m’s kth periodic returns. For example with k = 4, ρ_{4mj} equals the correlation between all annual returns of security j and index m. The cosθ_{kmj} is the in-phase or systematic component and sinθ_{kmj} is called the quadrature or unsystematic component where

cos 2 θ k m j + sin 2 θ k m j = 1 , k = 1 , 2 , ⋯ , K (14)

The phase-shift relative to another signal, particularly a strong signal such as the benchmark index is estimated at higher significance than inter-asset correlations used in MPT.

The normative DPT model prescribes what investors should do in the presence of mean-reversion risk in each revision period of their holding period. There are multiple single period efficient frontiers, one for each mean-reversion length. The statistical independence of the K mean-reversion risk components allows for multiple fund separation. Because the K mean-reversion risks are orthogonal, uncorrelated, and non-overlapping, K-fund separation applies. DPT implies positive models; calendar based or horizon based CAPM, and an autocovariance APT (Jones [

The DPT mean-reversion risk model is non-myopic. The one period solution includes the variance contributed by long-term mean-reversions. The DPT solution allows the application of hedging and speculative demands based on investor’s holding period. Investors with shorter holding periods will have zero hedging demand for the risk of longer length mean-reversions. On the other hand, investor’s with longer holding periods should have larger hedging demands since more mean-reversion lengths will be within the holding period. Reducing risk by holding a larger portion of a portfolio in high risk-return assets the longer the investment horizon is called time diversification. In addition to zero hedging demand for mean-reversion risks longer than the holding period, investors will have zero hedging demand for non-calendar length mean-reversion risks. Levels of mean-reversion active and passive risk are controlled for the 24 mean-reversion lengths given in _{k} in Equations (7) and (8), the level of systematic risk of mean-reversion length p_{k} is constrained in the optimal portfolio. The RHS constants cα_{k} in equations (9) and (10) limit levels of unsystematic risk.

Take the example of a particular investor with a holding period of four years. The investor needs a portfolio to hold over the first month. For each the 24 mean-reversion lengths in _{k} = cα_{k}) for all k. This will constrain all risk contributions to the portfolio variance equally. When all return processes in the economy are random walks this myopic solution will be optimal. The random walk process has no mean-reversion. The random walk variance has equal risk contributions at all mean-reversion lengths. DPT finds a myopic efficient frontier by changing the constant RHS value for all constraints. The myopic investor will select the efficient portfolio that satisfies his or her risk preference. Non-ca- lendar mean-reversion lengths have not been found to be different from a random walk (Jones [_{k }and cα_{k} for k = 1, 2, 4, 8, 16 and for k = 12, 20, and 24. Since we are sampling monthly, 1- month risk is reflected in harmonics of 2-months, 2.4-months and 4-months. Hedging will reduce our risk in the holding period while increasing expected return. If the investor prefers a passive portfolio, cβ_{k} can be set higher than cα_{k}. Since DPT assumes no short selling, the DPT optimizer cannot satisfy negative hedging demand.

Speculative demand results when the investor has a forecast of the phase of mean reverting returns for a particular mean-reversion length. Conditional speculative demand for the risk of a particular mean-reversion length will depend on forecasts of patterns within a given mean-reversion length. The investor can use the mean-reversion constraints (7) to (10) to speculate. For example, if the investor believes an 8-year market cycle is currently at the bottom, he or she can increase 8-year mean-reversion risk by increasing cβ_{k} and cα_{k} for k=6, 10, 14 and 22. Since the signal length we are using is four years, 8-year risk is reflected in the shorter harmonics; 2.2-month, 3.4-month, 4.8-month, and 8-month. Similarly if we believe 6-month mean-reversion is currently at a top we can speculate by tightening cβ_{k} and cα_{k} for k = 8. This will force 6-month risk out of the portfolio so that we can buy 6-month risk after returns fall.

At the end of the month the investor must find a new solution for the next month. The holding period has shortened to three years and 11 months. We can no longer use 4-year risk to hedge. We continue to hedge using 2-year, 1-year, 1/2-year, 1/4-year, and 1-month risk. As the holding period shortens fewer mean-reversion lengths can be used to hedge. In the last month of the holding period only 1-month risk is available to hedge. The question of how much mean-reversion risk we should use to hedge remains an open question.

The DPT model adopts the single index correlation structure proposed by Markowitz [

Because DPT is a linear programming problem, integer variables can be added to include options positions, fixed trading costs, and portfolio size constraints (e.g. Jones [

The signal processing representation of portfolio variance avoids the problem of over-lapping mean-reversions encountered in time domain models. The level of confidence is higher because low-frequency signal processing provides robust non-overlapping estimates of periodic mean-reversion risks. DPT allows geometric mean, growth optimal, or momentum objective functions. It allows additional constraints such as; options constraints, shortfall constraints, solvency constraints, and fundamental constraints as well as other portfolio management constraints (e.g. Jones [

DPT assumes an observation horizon and revision period of one month. Shorter estimation horizons are not possible using low-frequency digital signal processing solutions. While the Merton model suggests portfolio revision daily, a revision frequency shorter than one month is not meaningful using DPT. Longer observation periods such as quarterly or yearly do not provide enough time series data to estimate DPT parameters reliably. Practical implementation DPT requires longer return histories than are often considered relevant for myopic MPT, or high-frequency trading. Long time series are required to measure long horizon autocorrelations. While it is common practice to estimate parameters of MPT using five years of data, DPT requires a minimum of 16 years of monthly returns. We can estimate long-term mean-reversion variance contributions to single period risk at higher significance with longer time series. In addition, longer history allows measurement of longer period mean-reversion risks.

