Tutorials › Exam ALTAM - Advanced Long-Term Actuarial Mathematics › Decomposition of Total Pension EPV by Exit Cause

Home  /  Tutorials  /  Exam ALTAM / Valuation Under Competing Risks

Exam ALTAM · Valuation Under Competing Risks · Tutorial 346 of 999

Decomposition of Total Pension EPV by Exit Cause

Learn to value each mutually exclusive exit cause separately and combine the components into a total pension EPV without confusing cause probabilities with conditional shares.

Intermediate 10 min read

What You'll Learn

  • Define annual cause-specific exit probabilities and the probability of remaining active.
  • Derive the probability of exiting for a particular cause in a given year.
  • Decompose total EPV into retirement, withdrawal, disability, and death components.
  • Handle changing probabilities and benefit amounts by year.
  • Identify the error caused by treating exit probabilities as shares conditional on an exit.

Why Decompose a Pension EPV by Exit Cause?

A pension member may leave active service through retirement, withdrawal, disability, or death. Each exit can trigger a different benefit, and the probability of each exit depends on the member having remained active up to that point. Valuing the total benefit therefore requires tracking both the cause of exit and the time at which it occurs.

A cause-by-cause decomposition makes the calculation transparent: first value the retirement benefits, then the withdrawal benefits, disability benefits, and death benefits, each using its own probabilities and benefit amounts. The components can then be added because the member’s first exit from active service can occur by only one cause.

Definition: In a discrete annual competing-exit model, let \(q_{j,k}\) be the probability that an active member exits during year \(k\) for cause \(j\), conditional on being active at the start of that year. The causes \(j\) are mutually exclusive. The probability of remaining active throughout year \(k\), conditional on being active at its start, is \(p_k=1-\sum_j q_{j,k}\). Let \(B_{j,k}\) be the value, at the specified valuation date, of the benefit package triggered by cause \(j\) in year \(k\), and let \(v=1/(1+i)\) be the annual discount factor.

The definition uses cause-specific probabilities on a common basis: each \(q_{j,k}\) is conditional on being active at the start of year \(k\), not conditional on an exit having occurred. In the calculations below, benefits are valued at the end of the year of exit. The benefit amount \(B_{j,k}\) is the value at that payment date; if a benefit is itself a pension stream, its value can represent the value of that stream on the stated basis.

Deriving the Cause-Specific Decomposition

Let \(A_{k-1}\) denote the event that the member is still active at the start of year \(k\). The probability of this event is the product of the annual active probabilities in all preceding years. Conditional on being active at the start of year \(k\), the probability of exiting for cause \(j\) in that year is \(q_{j,k}\). Thus the probability of an exit for cause \(j\) in year \(k\) is the product of those two probabilities.

$$ P(\text{cause }j\text{ exit in year }k) = \left(\prod_{r=1}^{k-1}p_r\right)q_{j,k}. $$

If the cause-\(j\) benefit in year \(k\) is \(B_{j,k}\), its discounted contribution to expected present value is that event probability times the benefit, discounted for \(k\) years. Summing those contributions gives the cause-specific EPV. Since the first-exit events for different causes are mutually exclusive, the total expected present value is the sum of the cause-specific EPVs.

$$ \operatorname{EPV}_j = \sum_{k=1}^{n} v^k \left(\prod_{r=1}^{k-1}p_r\right) q_{j,k}B_{j,k}, \qquad \operatorname{EPV}_{\mathrm{total}} = \sum_j\operatorname{EPV}_j. $$

For year 1, the empty product is 1, so no prior-year survival factor is applied. For later years, the product is essential: a member who has already exited cannot generate another first-exit benefit from active service. The formula applies to a discrete model with mutually exclusive first exits and benefits valued at the stated payment dates. The horizon \(n\) may be finite or infinite; for an infinite horizon, the discounted expected contributions must converge.

1
Check the probability basis.
Confirm that each cause probability is conditional on being active at the start of the year, and that the cause probabilities plus the active probability sum to 1.
2
Find exposure to each year.
Multiply the active probabilities from earlier years to obtain the probability of reaching the start of the year still active.
3
Value each cause separately.
For every year and cause, multiply survival to that year, the cause-specific exit probability, the benefit value, and the appropriate discount factor.
4
Sum across causes.
Add the cause-specific EPVs to obtain the total expected present value.

