January finds many pondering the issue of what to do with their savings in the new year. There are two primary and distinct techniques of asset management: momentum and fair value.

- Momentum investors (‘the trend is your friend’) ignore fundamental value and follow the money, buying when prices are rising and selling when they reverse.
- Fair-value investors (‘a stock’s price is the present value of dividends’) disregard fund flows and trade securities based on their expected future cash flows.

A vast academic literature documents that both momentum and fair-value strategies are profitable; see, for example, Jegadeesh and Titman (1993) for momentum, and Fama and French (1992) for fair value. Investors use a mix of the two strategies to achieve their desired combination of risk and return.

Of course none of it makes sense from the perspective of canonical finance theory. As every trained economist learned at university, capital markets are efficient, so it is futile to pick stocks. The professionals have already priced in all the relevant information leaving no $100 bills lying on the sidewalk. Given this lack of guidance from traditional finance theory, investors have to rely on empirical observation of historic returns.

# Delegation matters

Standard theory assumes that investors look after their own affairs and invest directly in financial markets. An alternative approach is to recognise the practice of most asset owners to delegate responsibility to fund managers.

In new research (Vayanos and Woolley 2011a [1]; see also Vayanos and Woolley 2009 [2]) we take account of the impact of the resulting principal/agent relationships on asset price formation. The new framework can explain many features of capital market performance, such as momentum and value effects, that have so far proved problematic, while maintaining the assumption of rational agents. Moreover, it can be used to analyse the investment strategies that can be employed to exploit the inefficiencies. The model can be calibrated to show the risk-adjusted returns from the strategies used separately or in combination.

Significantly, it shows that the optimal mix of momentum and fair value depends on the investor’s horizon or term of his liabilities.

# Shedding new light

In our model, delegation is the key. For example, asset owners have imperfect knowledge of the ability of the fund managers they invest with. They are uncertain whether underperformance against the benchmark arises from the manager’s prudent avoidance of overpriced stocks or is a sign of incompetence. As shortfalls grow, investors conclude incompetence and react by transferring funds to outperforming managers. The investors’ gradual flows amplify the price changes that led to the initial underperformance and generate momentum. In this way, Bayesian updating can explain how some prices are pushed below fair value, simultaneously creating the well-documented value effect as well as momentum-trading opportunities.

# Calculating returns

The model incorporates multiple securities with each subjected to cash flow shocks and therefore changes in fair value, which are then amplified as investors move funds between managers according to accumulating evidence of their ability.^{1} The result is a theoretical model of a working market in which prices and returns on individual securities are observable over time. Into this marketplace we insert a new investor who, as price-taker, constructs portfolios with a variety of styles and execution. The Sharpe ratios (expected excess returns per unit of standard deviation) of these portfolios can then be compared over one period, and over multiple periods with continuous rebalancing, as in Vayanos and Woolley (2011b). The timing of the returns can also be observed. Below are the main findings. Some are intuitive but useful to have validated; others are far less obvious with striking implications for policy.

# Short horizon** **

The first step is to compare the Sharpe ratios for the two strategies over a one-period horizon.

- Based on optimal implementation of both strategies, and applying realistic calibrations, momentum dominates fair value.

This ties with intuition as well as the mechanism of the model. Amplification effects cause prices to continue trending down to the point where outflows dry up and prices begin to revert back to fair value; gradual inflows likewise explain amplified upward price movements. In this way, returns to the momentum style come quickly, whereas the fair-value investor has to be patient, often buying prematurely as he or she waits for reversal to occur.

Because it offers a higher Sharpe ratio over one period, momentum is the appropriate choice for investors seeking either short-term excess return or short-term risk reduction. But there are caveats.

Success for a momentum investor depends on getting the timing right, first with the buy signal and then selling before prices reverse. The model investigates combinations of ‘look backs’ – the length of time over which prices must rise for the security to be selected – and holding periods. Selecting the optimal lookback leads to strong, positive risk-adjusted returns. But as the investor moves away from this central peak, returns erode sharply to become negative for moving in too soon or selling out too late. (See Figure 1.)

