Structural analysis of reverse mortgages in Taiwan

Taiwan has rapidly transitioned into an aging society, with implications for older adults’ financial well-being amid subreplacement fertility and potential lack of familial support. Although medical advances extend life expectancy, retirees face employment and income challenges and may rely on insufficient pensions. Taiwan’s high homeownership rate (88.14% in 2008) leaves many older adults asset-rich but cash-poor, making home equity an important potential source of retirement income. In response, Taiwan piloted reverse mortgage (RM) programs in 2013 for adults over 65 with real estate (initially with stringent eligibility limits for low- and middle-low-income households, which limited early uptake). As thresholds eased and promotion increased, awareness grew; by July 2016 there were 809 commercial RM cases with an average applicant age of 73.32, and six banks offering programs. However, RM understanding is challenging due to multiple interacting factors (age, loan terms, rates, payment methods) and information asymmetries between debtors and creditors, risking misjudgments in negotiations. This study aims to address these gaps by applying established RM pricing frameworks (HECM pricing model) and structural analysis to evaluate lump-sum loanable amounts, rental income, remaining house value, creditor insurance protection, premium expenses, and creditor profits. Simulations and sensitivity analyses across occupations, ages, and sexes examine how parameters affect income replacement ratios and institutional risks, thereby providing guidance for both potential debtors and creditors in Taiwan.

Reverse mortgages (RMs) allow older homeowners to convert housing wealth into income while deferring repayment until death or exit. In the U.S., development accelerated after the 1989 HUD-backed Home Equity Conversion Mortgage (HECM) program, which insures lenders (nonrecourse) against loan balances exceeding property values. Research shows RMs can enhance retirees’ income and quality of life. Szymanoski (1994) introduced HECM pricing to derive principal limit factors and quantify insurance’s role in eliminating lender loss risk. Taiwan-focused studies adapted HECM-based models with local mortality, housing values, and interest rates (e.g., Wang et al., 2011). Risk to insurers from crossover (loan surpassing home value) has been analyzed using lattice and option-based methods (Huang et al., 2011; Li et al., 2010). Lee and Lo (2016) proposed a structural decomposition of house value into six components: loanable amount, rental income, remaining house value, creditor insurance protection, premium expense, and creditor profit—useful for assessing program profitability and risks. International literature continues to assess RM feasibility and risk across countries (China, U.S., Australia, Colombia, Korea), including securitization proposals and regulatory capital frameworks (Di Lorenzo et al., 2022; De la Fuente et al., 2023; Fuente et al., 2020; 2021). A consistent theme is that prior work emphasizes the supplier (lender/insurer) perspective, while information asymmetry persists regarding borrower-facing loan limits and benefits. This study responds by adopting Lee and Lo’s structural analysis to clarify program characteristics for debtors and risk implications for creditors.

Study design: The analysis follows Lee and Lo (2016) using a lump-sum RM to evaluate retirees across occupations (educator, public servant, military, laborer). It computes income replacement ratios post-retirement and decomposes the RM structure into lump-sum loan amount, rental income, remaining house value, creditor insurance protection, premium expenses, and creditor profit. Mortality is estimated via the Lee-Carter model calibrated to Taiwan data; house prices use the HPI with distributional tests (AD and KS). Sensitivity analyses vary age, sex, house value, and loan rates.

Dynamic house value model: House prices H(t) follow geometric Brownian motion. Under the physical measure P: dH(t)/H(t) = (μ − δ)dt + σ dW_P(t), where μ is expected return, δ rental (maintenance) yield, σ volatility. Under the risk-neutral measure Q: dH(t)/H(t) = (r − δ)dt + σ dW_Q(t). Rental rate δ is set to 1.5% based on market yields; house price volatility σ_H = 10%.

Mortality model: The Lee-Carter model ln m_{x,t} = α_x + β_x k_t + ε_{x,t} is used to estimate age-specific mortality over time, subject to identifiability constraints, and to derive survival probabilities p_x for ages 60–110. Survival and death probabilities by year feed into expected cash flow and option value calculations.

