--- title: "Portfolio Choice with Time Horizon Risk" authors: - name: "Alexis Direr" affiliation: "Université d'Orléans, LEO" orcid: "0000-0002-4459-7780" date: "2023-11-28" doi: "10.1142/S0219024923500267" journal: "International Journal of Theoretical and Applied Finance" volume: "26" issue: "06n07" publisher: "World Scientific Publishing" keywords: [portfolio choice, risk aversion, timing risk] jel_codes: [D8, E21] language: en type: research-article --- # Portfolio Choice with Time Horizon Risk **Authors** - Alexis Direr — Université d'Orléans, LEO — *alexis.direr@univ-orleans.fr* — ORCID: [0000-0002-4459-7780](https://orcid.org/0000-0002-4459-7780) **DOI**: [10.1142/S0219024923500267](https://doi.org/10.1142/S0219024923500267) **Published in**: *International Journal of Theoretical and Applied Finance* (IJTAF), vol. 26, no. 06n07, November 2023. World Scientific Publishing. **Keywords**: portfolio choice, risk aversion, timing risk. **JEL codes**: D8, E21. **Research framework**: Laboratoire d'Économie d'Orléans (LEO), Université d'Orléans. --- ## Abstract I study the allocation problem of investors who hold their portfolio until reaching a target wealth. The strategy suppresses final wealth uncertainty but creates a time horizon risk. I begin with a classical mean variance model transposed in the duration domain, then study a dynamic portfolio choice problem with Generalized Expected Discounted Utility preferences. Using long-term US return data, I show in the mean variance model that a large amount of time horizon risk can be diversified away by investing a significant share of equities. In the dynamic model, more impatient investors are also more averse to timing risk and invest less in equities. The optimal equity share is downward trending as accumulated wealth approaches its target. --- ## 1. Motivation and contributions The paper studies a two-asset portfolio problem in which the investor commits to a money goal and exits the market once that goal is reached. The setup reverses the classical investment frame: instead of fixing a horizon and bearing final-wealth risk, the investor fixes the terminal wealth and bears **time horizon (duration) risk**. The motivation is empirical: many household savings goals — a home purchase, retirement, helping children — are naturally expressed as wealth targets with a flexible date. The paper's contributions are the following. 1. **Empirical characterisation of duration risk.** Using US data 1871–2019, the paper provides five stylised facts on durations required to reach a wealth target with equities versus short-term bills, including a novel observation that bills are riskier than equities not only in expected duration but also in duration variance and skewness. 2. **Mean–variance portfolio theory in the duration domain.** A CAPM-style model is recast with mean duration and duration variance replacing expected return and return variance. The paper derives a duration efficient frontier and shows that a substantial share of equities (≥ 35%) is optimal for any timing-risk-averse investor. 3. **Dynamic portfolio choice with Generalized Expected Discounted Utility (GEDU).** A microfounded intertemporal model in which investors maximise $E[\phi(D(t)u(1))]$, following Dejarnette et al. (2020). GEDU preferences disentangle attitude towards intertemporal utility risk from the discount function and accommodate the timing-risk aversion documented in the experimental literature, which the standard Expected Discounted Utility (EDU) model cannot do (it predicts a counter-intuitive *preference* for timing risk under decreasing convex discounting). 4. **Impatience as a driver of timing-risk aversion.** The paper shows that in the duration domain, impatience plays a dual role: it makes equities more attractive (shorter expected durations) but also makes the investor more averse to duration risk (long delays needed to recoup losses are more costly). The second effect dominates, so **more impatient investors invest less in equities** — a result opposite to the standard intuition that impatient agents take more risk. 5. **A non-linear distance-to-target glide path.