Gender Gap in Transition Probabilities and Roll-Up Sensitivity

Response to Reviewer Comment

1 Executive Summary

We quantify the gender gap in healthy-state persistence and its impact on the GLWB-LTC contract value sensitivity to the roll-up rate.

Transition probability gap. Starting from age 50, we compute the cumulative probability of remaining in the healthy state as the product of one-year healthy-to-healthy transition probabilities \(p^{HH}_x\). A 50-year-old woman has a cumulative probability of reaching age 120 while still healthy of 0.31%, compared to only 0.0024% for a man — a gap of roughly two orders of magnitude (the female probability is approximately 130 times the male one). The gap emerges gradually: at age 65 the female cumulative probability exceeds the male one by 3.2%, but by age 80 the difference reaches 10%, by age 100 it exceeds 79%, and it grows exponentially thereafter as male healthy-state mortality accelerates faster.

Roll-up sensitivity gap. Using the GLWB-LTC pricing engine, we sweep the roll-up rate from 0% to 7% and measure the percentage increase in the contract value \(V_0\) for each gender. The results show:

• Male: \(V_0\) rises from 94.84 to 109.32, a +15.27% increase.
• Female: \(V_0\) rises from 94.87 to 118.09, a +24.48% increase.
• Sensitivity gap: 9.21 percentage points — females exhibit roughly 60% higher relative sensitivity to roll-up variations (\(24.48\% / 15.27\% \approx 1.60\)).

Conclusion. The economic mechanism is straightforward: since women are more likely to remain in the healthy state, they collect the guaranteed withdrawal \(w \cdot A_t\) (where \(A_t = P \cdot e^{\rho t}\) is the roll-up-compounded benefit base) over a longer horizon. A marginal increase in the roll-up rate \(\rho\) therefore compounds over more years for females, producing a measurably larger impact on the contract value. This confirms that the greater female propensity to remain healthy directly translates into heightened sensitivity of the GLWB-LTC fair value to the roll-up rate.

2 Part 1: Transition Probability Gap

2.1 Methodology

For each gender \(g \in \{\text{Male}, \text{Female}\}\), we compute the cumulative probability that a 50-year-old individual remains in the healthy state up to age \(50 + k\):

\[ {}_{k}p^{HH}_{50}(g) = \prod_{i=0}^{k-1} p^{HH}_{50+i}(g), \quad k = 1, \ldots, 70, \]

where \(p^{HH}_{x}(g)\) is the one-year probability of staying healthy at age \(x\) for gender \(g\).

The relative gap at age \(50 + k\) is defined as:

\[ \text{Gap}(k) = \frac{{}_{k}p^{HH}_{50}(\text{Female}) - {}_{k}p^{HH}_{50}(\text{Male})}{{}_{k}p^{HH}_{50}(\text{Male})} \times 100. \]

2.2 Results

Cumulative probability of remaining healthy from age 50, by gender.

	Age	Cum. P(H) Male	Cum. P(H) Female	Relative Gap (%)
5	55	0.983492	0.989889	0.65
10	60	0.961053	0.977261	1.69
15	65	0.931113	0.961303	3.24
20	70	0.889205	0.936047	5.27
25	75	0.826329	0.889915	7.70
30	80	0.727809	0.801630	10.14
35	85	0.581836	0.656624	12.85
40	90	0.393285	0.466599	18.64
45	95	0.221876	0.300262	35.33
50	100	0.102399	0.183089	78.80
55	105	0.034541	0.100242	190.21
60	110	0.007162	0.045411	534.02
65	115	0.000710	0.015112	2029.15
70	120	0.000024	0.003127	13041.59

Cumulative probability of remaining in the healthy state from age 50.

Relative gap in cumulative healthy survival probability (Female vs Male).

3 Part 2: Roll-Up Sensitivity

3.1 Methodology

We price the GLWB-LTC contract for each gender using the GLWB_LTC_Engine, sweeping the roll-up rate \(\rho\) from 0% to 7% in steps of 0.5%. The base contract value at \(\rho = 0\) is denoted \(V_0^{(g)}(0)\). The percentage increase in contract value is:

\[ \Delta V^{(g)}(\rho) = \frac{V_0^{(g)}(\rho) - V_0^{(g)}(0)}{V_0^{(g)}(0)} \times 100. \]

The sensitivity gap is defined as:

\[ \text{SensGap}(\rho) = \Delta V^{(\text{Female})}(\rho) - \Delta V^{(\text{Male})}(\rho). \]

3.2 Results

Contract value and roll-up sensitivity by gender.

Roll-up (%)	V₀ Male	V₀ Female	% Incr. Male	% Incr. Female	Sens. Gap (pp)
0.0	94.84	94.87	0.00	0.00	0.00
0.5	94.86	94.91	0.02	0.04	0.02
1.0	94.89	94.98	0.05	0.12	0.07
1.5	94.94	95.11	0.10	0.25	0.15
2.0	95.04	95.32	0.20	0.48	0.27
2.5	95.20	95.65	0.38	0.83	0.45
3.0	95.47	96.19	0.66	1.39	0.73
3.5	95.89	96.97	1.10	2.21	1.11
4.0	96.53	98.05	1.78	3.35	1.57
4.5	97.43	99.48	2.73	4.87	2.14
5.0	98.67	101.37	4.04	6.85	2.81
5.5	100.33	103.85	5.79	9.46	3.68
6.0	102.49	107.15	8.07	12.95	4.88
6.5	105.36	111.82	11.09	17.87	6.78
7.0	109.32	118.09	15.27	24.48	9.21

Contract value V₀ as a function of the roll-up rate.

Percentage increase in V₀ relative to the zero roll-up baseline.

Sensitivity gap (Female minus Male percentage increase) in percentage points.