Lump-sum RM design: Risk-free rate r, loan rate r_ℓ = r + π. Initial house value H(0). The debtor receives a net lump-sum BAL(0) after fees. The debtor pays mortgage insurance premiums but accrues them onto the balance. Initial application fee/premium rate π₀ applies to H(0) at t=0; annual renewal premium π_m applies to the outstanding balance BAL(t). Loan balance evolves by compounding interest and adding premiums: BAL(1) = [π₀ H(0) + BAL(0)] e^{r}, and recursively BAL(t) = BAL(t−1) (1 + π_m) e^{r}, yielding BAL(t) = [π₀ H(0) + BAL(0)] (1 + π_m)^{t−1} e^{rt}.

Rental income: Under Q, H(t) = H(0) exp[(r − δ − 0.5σ²)t + σ W_Q(t)]. The present value of rental income VR(0) is the discounted expected rental cash flows weighted by survival/death probabilities until lease expiry T.

Remaining house value: Using real option logic, the expected value to the estate on termination is the expectation of max(H(t) − BAL(t), 0) weighted by mortality timing and discounted. Closed forms via Black–Scholes components under lognormality are employed (per Lee and Lo, 2016).

Insurance protection (put) and premium expense: From the creditor’s perspective, each year’s loss when BAL(t) > H(t) is a put option on the house, valued and aggregated over death timing. Premium expense includes π₀ H(0) at origination and expected discounted stream of renewal premiums π_m on BAL(t) while the borrower survives.

Principal calibration: The lump-sum BAL(0) is solved such that expected premium income equals expected insurance cost (Vp(0) = V₁(0)), using nonlinear root-finding. BAL(0) is also converted to a level lifetime annuity TP = BAL(0) / Σ_t (1+r)^{-t} up to maximum age w=110 to compute income replacement ratios (annuity divided by pre-retirement monthly salary).

House Price Index (HPI) and tests: HPI (base year 2021=100) is used to calibrate H(0) and volatility. Quarterly HPI returns (2016Q1–2023Q3) are annualized; volatility estimated at 0.0709. AD and KS tests assess normality of HPI change rates and lognormality of the index. A two-sample KS compares empirical HPI paths and GBM-simulated paths (10,000 simulations) for goodness-of-fit.

Parameters: Ages 60–95 (5-year steps); sexes male/female; risk-free rate r=1.04%; loan rates 1.86%, 2.0775%, 2.295%, 2.5125%, 2.73%; house values NT$6m, NT$8m, NT$10m; δ=1.5%; σ_H=10%; π₀=2%; π_m=1.25%; maximum age w=110. Occupations: educator, public servant, military, laborer.

• Distributional validation of HPI and GBM:

  • Anderson–Darling (AD) test for HPI change rate (2016Q1–2023Q3): statistic 0.6436, p=0.0844; fail to reject normality at 5%.
  • AD test for HPI level lognormality: statistic 1.3625, p=0.0013; reject lognormality of the index level.
  • Kolmogorov–Smirnov (KS) test for HPI change rate: statistic 0.1079, critical value 0.2379, p=0.8257; fail to reject normality.
  • Two-sample KS on GBM-simulated change rates vs observed: average p-value across 10,000 simulations ≈ 0.783; simulated change rates consistent with normality, supporting GBM for returns.

• Maximum loanable amounts and structural decomposition:

  • Loanable amounts increase with borrower age due to shorter expected durations, while present value of rental income declines with age.
  • Example (model-based, NT$1,000,000 house, age 70): lump-sum NT$509,710; expected rental income NT$223,510; remaining house value NT$189,150; creditor profit NT$77,630; premium paid NT$143,210 with equal-valued insurance protection (insurer break-even by construction).
  • For identical settings, women have higher maximum loanable amounts than men, reflecting mortality differences; however, creditor profit is lower when lending to women due to longer life expectancy.
  • Using HPI-based initial prices (e.g., aligning to 2016 levels) reduces overall amounts compared to baseline Table 5 assumptions; creditor profit increases while RM insurance cost decreases, consistent with relatively lower indexed house prices (2021=100 benchmark higher than 2016 level by about 28%).