** The optimal equity share is a function of the *remaining wealth gap* rather than the remaining horizon. The share plateaus over a wide range of intermediate wealths and converges to zero only as accumulated wealth approaches the target, in contrast to the linear glide paths recommended by mainstream financial advice. --- ## 2. Data **Source.** Annual US data, 1871–2019, all returns annualized, real (deflated by US CPI inflation), inclusive of dividends. - **Equities**: Cowles (1939) for 1871–1925 (12 to 258 NYSE-listed value-weighted securities); S&P 90 index from 1926 to 1957; S&P 500 index from 1957 onwards. - **Bills**: Shiller (1989, 2015) short-term bond annualized rates — 6-month commercial paper rate (Federal Reserve Board) until 1997, 6-month certificate of deposit rate from 1997 to 2012, completed to early 2020 from macrotrends.net. Treasury bills would have been a cleaner proxy for the riskless return but are unavailable before 1920. **Average real returns over 1871–2019**: equities exceed bills by about **3.7 percentage points** per year (the historical equity risk premium). ### Empirical findings (Table 1 of the paper) Statistics are computed by rolling-window over starting investment dates, conditional on the target wealth being reached before the end of the sample. **Equities.** | Target wealth ($1 invested) | 1.25 | 1.5 | 2 | 2.5 | 3 | 3.5 | 4 | |:---:|:---:|:---:|:---:|:---:|:---:|:---:|:---:| | Mean duration (years) | 4.5 | 7.3 | 11.2 | 13.8 | 16.3 | 18.6 | 20.4 | | Duration std. dev. | 4.0 | 5.1 | 6.0 | 6.3 | 6.5 | 7.0 | 6.9 | | Duration skewness | 1.60 | 1.07 | 0.60 | 0.47 | 0.31 | 0.20 | 0.02 | | Rolling windows | 147 | 142 | 128 | 126 | 125 | 124 | 124 | **Bills.** | Target wealth ($1 invested) | 1.25 | 1.5 | 2 | 2.5 | 3 | 3.5 | 4 | |:---:|:---:|:---:|:---:|:---:|:---:|:---:|:---:| | Mean duration (years) | 7.1 | 11.5 | 18.0 | 22.8 | 26.7 | 30.0 | 32.9 | | Duration std. dev. | 4.7 | 6.3 | 8.1 | 9.0 | 9.8 | 10.3 | 10.9 | | Duration skewness | 1.70 | 1.37 | 0.98 | 0.68 | 0.45 | 0.26 | 0.12 | | Rolling windows | 136 | 128 | 123 | 118 | 115 | 113 | 113 | **Comparative statistics.** | Target wealth | 1.25 | 1.5 | 2 | 2.5 | 3 | 3.5 | 4 | |:---:|:---:|:---:|:---:|:---:|:---:|:---:|:---:| | Bill delay premium (%) | 58 | 58 | 61 | 65 | 64 | 61 | 61 | | Duration correlation (eq/bills) | −0.08 | −0.15 | −0.19 | −0.29 | −0.31 | −0.36 | −0.33 | The yearly real return correlation between equities and bills over the full period is **−0.129**. ### Five empirical findings 1. **Bill delay premium ≈ 60%.** Mean duration with bills is on average about 60% longer than with equities, a direct reflection of the historical equity risk premium. 2. **Duration risk increases with target wealth.** Both standard deviation and time variability of durations grow as the wealth target grows, paralleling the well-known fact that final-wealth risk grows with horizon. 3. **Bills are riskier than equities in the duration domain.** Bills dominate equities in *neither* mean *nor* standard deviation of duration. This breaks the classical mean–variance trade-off: there is no compensation for higher duration risk in the form of shorter expected duration when holding bills. 4. **Duration risk is positively skewed.** Duration distributions have long, fat right tails — durations that are unusually long but rarely unusually short. Skewness is uniformly larger for bills than for equities, strengthening Finding 3. 5. **Duration correlations between equity and bills are negative.** A long horizon for one asset tends to coincide with a short horizon for the other, with the (absolute) correlation increasing in target wealth — implying significant horizon-risk diversification gains, especially for high-target investors. --- ## 3. Mean–variance approach in the duration domain ### 3.1 Setup Investors care only about expected duration $E(t)$ and duration standard deviation $\sigma(t)$, with utility $U(E(t), \sigma(t))$, $U_1 > 0$, $U_2 < 0$. With initial wealth $w = 1$, target wealth $W$, and equity share $\alpha$, the cumulative portfolio return is $$ R_{0\to t}(\alpha) = \prod_{s=0}^{t-1} \big[\alpha R^r_s + (1-\alpha) R^f_s\big] $$ The probability of first reaching $W$ at date $t$ given $\alpha$ is $\pi(t,\alpha)$. Investors solve $$ \max_\alpha\; U\!