**Figure 1.** Sharpe ratio of momentum as a function of the lookback period^{2}

*Note: *The vertical axis is the excess return as measured by the Sharpe ratio; the horizontal axis is the lookback measured in years.

By contrast, success with fair-value investing is relatively insensitive to execution. The model shows that crude value models, such as those based on price to current earnings, deliver results only slightly below those using more refined estimates of future cash flows.

# Long horizon

The second step is to compare the risk-adjusted returns in a dynamic framework over multiple periods with continuous rebalancing. Here the results are very different and the bottom line is that fair value dominates momentum for long horizons. This result hinges on the difference in risk characteristics.

Because momentum stocks are selected without reference to their value, the outcome of each momentum play is independent of the last. Momentum investing is a series of uncorrelated bets which means that the long-run risk of the strategy equates to the sum of risks for the intervening sub-periods. On the other hand, fair-value investors bide their time waiting to buy low and sell high. Purchases that disappoint will in many cases have become cheaper and are retained. The strategy benefits from the negative serial correlation of returns that causes the annualised long-run risk to decline over time.

All this means that the Sharpe ratio of momentum is approximately constant over time but the ratio for fair value rises with the lengthening term over which the strategy is employed (Figure 2).

**Figure 2.** Annualised Sharpe ratios of momentum and fair value as a function of the investment horizon^{2}

# Investment horizon

Our analysis offers a solution to one of investment's longstanding puzzles: Is the long-run equivalent to the succession of intervening short-runs? Operationally, the issue is whether investors should focus on getting the best return each year, or the best return over the long-run, or is there no difference? The answer is relevant to the current debate about short-termism and the so far unsubstantiated impression that long-run investing is privately, as well as socially, beneficial.

The answer lies in the choice and mix of strategies. It is well-known that returns to momentum and fair value display low or negative correlation (*eg* Asness *et al* 2009).

- This lack of positive correlation means that combining the two strategies improves a portfolio's risk-adjusted return.

Using relative returns from the model permits a definitive response. The model shows that the optimal mix depends on the term of the liabilities and therefore on the horizon of the individual fund.

- The longer the horizon, the more the investor should commit to fair value and the less to momentum.
- The shorter the horizon, the more the investor should commit to momentum and the less to fair value.

These findings are robust for all calibrations and it is the difference in risk over time that is the decisive factor. Figure 2 is based on optimal execution of both strategies and any departure from this will naturally shift the curves and crossover point.

# Practice and policy

The analysis opens up a world of questions and challenges to conventional practice and policy.

- Most large asset pools, sovereign-wealth, pension and charitable funds, have long-term liabilities or objectives. The model shows they are best served by majoring on fair value and limiting the use of momentum.
- Yet several practices and conventions draw them in the opposite direction. These include the magnetism of the herd, tracking error constraints in relation to momentum-distorted indices, short-term performance assessment and reward, and regulatory or actuarial mark-to-market valuations.

Lack of awareness of the drawbacks of short-termism no doubt contributes to these costly mistakes.

# References

Asness, C, T Moskowitz, and L Pedersen (2009), “Value and Momentum Everywhere”, Working Paper, New York University.

Fama, E and K French (1992), "The Cross-Section of Expected Stock Returns", *Journal of Finance*, 47:427-465.

Jegadeesh, N and S Titman (1993), "Returns to Buying Winners and Selling Losers: Implications for Stock Market Efficiency", *Journal of Finance*, 48:65-91.

Vayanos, D and P Woolley (2009), “An Institutional Theory of Momentum and Reversal [2]”, CEPR DP 7068.

Vayanos D and P Woolley (2011a), "An Institutional Theory of Momentum and Revers [1]al", Working paper, London School of Economics.

Vayanos D and P Woolley (2011b), “A Theoretical Analysis of Momentum and Value Strategies", Working paper, London School of Economics.

^{1} For simplicity, the basic model has only one active manager and one investor. The investor responds to the performance of the manager by moving his funds between the active fund and an index fund.

^{2} Figures 1 and 2 are based on data drawn from a conservative calibration in which only a subset of flows are considered. The actual flows will be augmented by investors external to the model piggybacking on momentum.