Reference Dependence in Reaching for Yield

Reference Dependence in Reaching for Yield

The reference-dependence mechanism for reaching for yield rests on Prospect-Theory machinery: investors form a reference rate (the rate they expect to earn or have historically earned), and evaluate the safe asset relative to this reference. When the safe rate falls below the reference, the safe asset is experienced as a loss, and the investor’s loss-aversion makes her unwilling to accept this certain loss. She substitutes toward the risky asset, where the variability creates some chance of avoiding the loss.

The formal model. Lian, Ma, and Wang ($2019$) write the investor’s utility as a piecewise-linear loss-averse function over the portfolio return $r_p$ relative to a reference rate $r_{\text{ref}}$:
$u(w(1+r_p))=\{\begin{array}{ll}w(r_p-r_{\text{ref}})& r_p\ge r_{\text{ref}}\\ -\lambda w(r_{\text{ref}}-r_p)& r_p<r_{\text{ref}}\end{array}$
where $\lambda >1$ captures loss aversion (typically $\lambda \approx 2$). The optimal allocation ${\varphi }^{\ast }$ to the risky asset depends on $(r_f,r_{\text{ref}})$:

• When $r_f\ge r_{\text{ref}}$: the safe asset already exceeds the reference, so the investor has no reason to take risk; ${\varphi }^{\ast }$ is small and approximately independent of $r_f$.
• When $r_f<r_{\text{ref}}$: the safe asset is below the reference (a loss); the investor substitutes toward the risky asset, where the variability creates some chance of beating the reference. As $r_f$ falls further below $r_{\text{ref}}$, the certain-loss size grows and the substitution intensifies.

Two predicted patterns.

1. Non-linearity. The reaching-for-yield effect is small or zero when $r_f$ is well above $r_{\text{ref}}$ (the safe asset is fine; no need to take risk). The effect grows as $r_f$ falls below $r_{\text{ref}}$, becomes maximal at very low $r_f$, and may even strengthen as $r_f$ approaches zero. Empirically: Lian-Ma-Wang’s Test 1 finds substantial non-linearity in the rate-allocation relationship, with the effect concentrated at low rates.
2. History dependence. The reference rate is shaped by past experience. A subject who experienced a high rate in a previous round adjusts her reference rate upward; when she then faces a low rate, her certain-loss perception is sharper and her reaching-for-yield is stronger. Empirically: Lian-Ma-Wang’s Test 2 finds that subjects exposed to high then low rates allocate $66\%$ to the risky asset in the low condition, versus $48\%$ for subjects exposed to low then high then low again. The history-dependence effect is large and confirms the reference-dependence interpretation.

Where the reference rate comes from. The Lian-Ma-Wang paper does not commit to a single source; they discuss several possibilities supported by separate literatures:

• Past experience — Kahneman-Miller ($1986$); Simonsohn-Loewenstein ($2006$); Malmendier-Nagel ($2011$); Bordalo-Gennaioli-Shleifer ($2017$); DellaVigna et al. ($2017$). Investors anchor on rates they have observed in the past.
• Status quo — the rate the investor was earning before the current decision.
• Forward-looking rational expectations — a fully rational expectation of future rates.

The history-dependence test (Test 2) is most consistent with past-experience as the source. The “rational expectations” version of the reference rate would not show the strong history dependence the data exhibit.

Intuition

The reference-dependence mechanism converts the standard mean-variance investor — who cares about absolute return levels — into a Prospect-Theory investor who cares about gain-loss outcomes relative to a reference. The reference rate plays the role that the reference wealth plays in the original Kahneman-Tversky formulation: outcomes above the reference register as gains, outcomes below as losses, and losses are weighted more heavily than gains by the loss-aversion factor $\lambda$.

The applied implication for portfolio choice runs through a single observation: when the safe rate falls below the reference rate, the safe asset guarantees a loss. Loss aversion makes the investor strongly averse to this guaranteed loss; she therefore prefers the risky asset’s chance of avoiding the loss, even though the risky asset’s variability would normally be unattractive. The mechanism is reminiscent of the risk-seeking-in-the-loss-domain property of the Prospect-Theory value function: rather than accept a sure loss, the agent prefers a gamble that might (or might not) avoid the loss entirely.