\left(E(t,\alpha),\, \sqrt{V(t,\alpha)}\right) $$ with $E(t,\alpha) = \sum_t \pi(t,\alpha)\, t$ and $V(t,\alpha) = \sum_t \pi(t,\alpha) (t - E(t,\alpha))^2$. ### 3.2 Calibration on US data Using the empirical joint distribution of yearly real returns (1871–2019) and varying $\alpha$ from 0 to 100% in 5% steps, the paper traces the duration mean–variance frontier (Figure 2 in the paper). **Doubling-wealth target (W = $2):** - The frontier is ellipse-shaped, as in classical Markowitz theory. - The **100% equity portfolio dominates the 100% bill portfolio** on both axes — a striking departure from the wealth-domain CAPM intuition. - The **minimum variance portfolio** holds **35% equities / 65% bills** (the abstract figure caption reads "55% of equity" but the table for the minimum-variance portfolio at W = $2 reports **38%** equities — see flagged inconsistency below). - Reducing equity from 100% to the minimum-variance allocation cuts duration standard deviation from **6.1 to 3.7** years. - The bold "efficient frontier" excludes the upper arc rejected by timing-risk-averse investors. Efficient portfolios contain at least **35% equities**. ### 3.3 Sensitivity to target wealth — minimum-variance portfolios (Table 2) | Target wealth ($1 invested) | 1.5 | 2 | 2.5 | 3 | 3.5 | 4 | |:---:|:---:|:---:|:---:|:---:|:---:|:---:| | Mean duration (years) | 8.5 | 12.9 | 16.8 | 19.3 | 21.9 | 24.1 | | Duration std. dev. | 3.2 | 3.7 | 3.9 | 3.9 | 4.1 | 4.0 | | Equity share (%) | 32 | 38 | 36 | 44 | 44 | 47 | A 166% increase in money goal (from $1.5 to $4) yields a 188% rise in mean duration but only a **25%** rise in duration standard deviation. This time-diversification effect — duration risk grows much more slowly than expected duration — explains why the optimal equity share *increases* with target wealth and, equivalently, **decreases as accumulated wealth approaches the target**. --- ## 4. Dynamic model with Generalized Expected Discounted Utility ### 4.1 Preferences Intertemporal utility over a (possibly random) consumption stream $C = (c(0), \ldots, c(\tau))$ is $$ U(C) = \sum_{t \in T} D(t)\, u(c(t)) $$ with $u$ strictly increasing on $[\underline c, \bar c] \subset \mathbb R_+$ and $D$ strictly decreasing from $T$ to $[0,1]$. Investors are GEDU maximisers: $$ \max\; E\big[\phi(U(C))\big] $$ with $\phi$ strictly increasing. When $\phi$ is affine, the model collapses to Expected Discounted Utility (EDU). The curvature of $\phi$ controls aversion to *intertemporal-utility* risk separately from the curvature of $u$ (consumption-smoothing). When the only source of risk is the *date* of consumption — not its level — preferences reduce to $E[\phi(D(t) u)]$ for fixed $u$. EDU implies a *preference* for date randomization under standard conditions on $D$ (decreasing and convex), contradicting the experimental evidence in Onay & Öncüler (2007) and Dejarnette et al. (2020) that subjects are timing-risk *averse*. ### 4.2 Investment problem With two assets, investors choose a sequence $(\alpha_0, \alpha_1, \ldots)$ to solve $$ \max_{\alpha_0}\; \sum_{t=0}^{\tau} \pi_t(\alpha_0, \alpha_1^\star, \ldots, \alpha_{t-1}^\star)\, \phi(D(t)u(1)) $$ subject to $R_{0 \to t}(\cdot)w \geq 1$ for the first time at $t$. Returns are assumed serially independent (a simplification — the paper notes that empirical countercyclicality of expected returns is a natural extension). Borrowing and short-selling are forbidden; transactions costs are ignored. The state variable is current wealth $x \in [w_0, 1)$. The optimal policy is a **timeless function** $\alpha(x)$ derived from the value function $$ V(x) = \max_\alpha \int_{\underline R}^{\bar R} V(R(\alpha)x)\, f(R(\alpha))\, dR(\alpha)\, \phi(D(t)u(1)) $$ The non-exponential