---
title: "La stratégie de désinvestissement graduel des marchés financiers sécurise-t-elle réellement l'épargne ?"
title_en: "Does the gradual disinvestment strategy really secure savings?"
authors:
  - name: "Alexis Direr"
    affiliation: "Université d'Orléans, CNRS, LEO (UMR 7322) & Paris School of Economics"
  - name: "Eric Yayi"
    affiliation: "Université d'Orléans, CNRS, LEO (UMR 7322)"
date: "2017-05-31"
doi: "10.3917/reco.693.0505"
keywords: [portfolio choice, asset allocation, lifecycle investing, target-date funds, retirement saving, gradual disinvestment, risk measures]
keywords_fr: [choix de portefeuille, allocation d'actifs, sécurisation des plus-values, épargne]
jel_codes: [G11, D14, D91, J14]
language: [fr, en]
type: research-article
---

# La stratégie de désinvestissement graduel des marchés financiers sécurise-t-elle réellement l'épargne ?

**Authors**
- Alexis Direr — Université d'Orléans, CNRS, LEO (UMR 7322), F-45067 Orléans, and Paris School of Economics — *alexis.direr@univ-orleans.fr*
- Eric Yayi — Université d'Orléans, CNRS, LEO (UMR 7322), F-45067 Orléans — *eric.yayi@univ-orleans.fr*

**DOI**: [10.3917/reco.693.0505](https://doi.org/10.3917/reco.693.0505)

**Keywords (EN)**: portfolio choice, asset allocation, lifecycle investing, target-date funds, retirement saving.
**Mots-clés (FR)**: choix de portefeuille, allocation d'actifs, sécurisation des plus-values, épargne.
**JEL codes**: G11, D14, D91, J14.

---

## Abstract

**Français (verbatim).** Cet article étudie la validité empirique du mécanisme de sécurisation du capital final fondé sur le désinvestissement graduel en actifs risqués. Nous comparons ce mécanisme à une politique de constance de la part investie en actifs risqués à partir des données financières de cinq pays, pour des horizons variant de 5 à 30 ans. Nous utilisons différentes définitions du risque : l'écart-type du rendement cumulé final, son degré de dissymétrie négative, la probabilité de perte ou la prime de risque réclamée par un investisseur pour passer d'une stratégie à l'autre. Contrairement aux recommandations habituelles des conseillers financiers, nos résultats ne montrent pas d'avantages tangibles de la stratégie de désinvestissement graduel en termes de réduction du risque à l'échéance du placement. Ces résultats sont robustes à différentes variantes d'estimation du risque.

**English (indicative translation, not from the source).** The paper tests the empirical validity of the "capital securing" mechanism based on gradual disinvestment from risky assets. The mechanism is compared to a constant-risky-share policy using historical financial data from five countries and horizons from 5 to 30 years. Four risk measures are considered: standard deviation of the cumulative real return, skewness, loss probability, and the calibrated risk premium an investor would require to switch from one strategy to the other. Contrary to standard financial-advisor recommendations, the results show no tangible advantage of gradual disinvestment in reducing terminal risk. The conclusions are robust to several alternative risk specifications.

---

## 1. Motivation and contributions

The paper is motivated by the practical and regulatory prominence of gradual-disinvestment savings products — French *assurance-vie* and PERP contracts (2.2 million subscribers in 2013), and U.S. target-date funds, which by 2011 captured more than 90% of assets in 23% of 401(k) plans and were endorsed by the 2006 Pension Protection Act as a default option. Despite their widespread adoption, the empirical case that progressively reducing the risky share actually lowers terminal risk has not been tested at scale on real returns. The closest theoretical benchmarks (Merton 1969; Mossin 1968; Samuelson 1969) deliver a constant share under iso-elastic preferences and i.i.d. returns; departures (Bodie and Crane 1997; Samuelson 1991) hinge on negative serial correlation, which is empirically weak.

The paper makes the following contributions.

1. **A multi-country, multi-horizon empirical comparison.** Real historical returns for five countries — Germany (1959–2014), Denmark (1922–2014), the United States (1871–2012), France (1947–2013) and Sweden (1874–2012) — are used to compare the gradual-disinvestment strategy against a constant-share strategy with the *same average risky share* across the accumulation period. Sliding windows generate $N = n - T$ realisations per country for each horizon $T \in \{5, 10, 15, 20, 25, 30\}$ years; a pooled database equal-weights all sub-periods across countries.

2. **Four risk measures applied jointly.** Standard deviation of the cumulative real return; skewness coefficient (Harvey and Siddique 2000; Kimball 1990); loss probability at maturity; and the calibrated risk premium an investor with CRRA preferences would require to accept the gradual profile rather than the constant one. Each is tested for statistical significance (Student's $t$ on returns, Fisher on variances, asymptotic $Z$ on skewness).