The history-dependence prediction is the cleanest signature. Standard expected-utility models with state-independent preferences predict that today’s allocation should depend only on today’s $(r_f,\mu ,\sigma )$ — past rates should not matter. Reference dependence with history-shaped references predicts the opposite: today’s allocation depends on the path of rates the investor has experienced, because that path shapes the reference. Lian-Ma-Wang’s Test 2 confirms this prediction sharply, providing strong support for the reference-dependence mechanism over rational alternatives.

Properties

(Loss-Averse Utility Over the Portfolio Return) Reference is the rate the investor anchors on.

$u(w(1+r_p))=\{\begin{array}{ll}w(r_p-r_{\text{ref}})& r_p\ge r_{\text{ref}}\\ -\lambda w(r_{\text{ref}}-r_p)& r_p<r_{\text{ref}}\end{array}$
The piecewise-linear function captures the two key Prospect-Theory features: loss aversion (the slope $\lambda >1$ on the loss side exceeds the slope on the gain side, so equivalent gains and losses are not equally valued) and kink at the reference (the function is non-differentiable at $r_p=r_{\text{ref}}$). The kink is essential for generating the corner-solution behavior at low rates.

(Prediction Direction) When $r_f<r_{\text{ref}}$, ${\varphi }^{\ast }$ rises as $r_f$ falls.

The model predicts $\partial {\varphi }^{\ast }/\partial r_f<0$ in the regime $r_f<r_{\text{ref}}$. Intuitively, lower $r_f$ means the safe asset’s certain-loss is larger, the investor’s loss-aversion bites harder, and she substitutes more toward the risky asset to escape the loss. The prediction is the opposite of the standard mean-variance benchmark prediction (independence) and of the standard DARA refinement (positive elasticity).

(Non-Linearity) Effect concentrated at low $r_f$.

When $r_f$ is well above $r_{\text{ref}}$, the safe asset already exceeds the reference and there is no loss; the investor’s behavior is approximately the mean-variance baseline. When $r_f$ falls below $r_{\text{ref}}$, the loss-aversion mechanism kicks in. The transition produces a kink in the rate-allocation relationship at $r_f=r_{\text{ref}}$. Lian-Ma-Wang’s Test 1 finds substantial non-linearity in the empirical relationship, with allocations rising sharply only as $r_f$ falls below a threshold (consistent with $r_{\text{ref}}\approx 3$–$5\%$ in the U.S. context).

(History Dependence — Test 2 Result) Past-rate exposure shapes today’s allocation.

Lian-Ma-Wang’s Test 2 randomly assigns subjects to one of two sequences:

• Group G1: High rate condition first ($5$–$10\%$), then low rate condition ($1$–$6\%$).
• Group G2: Low rate condition first, then high rate condition, then low rate condition again.

In the low-rate condition, G1 subjects allocate $66.33\%$ to the risky asset; G2 subjects allocate $55.75\%$ — a $10.58$ pp difference ($t=3.44$). The history-dependence effect is large and statistically clean. The interpretation: G1’s reference rate has been pulled upward by their high-rate experience; when they then face the low rate, their certain-loss perception is sharper. G2’s reference rate is closer to the low rate. The pattern is what reference-dependence predicts and what no rational-expectations alternative can match.

(Source of Reference) Past experience is the leading candidate.

Lian-Ma-Wang discuss several possible sources for the reference rate. Past-experience-based references — drawing on Kahneman-Miller ($1986$); Simonsohn-Loewenstein ($2006$); Malmendier-Nagel ($2011$); Bordalo-Gennaioli-Shleifer ($2017$); DellaVigna-Lindner-Reizer-Schmieder ($2017$) — best fit the history-dependence evidence. A status-quo-based reference rate is harder to falsify. Forward-looking rational-expectations references would not produce the history dependence Test 2 documents.

(Implication for Reference Rate Updates) Persistent low rates should eventually weaken reaching-for-yield.