generalised discount function makes the problem **time-inconsistent** in the Strotzian sense; the paper adopts the naïve consumer assumption (in line with much of the behavioural finance literature) rather than computing the Strotzian intra-personal equilibrium of Luttmer & Mariotti (2003) or the time-consistent mean-variance equilibrium of Basak & Chabakauri (2010). Existence and uniqueness of the optimum are not formally established but simulations show fast convergence to a unique solution. ### 4.3 Functional forms and calibration - Discount function: exponential, $D(t) = \beta^t$. - Risk aggregator: CRRA, $\phi(U) = U^{1-\gamma}/(1-\gamma)$ for $\gamma \neq 1$, $\phi(U) = \ln U$ for $\gamma = 1$. - Calibrated parameters: - $\beta = 0.953$, matching the average real annualized riskless rate of **4.88%** over 1871–2019. - $\gamma$ swept over a broad range from **1 to 31**. Restricting to $\gamma \geq 1$ corresponds to timing-risk aversion. $\gamma = 1$ (log) corresponds to *timing-risk neutrality*: the investor minimises expected duration regardless of dispersion. $\gamma > 1$ ($\phi$ more concave than log): timing-risk aversion. $\gamma < 1$: timing-risk seeking (excluded by the experimental evidence). The simulation discretizes wealth on a grid of 800 points over $[0.2, 1]$ with step 0.001 and the equity share over 1001 values from 0 to 1 (step 0.001). To handle the case where the last-period return overshoots the target, the consumption date is advanced proportionally to an "excess wealth ratio" $\varepsilon(\alpha) = (R(\alpha)x - 1)/(R(\alpha)x - x) \in [0,1)$, with $V(R(\alpha)x) = \beta^{-(1-\gamma)\varepsilon(\alpha)} V(1)$. Equity shares are reported only for $x \geq 0.25$ to avoid the lower-bound bias documented in the appendix. --- ## 5. Results ### 5.1 Distance-to-target glide path (Figure 4) For $\beta = 0.953$ and $\gamma$ ranging from 1 to 31, the optimal equity share is plotted as a function of $x$, the relative distance to target. - $\gamma = 1$ (timing-risk-neutral): equity share stays near **100%** over a broad wealth interval, dipping only as $x \to 1$. - $\gamma \in [2, 5]$: balanced portfolios with equity share starting near 100% for low $x$, declining to a long plateau in the **45–70%** range, then converging to zero near the target. - $\gamma = 6$: equity share around **20%** over most of the wealth range. - $\gamma \geq 11$: heavy bill dominance, equity share around **5–15%**. The shape is **non-linear with a broad plateau**: equity is rebalanced sharply downwards once at low $x$ (depending on $\gamma$), then held roughly constant until the close vicinity of the target, where it converges to zero. Convergence to zero near the target reflects the asymmetry of the payoff: the investor bears the downside but cannot benefit from upside beyond the goal. ### 5.2 Impatience (Figure 5) Fixing $\gamma = 11$ and varying $\beta$ from 0.900 to 0.999: - $\beta = 0.999$ (near-time-neutral): equity share near **100%** over most of the wealth range. - $\beta = 0.953$ (calibrated to historical riskless rate): plateau around **15%**. - $\beta = 0.900$: equity share around **5–10%**. > **Result.** More impatient investors invest *less* in equities. Two opposing channels operate. Impatience increases the demand for equities (shorter expected delays) but also increases aversion to duration risk (long recoveries are heavily discounted). The second effect dominates. ### 5.3 Implications for financial advice - Optimal portfolios condition on **remaining wealth gap**, not remaining time. - The glide path is **non-linear** with a wide plateau, contradicting the linear glide paths used by many target-date and target-wealth products. - Risk-profiling questionnaires should assess timing-risk tolerance and patience separately from financial-risk tolerance, since impatience is a key driver of timing-risk aversion. --- ## 6. Conclusion The paper makes four claims supported by US 1871–2019 data and a calibrated dynamic model. 1. Fixed-income assets are not safer than equities in the duration domain. Low yields amplify small return variations into large duration swings. 