3. **A real-world disinvestment grid.** The decreasing profile is calibrated on the average grid of 20 U.S. retirement contracts and target-date funds (Allianz, Vanguard, Fidelity, BlackRock, T. Rowe Price, etc.): the risky share starts near 85% at age 35 (30 years before a 65-year liquidation) and falls roughly linearly to 40% at retirement. A French calibration (14 *assurance-vie* and PERP contracts) is used as a robustness check; French contracts feature a markedly lower starting share (≈ 65% vs. 85%) and a steeper decline.

4. **Headline empirical finding.** No tangible difference between the two strategies emerges on standard deviation, skewness, loss probability or risk premium at any horizon up to 30 years. The result is robust to (i) Saha (1993) expo-power preferences with increasing relative risk aversion (Holt and Laury 2003), (ii) equalisation of expected returns across strategies, and (iii) the French disinvestment grid.

5. **Conditional advantage under illiquid intertemporal wealth.** When the saver invests gradually period-by-period (no leverage against future labour income), a constant share applied to *current* financial wealth implicitly produces a *rising* exposure of intertemporal wealth to equity risk. The gradual contractual profile dampens this drift and yields a slightly lower standard deviation of terminal wealth — but the gain is quantitatively small and disappears once expected returns are equalised.

---

## 2. Data and disinvestment profile

### 2.1 Return series

| Country | Risky asset | Risk-free asset | Period | Source |
|---|---|---|---|---|
| Germany | DAX | German government bonds | 1959–2014 | Bloomberg, FRED |
| Denmark | OMX Copenhagen 20 (extended) | Danish government bonds | 1922–2014 | Nielsen and Risager (2001), Bloomberg, FRED |
| United States | S&P composite (from 1926) | 3-month T-bills | 1871–2012 | Robert Shiller (Yale) |
| France | CAC 40 | 10-year Treasuries | 1947–2013 | Le Bris and Hautcœur (2010), IFS 2014 |
| Sweden | Stock basket | Long-term government bonds | 1874–2012 | Waldenström (2014) |

Real returns are derived from total return indices and contemporaneous inflation (Appendix A). Equity returns vary considerably across business cycles and crises; bond returns are markedly less volatile. Headline descriptive statistics for the 30-year cumulative real return in the pooled database: equity mean 7.09% annualised, risk-free mean 2.52%, equity premium ≈ 4.57%, equity standard deviation 6.60 (cumulative scale).

### 2.2 The disinvestment grid

The reference profile is the cross-contract average of 20 U.S. target-date funds and retirement accounts (Allianz, American Funds, BlackRock, Fidelity, Invesco, John Hancock, PIMCO Real Return, Principal, JP Morgan, T. Rowe Price, Vanguard Target Retirement, Wells Fargo, American Century, Franklin Templeton, Manning & Napier, Putnam, ClearTrack Retirement Income, MassMutual RetireSmart, Russell, TIAA-CREF). Liquidation is normalised to age 65. The risky share is approximately 85% at age 35 and declines roughly linearly to 40% at age 65.

The constant alternative invests the *arithmetic mean* $\bar{\alpha}$ of the gradual profile in every period:

$$\bar{\alpha} = \frac{\alpha(1) + \dots + \alpha(T)}{T}.$$

By construction the two strategies share the same average exposure to the risky asset; they differ only in the time profile of that exposure.

---

## 3. Methodology

For a country with $n$ years of data and a horizon $T \in \{5, 10, 15, 20, 25, 30\}$, the cumulative real return in window $s = 1, \dots, n - T$ under the gradual profile is

$$R_s^d = \prod_{t=1}^{T} \big[1 + \alpha(t)\,r_s(t) + (1 - \alpha(t))\,r_s^f(t)\big],$$

and under the constant profile

$$R_s^c = \prod_{t=1}^{T} \big[1 + \bar{\alpha}\,r_s(t) + (1 - \bar{\alpha})\,r_s^f(t)\big].$$

Sliding the window yields $N$ paired observations $\{(R_s^d, R_s^c)\}_{s=1}^{N}$ per country per horizon. The pooled database equal-weights all sub-periods across all countries, implicitly drawing both the country and the sub-period uniformly.

The risk-premium measure of §4.4 calibrates a CRRA utility $u(W) = W^{1-\lambda}/(1-\lambda)$ with relative risk aversion $\lambda$, and defines the per-window premium $P$ that equalises expected utilities:

$$\frac{1}{N}\sum_{s=1}^{N} u(R_s^d + P) = \frac{1}{N}\sum_{s=1}^{N} u(R_s^c).$$

A positive $P$ means the constant profile is preferred; a negative $P$ means the gradual profile is preferred. The baseline is $\lambda = 2$.