• Sensitivity to loan rates:

  • Higher loan rates reduce lump-sum loanable amounts (Table 8) across ages/sexes.
  • Consequently, income replacement ratios decline as loan rates rise (Tables 10–11).

• Income replacement ratios by occupation and demographics:

  • Given NT$8m loan and 1.86% rate, older applicants have higher income replacement ratios than younger ones.
  • Laborers exhibit the highest replacement ratios across ages because their monthly pensions are lower than those of educators, public servants, and military personnel; thus, the same annuity constitutes a larger share of pre-retirement income (Table 9).
  • By sex, for identical parameters, men show higher income replacement ratios than women across ages (Fig. 3; Table 10), even though women’s loanable amounts are higher; interest rate increases narrow the sex gap in replacement ratios.

• Practical implications:

  • Applying after age 75 is advantageous for debtors, yielding higher annuities/replacement ratios; creditor profits tend to decrease with older borrowers.
  • Debtors with stronger credit profiles have advantages in obtaining RM on favorable terms.

The study addresses information asymmetries in Taiwan’s RM market by dissecting RM contracts into borrower- and lender-relevant components. Validating GBM for HPI change rates supports the modeling framework used to price loan limits, premiums, and insurance protection. Findings that loanable amounts increase with age (despite falling rental PV) and that women qualify for higher loan amounts (but may produce lower creditor profit) explain observed program dynamics and help align expectations in negotiations. Sensitivity analyses reveal the strong role of interest rates: higher rates compress both lump-sum values and income replacement ratios, underscoring the importance of rate shopping and timing. Occupational comparisons show laborers derive the greatest relative income support from RM, highlighting RM’s potential social welfare value for lower-pension groups. From the creditor perspective, structural decomposition clarifies profitability drivers—remaining house value, premium income, and insured crossover risk—and how indexed house prices affect insurance costs and margins. These insights inform product design, underwriting, and risk management. Policy-wise, recommending independent third-party counseling for borrowers can reduce misjudgments and improve program suitability, directly responding to the study’s initial problem of information imbalance.

This study applies HECM-based pricing and structural analysis to Taiwan’s reverse mortgage market, simultaneously evaluating debtor benefits (lump-sum, annuity income, replacement ratios) and creditor outcomes (insurance protection, premiums, profits). Contributions include: (1) a unified framework bridging borrower and lender perspectives; (2) empirical calibration to Taiwan mortality and housing data with distributional validation; and (3) practical guidance by age, sex, occupation, house value, and loan rate. Key conclusions: older applicants (especially 75+) and lower interest rates materially improve debtor outcomes; women qualify for higher loan amounts, though income replacement ratios are higher for men under identical settings; laborers gain the most in relative terms due to lower pensions. Policy recommendations include expanding maximum loan amounts, reducing interest rates, widening eligibility, and implementing risk-sharing mechanisms to protect public finances, alongside mandatory independent counseling to mitigate information asymmetry. Future research should refine house price dynamics and contract risk modeling with richer stochastic processes and data, and evaluate regulatory and securitization designs in Taiwan’s context.

Key limitations stem from data and modeling simplifications. Parameter assumptions (e.g., house price volatility, rental yield) and the use of GBM for house price dynamics, while supported for returns, may not capture abrupt shifts, regime changes, or macroeconomic covariates. Limited availability of granular Taiwanese RM data required assumptions on contract termination timing (end-of-year mortality timing) and premium accrual. These factors may affect the precision and generalizability of results beyond the studied settings. The authors suggest future work incorporate richer time-series models (e.g., ARIMA/GARCH), stochastic differential frameworks with macro factors, and broader datasets to better capture market fluctuations and behavioral responses.