2. A CAPM-style mean–variance analysis shows substantial diversification gains from including a significant equity share — at least 35% on the efficient frontier. 3. Investors hold less equity not only because they are risk-averse but because they are impatient: timing-risk aversion is driven by the discount function, not by static risk aversion. 4. The optimal target-wealth glide path is non-linear: a broad plateau of equity share over intermediate wealths, with sharp de-risking only near the target. ### Limitations and research extensions The paper points to several extensions: - Adding richer asset menus (bonds, size/value-sorted equity portfolios, a risk-free rate); a market-portfolio existence and two-fund separation analysis is left for future work. - Allowing serially correlated returns and conditioning the equity share on return predictors such as the dividend–price ratio (Golez & Koudijs, 2018), which may forecast investment durations as well as returns. - Existence and uniqueness of the equilibrium policy are not formally proved (cf. Ekeland et al., 2012; Tan et al., 2021); current evidence is numerical. - Time-inconsistency is handled with the naïve consumer assumption rather than with a Strotzian intra-personal equilibrium. --- ## Acknowledgments The author thanks Sébastien Galanti, Christophe Hurlin, Louis Raffestin and seminar participants at the Laboratoire d'Économie d'Orléans for useful comments, and two anonymous Journal referees for suggesting valuable improvements. --- ## Main references Barberis, N. (2000). Investing for the long run when returns are predictable. *Journal of Finance* 55(1), 225–264. Basak, S., Chabakauri, G. (2010). Dynamic mean-variance asset allocation. *Review of Financial Studies* 23, 2970–3016. Blanchet-Scalliet, C., El Karoui, N., Jeanblanc, M., Martellini, L. (2008). Optimal investment and consumption decisions when time horizon is uncertain. *Journal of Mathematical Economics* 44(11), 1100–1113. Bodie, Z., Merton, R. C., Samuelson, W. (1991). Labor supply flexibility and portfolio choice in a life cycle model. *Journal of Economic Dynamics and Control* 16, 427–449. Campbell, J. Y., Viceira, L. M. (2002). *Strategic Asset Allocation: Portfolio Choice for Long-Term Investors*. Oxford University Press. Dejarnette, P., Dillenberger, D., Gottlieb, D., Ortoleva, P. (2020). Time lotteries and stochastic impatience. *Econometrica* 88(2), 619–656. Dillenberger, D., Gottlieb, D., Ortoleva, P. (2020). Stochastic impatience and the separation of time and risk preferences. Princeton University working paper. Ebert, S. (2021). Prudent discounting: Experimental evidence on higher-order time risk preferences. *International Economic Review* 62(4), 1489–1511. Epstein, L., Zin, S. (1989). Substitution, risk aversion, and the temporal behavior of consumption and asset returns: A theoretical framework. *Econometrica* 57, 937–969. Karatzas, I., Wang, H. (2001). Utility maximization with discretionary stopping. *SIAM Journal on Control & Optimization* 39, 306–329. Liu, H., Loewenstein, M. (2002). Optimal portfolio selection with transaction costs and finite horizons. *Review of Financial Studies* 15(3), 805–835. Markowitz, H. M. (1952). Portfolio selection. *Journal of Finance* 7(1), 77–91. Martellini, L., Urošević, B. (2006). Static mean-variance analysis with uncertain time horizon. *Management Science* 52(6), 955–964. Merton, R. C. (1969). Lifetime portfolio selection under uncertainty: The continuous-time case. *Review of Economics and Statistics* 51, 247–257. Onay, S., Öncüler, A. (2007). Intertemporal choice under timing risk: An experimental approach. *Journal of Risk and Uncertainty* 34, 99–121. Samuelson, P. A. (1969). Lifetime portfolio selection by dynamic stochastic programming. *Review of Economics and Statistics* 51, 239–246. Viceira, L. M. (2001). Optimal portfolio choice for long-horizon investors with nontradable labor income. *Journal of Finance* 56, 433–470. *The full reference list appears in the PDF.*