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## 4. Results

### 4.1 Expected returns

Annualised real returns are $\approx 5\%$ for Sweden and the United States, $\approx 4.5\%$ for Germany and Denmark, $\approx 3\%$ for France, and $\approx 4.5\%$ for the pooled database. Strategy-level differences are small and disappear in countries with the longest series. The Student test on the paired return differential (Table 2) rejects the null of equal mean returns only for Germany at 25 and 30 years (5% level); all other country/horizon combinations are insignificant.

### 4.2 Dispersion (Table 3)

Standard deviation rises with horizon — consistent with i.i.d. returns where the cumulative dispersion grows roughly with $\sqrt{T}$ — and varies sharply by country (high in Sweden, low in Germany and the U.S.). Strategy-level differences are barely visible: the constant profile is marginally better at long horizons in Germany, France and Sweden, equivalent in the U.S., and marginally worse in Denmark. The Fisher test of variance equality fails to reject the null for any country or horizon at the 5% level. The "securing" mechanism delivers no measurable variance reduction at maturity.

### 4.3 Skewness (Table 4)

Skewness coefficients are positive (right-tailed cumulative returns) with no clear horizon profile, except in the pooled database where they trend upward. Strategy-level differences are visually small but the asymptotic $Z$ test rejects equality of skewness for Denmark across most horizons, the U.S. at 10 years, Sweden at 25–30 years, and the pooled database at 10–30 years (5% level). Where present, the difference modestly favours the constant profile.

### 4.4 Risk premium (Tables 5, Figures 6 & 9)

At $\lambda = 2$, the risk premium of the gradual profile is approximately zero for Denmark, the U.S. and Sweden at all horizons, and for the others at horizons under 20 years. It becomes mildly positive (constant profile preferred) for Germany, France and the pooled database at long horizons, but the magnitudes remain quantitatively small relative to country and horizon effects. Table 5 shows the premium is essentially flat in $\lambda \in \{1, 1.5, 2, 2.5, 3, 4\}$, ruling out a strong dependence on the calibration of risk preferences.

> **Authors' critical reading.** A nonzero risk premium can mechanically reflect differences in *expected return* between the two strategies rather than differences in risk per se. The robustness check of §5.2 isolates the latter.

### 4.5 Loss probability (Figure 7)

Loss probability falls sharply with horizon — from $\approx 15\%$ at 5 years to under $4\%$ at 30 years in every country except France (whose equity premium is the lowest in the sample). Strategy-level differences are negligible and unsystematic in sign. The data confirm horizon diversification of equities but provide no support for the inference that gradual disinvestment outperforms a constant share on this dimension.

---

## 5. Robustness and extensions

### 5.1 Alternative preferences (Figure 8)

Saha (1993) expo-power utility with $r = 0.269$ and $\alpha = 0.029$ (Holt and Laury 2003) gives increasing relative risk aversion as wealth rises. Even though one might expect this to mechanically favour the gradual profile (less equity exposure when wealth is largest), the simulations across initial wealth levels $W_0 \in \{1, 100, 1000, 10000\}$ on the pooled database show the gradual profile *does not* outperform the constant one. The investor only values risk on terminal wealth, not the time profile of exposure.

### 5.2 Equalising expected returns (Figure 9)

The constant share is re-set country-by-country and horizon-by-horizon to deliver the *same expected cumulative return* as the gradual profile, isolating the risk dimension. Risk premia change negligibly relative to §4.4: the near-equivalence of the two strategies is not an artefact of expected-return differences.

### 5.3 French disinvestment grid (Figures 10, 11)

Replacing the U.S. grid with the average of 14 French *assurance-vie* and PERP contracts (AXA Arpège/Clèr/Far/PERP, BNP PERP, Solésio Banque Postale, Générali, HSBC Élysées, MAAF Winalto, MAIF, MMA, Société Générale Epicéa) yields a profile that starts at $\approx 65\%$ risky share rather than $\approx 85\%$ and declines more steeply. Under this grid the constant profile shows a slightly larger advantage — clearer in Germany and France, marginal elsewhere — but the conclusion of near-equivalence still holds.

### 5.4 Illiquid intertemporal wealth (Figures 12–14, Table 6)

The benchmark assumes a single initial capital invested for $T$ periods. In reality, savers contribute periodically. With perfect markets the saver could borrow against future income at the risk-free rate and the problem reduces to the benchmark. With *imperfect* markets and zero borrowing capacity, the saver invests progressively as savings accumulate.