MPT is a short-term volatility model while DPT is a long-term variance model. In MPT returns are independently distributed over time (IID), or constant and stationary resulting in myopic solutions. The random walk model suggests that shorter-term volatility (hourly or daily) generates one-month risk while the DSP model suggests that variances of multiple longer-term holding period returns generate one-month risk. While MPT researchers attempt to add tail risk to account for mean-reversion, DPT explicitly, accounts for mean-reversions up to sixteen years and can be formulated to include longer mean-reversions. In the DPT paradigm, stock returns are stationary but can have multiple length mean reverting patterns. Glover and Jones [

Most intertemporal portfolio choice studies do not allow the multi-period investor to differentiate using a large universe of assets. Multiple period models are not able to identify buy-and-hold allocations to specific styles, sectors, or securities. Non-myopic single period DPT is able to identify buy-and-hold allocations to specific assets classes, or securities, over alternative horizons within optimal portfolios. The DPT model does not require conditioning predictive state variables or forecasts of expected returns but does allow intertemporal hedging by incorporating forecasts of unconditional longer-term mean-reversion variances. While MPT and intertemporal theory evolved into volatility models. In low- frequency DPT, the shortest estimation period possible is a monthly period. DPT is a long horizon stationary variance model. By quantifying the risk of unconditional mean-reversions of different lengths, DPT can satisfies hedging and speculative demands that are consistent with the investor’s holding period and expectations.

MPT assumes investors make decisions myopically in a static single period framework assuming no mean-reversion. Intertemporal portfolio choice assumes investors make intertemporal decisions dynamically based on conditional mean-reversions. DPT assumes investors make long-horizon decisions dynamically using static single period solutions that incorporate unconditional mean- reversion risks. The DPT solution satisfies both myopic demand generated by the return maximizing objective and the covariance of returns, and satisfies non- myopic demand generated by hedging and speculative demand for mean-rever- sion risks of different lengths in relation to the holding period.

DPT is linear with a linear programming solution, MPT requires quadratic programming, and intertemporal portfolio choice requires dynamic programming, or numerical methods. MPT does not provide for predictable time variation in the investment returns and the quadratic solution is ill-conditioned and unstable. Continuous-time portfolio choice provides useful understanding of strategic asset allocation in terms of hedging and speculative demands in the presence of mean-reversion. It is useful for daily revisions over yearly or longer holding periods, or monthly revisions for holding periods longer than twenty years. Intertemporal portfolio choice is hard to solve, is restricted to maximum utility formulations, and is limited to small asset universes. Intertemporal portfolio choice is difficult to implement because long-term forecasts of the dynamics of state variables and their relation to asset returns is subject to considerable uncertainty. The DPT solution includes hedging and speculative demand for multiple mean-reversion length risks and has a stable solution based on stationary mean-reversion variance. The DPT long-horizon model does not require forecasts of state variables or risk premiums but uses long-term forecasts of unconditional autocorrelations to allow multiple period strategic decisions in the single period solution. DPT uses long-term mean-reversion return variances to describe monthly risk. Investors will have zero hedging demand for mean-re- version risks with lengths longer than their holding period and zero hedging demand for non-calendar length mean-reversion risks. The application of low- frequency digital processing in DPT gives investors a framework and effective model for constructing long-term portfolios in single period solutions. A better understand of DPT will help portfolio theory move from the age of volatility to the digital age of long-term reliability and stability.

MPT and intertemporal portfolio choice have serious theoretical and practical weakness. DPT improves on some of these shortcomings by incorporating long-term risk in a linear optimization. The main innovation and contribution of this research is to provide a single period solution that gives investors control over their exposure to the long-term distribution of portfolio returns as well as to passive and active portfolio returns. It provides a long horizon risk model and its linear formulation offers a significant potential for further innovation. DPT adds financial theory as a new field of digital technology. DPT is limited to long-term stationary mean-reversion components of risk. It does not capture short-term black swan risk. The method depends on using monthly time signals. Further research is needed to clarify how levels of hedging demand and speculative demand for mean-reversion risks affect long-term optimal solutions over various holding periods. DPT can be empirically tested to shed light on these and other issues.

I am grateful for the comments of Peter L. Bernstein, Dimitris Bertsimas, Mark N. Broadie, John Y. Campbell, John S. Cone, George B. Dantzig, Fred W. Glover, Clive W.J. Granger, Rustam Ibragimov, Steven M. Kay, Andrew W. Lo, David G. Luenberger, Harry M. Markowitz, Richard O. Michaud, Merton H. Miller, William T. Moore, Panos M. Pardalos, Stephen A. Ross, James Stock, Mark Rubinstein, Paul A. Samuelson, Charles S. Tapiero, John N. Tsitsiklis and Israel Zang. All errors are the responsibility of the author.

Jones, C.K. (2017) Modern Portfolio Theory, Digital Portfolio Theory and Intertemporal Portfolio Choice. American Journal of Industrial and Business Management, 7, 833-854. https://doi.org/10.4236/ajibm.2017.77059