A long-run prediction: if rates remain persistently low, the reference rate eventually adjusts downward through experience, and the certain-loss perception fades. The reaching-for-yield effect should attenuate over time as the new low-rate environment is incorporated into the reference. This is harder to test directly because it requires a long sample of stable low rates; partial evidence from the post-$2008$ U.S. environment (where rates remained near zero for nearly a decade) suggests that retail-investor reaching-for-yield behavior did persist substantially through the period rather than fully attenuating.

(Robustness to Alternative Risk-Preference Specifications) DARA reverses the prediction; reference dependence is needed.

A natural rational refinement of the mean-variance baseline is decreasing absolute risk aversion (DARA), under which the investor’s risk aversion falls as her wealth rises. Under DARA, lower $r_f$ makes the investor effectively poorer (her future wealth from $r_f$ is lower), which makes her more risk-averse and pushes her away from the risky asset. The data show the opposite. The reaching-for-yield pattern therefore cannot be rationalized within standard expected-utility plus DARA; reference dependence (or salience, or both) is required.

Examples

A retiree facing a sub-historical-rate Treasury yield

A retired investor has historically earned $4$–$5\%$ on safe Treasury holdings throughout her working life. Her reference rate is anchored at approximately $4.5\%$. The current $10$-year Treasury yield is $1.5\%$. The reference-dependence mechanism predicts that she experiences the safe asset as a $3$ pp certain loss relative to her reference, and substitutes toward the risky asset (high-yield bonds, dividend stocks, REITs) to escape this loss. The substitution is particularly strong because she is loss-averse over the whole portfolio. Empirically, retiree allocations to fixed income fell substantially during the post-$2008$ low-rate period and shifted toward higher-risk dividend-paying equities and high-yield bonds — exactly the pattern the model predicts.

Walk-through of the Test 2 history-dependence design

Subjects in G1 do two rounds:

• Round 1 (high condition): safe $5\%$, risky $10\%$ avg / $18\%$ vol. They allocate, on average, $48\%$ to the risky asset (consistent with mean-variance baseline).
• Round 2 (low condition): safe $1\%$, risky $6\%$ avg / $18\%$ vol. They now allocate $66\%$ to the risky asset.

Subjects in G2 do three rounds:

• Round 1 (low condition): safe $1\%$, risky $6\%$. Allocate $56\%$ to risky.
• Round 2 (high condition): safe $5\%$, risky $10\%$. Allocate $47\%$ to risky.
• Round 3 (low condition again): safe $1\%$, risky $6\%$. Allocate $56\%$ to risky.

The G1 vs. G2 comparison in the low condition: G1 allocates $66\%$, G2 allocates $56\%$ — a $10$ pp difference. The interpretation: G1 has just experienced the high rate ($5\%$) which has pulled its reference up; the subsequent low rate ($1\%$) feels much further below the reference and triggers stronger reaching-for-yield. G2’s reference has been established by repeated exposure to the low rate, so the same low rate triggers less aversion. The result cannot be rationalized within any rational benchmark; it is a clean empirical signature of reference dependence.

Why an MBA-trained investor still exhibits the bias

Lian-Ma-Wang test their results on Harvard Business School MBA students with real-money stakes (\200$ payouts to randomly selected subjects). The HBS sample exhibits the same reaching-for-yield pattern as the Mechanical Turk sample, with magnitudes if anything slightly larger. The financial sophistication of the population does not eliminate the bias. The interpretation: reference dependence is a structural cognitive feature, not a sophistication-correlated bias that training can eliminate. This makes the result particularly relevant for policy because the macro-implications of reaching-for-yield apply to sophisticated as well as unsophisticated investors.

See Also

• 8.15. Reaching for Yield — parent hub.
• 8.15.2. Salience and Proportional Thinking in Reaching for Yield — sibling: alternative behavioral mechanism that operates jointly with reference dependence.
• 8.5. Prospect Theory — the loss-aversion framework that the mechanism rests on.
• 8.5.1. Prospect Theory — the underlying value-function machinery.
• 8.6.8. Experience Effects — the closely related theory of how past experience shapes investor expectations and references.
• 8.10.6. Hyperbolic Discounting — closely related preference bias affecting consumption-savings decisions.