The implicit share of intertemporal wealth invested in equities at date $t$ is

$$\gamma(t+1) = \frac{A(t)}{A(t) + \sum_{s=t+1}^{T} a_s / \prod_{j=t+1}^{s}(1 + r^f(j))} \, \alpha(t+1).$$

Two findings follow:

- **A constant contractual share applied to current financial wealth produces a *rising* implicit exposure of intertemporal wealth** — from $\approx 4\%$ at age 35 to $> 50\%$ at age 65 over a 30-year horizon. The gradual contractual profile dampens this drift; its implicit profile rises in the first half, plateaus, then declines slightly.
- **The gradual profile produces a lower terminal-wealth standard deviation than the constant profile** in this setting (Figure 14b), and Table 6 rejects the null of equal variances at long horizons (≥ 20 years, 10–5% level). Mean terminal wealth and skewness also differ significantly.

After equalising expected terminal wealth across the two strategies (Figure 14d, last column of Table 6), the variance test no longer rejects equality at any horizon. The advantage of the gradual profile in this case is therefore quantitatively small and partly attributable to expected-return differences.

---

## 6. Conclusion

Three takeaways follow.

1. **The gradual-disinvestment strategy does not measurably reduce terminal risk** relative to a constant-share strategy with the same average risky exposure. The conclusion holds across five countries, six horizons (5–30 years), four risk measures, two utility specifications, and two real-world disinvestment grids.
2. **A common rationale for the gradual profile confuses two notions of risk.** Risk on the *annualised* return decreases with horizon (time diversification of equities); risk on the *cumulative* return increases. Only the latter is relevant for a saver liquidating at maturity.
3. **The gradual profile is not a natural hedge against late-period crashes either.** Once compared to a constant strategy with matched average exposure, the gradual profile concentrates risk in the early portion of the accumulation period, exposing the saver more strongly to early bear markets.

The one setting in which the gradual profile shows an advantage — illiquid human capital with progressive savings — yields a small standard-deviation gain that survives expected-return equalisation only at long horizons and only marginally. The empirical case for systematic gradual disinvestment as a risk-management device is therefore weak.

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## Acknowledgments

The authors thank Christophe Hurlin, Najat El Mekkaoui and Bruno Séjourné for comments on the preliminary version, as well as David Crainich and participants at the Journées Internationales du Risque (Niort), the GdRE 2016 conference and the AFSE 2016 conference. All remaining errors are theirs.

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## Main references

Ameriks, J., Zeldes, S. (2004). How do household portfolio shares vary with age? *Working paper*.

Bodie, Z., Crane, D. B. (1997). Personal investing: advice, theory, and evidence from a survey of TIAA-CREF participants. *Working paper*.

Campbell, J. Y., Hentschel, L. (1992). No news is good news: an asymmetric model of changing volatility in stock returns. *Journal of Financial Economics* 31(3), 281–318.

Cochrane, J. H. (2008). The dog that did not bark: a defense of return predictability. *Review of Financial Studies* 21(4), 1533–1575.

Fama, E. F., French, K. R. (1988). Permanent and temporary components of stock prices. *Journal of Political Economy* 96, 246–273.

Harvey, C. R., Siddique, A. (2000). Conditional skewness in asset pricing tests. *Journal of Finance* 55(3), 1263–1295.

Holt, C. A., Laury, S. K. (2002). Risk aversion and incentive effects. *American Economic Review* 92(5), 1644–1655.

Jagannathan, R., Kocherlakota, N. R. (1996). Why should older people invest less in stock than younger people? *Federal Reserve Bank of Minneapolis Quarterly Review* Summer, 11–23.

Kimball, M. S. (1990). Precautionary saving in the small and in the large. *Econometrica* 58(1), 53–73.

Le Bris, D., Hautcœur, P.-C. (2010). A challenge to triumphant optimists? A blue chips index for the Paris stock exchange, 1854–2007. *Financial History Review* 17(2), 141–183.

Markowitz, H. (1952). Portfolio selection. *Journal of Finance* 7(1), 77–91.

Merton, R. (1969). Lifetime portfolio selection under uncertainty: the continuous time case. *Review of Economics and Statistics* 51(3), 247–257.

Mitchell, O. S., Utkus, S. (2012). Target-date funds in 401(k) retirement plans. *NBER Working Paper* 17911.

Mossin, J. (1968). Optimal multiperiod portfolio policies. *Journal of Business* 41(2), 215–229.

Saha, A. (1993). Expo-power utility: a flexible form for absolute and relative risk aversion. *American Journal of Agricultural Economics* 75(4), 905–913.

Samuelson, P. (1991). Long-run risk tolerance when equity returns are mean-regressing: pseudo paradoxes and vindication of businessman's risk. *MIT Press*, 181–200.

Samuelson, P. A. (1969). Lifetime portfolio selection by dynamic stochastic programming. *Review of Economics and Statistics* 51, 239–243.

Waldenström, D. (2014). Swedish stock and bond returns, 1856–2012.

Yayi, E. (2015). A mean-variance evaluation of gradual disinvestment strategy. *SSRN working paper*.

*The full reference list appears in the PDF.*