Worked Valuations

Worked Example: Constant Annual Exit Probabilities

A hypothetical pension plan values benefits over two years. At the start of each year, an active member has annual exit probabilities \(q_R=0.10\) for retirement, \(q_W=0.08\) for withdrawal, \(q_D=0.02\) for disability, and \(q_M=0.01\) for death. These probabilities are unchanged in year 2. The annual effective interest rate is \(i=5\%\). At the end of the year of exit, the respective benefit values are $30,000, $8,000, $50,000, and $20,000. Find each cause-specific EPV and the total.

Method. First find the probability of remaining active in each year:

$$ p=1-(0.10+0.08+0.02+0.01)=0.79, \qquad v=\frac{1}{1.05}=\frac{20}{21}. $$

For any cause, the year-1 contribution is \(v q_j B_j\), and the year-2 contribution is \(v^2p q_jB_j\). The combined multiplier on \(q_jB_j\) is therefore

$$ v+v^2p = \frac{20}{21} + \left(\frac{20}{21}\right)^2(0.79) = \frac{20}{21}+\frac{316}{441} = \frac{736}{441} = \frac{184}{105}. $$

Retirement. The annual cause-weighted benefit is \(0.10(30{,}000)=\$3{,}000\). Hence

$$ \operatorname{EPV}_R = 3{,}000\left(\frac{184}{105}\right) = \$5{,}257.14. $$

Withdrawal. The annual cause-weighted benefit is \(0.08(8{,}000)=\$640\), giving

$$ \operatorname{EPV}_W = 640\left(\frac{184}{105}\right) = \$1{,}121.52. $$

Disability. The annual cause-weighted benefit is \(0.02(50{,}000)=\$1{,}000\), so

$$ \operatorname{EPV}_D = 1{,}000\left(\frac{184}{105}\right) = \$1{,}752.38. $$

Death. The annual cause-weighted benefit is \(0.01(20{,}000)=\$200\), so

$$ \operatorname{EPV}_M = 200\left(\frac{184}{105}\right) = \$350.48. $$

The total, rounded to the nearest cent, is

$$ \operatorname{EPV}_{\mathrm{total}} = 5{,}257.14+1{,}121.52+1{,}752.38+350.48 = \$8{,}481.52. $$

Check. The total cause-weighted benefit in either year, conditional on being active at the start of that year, is \(3{,}000+640+1{,}000+200=\$4{,}840\). The total EPV is \(\$4{,}840(v+v^2p)=\$8{,}481.5238\ldots\), which rounds to the same result. This check confirms that the cause components add to the total.

Worked Example: Probabilities and Benefits Vary by Year

A hypothetical plan uses an annual effective interest rate of \(3\%\) and a two-year projection. In year 1, the cause probabilities for retirement, withdrawal, disability, and death are \(0.05, 0.10, 0.02,\) and \(0.03\). In year 2, conditional on remaining active, they are \(0.10, 0.05, 0.03,\) and \(0.02\). The year-1 benefit values, in the same cause order, are $40,000, $5,000, $60,000, and $25,000. The year-2 values are $44,000, $6,000, $66,000, and $27,500. Find the total EPV and its components.

Method. In each year the cause probabilities sum to \(0.20\), so \(p_1=p_2=0.80\). Also \(v=1/1.03\). The year-1 retirement contribution, for example, is \(v(0.05)(40{,}000)\). Every year-2 contribution includes the probability \(p_1=0.80\) of reaching that year still active, as well as two years of discounting.

For clarity, the year-2 factor applied to each year-2 cause-weighted benefit is

$$ v^2p_1=\frac{0.80}{1.03^2}=\frac{800}{10609}. $$

The four year-1 cause-weighted benefits before discounting are \(2{,}000\), \(500\), \(1{,}200\), and \(750\). The four year-2 amounts before survival and discounting are \(4{,}400\), \(300\), \(1{,}980\), and \(550\). Therefore:

  • Retirement: \(2{,}000/1.03+4{,}400(800/10609)=\$5{,}384.62\).
  • Withdrawal: \(500/1.03+300(800/10609)=\$730.77\).
  • Disability: \(1{,}200/1.03+1{,}980(800/10609)=\$2{,}723.08\).
  • Death: \(750/1.03+550(800/10609)=\$1{,}173.08\).

Adding the unrounded component values and rounding the total to the nearest cent gives

$$ \operatorname{EPV}_{\mathrm{total}}=\$10{,}011.54. $$

Check. The year-1 cause-weighted benefits sum to \(2{,}000+500+1{,}200+750=\$4{,}450\). The year-2 amounts sum to \(4{,}400+300+1{,}980+550=\$7{,}230\). Thus the total is \(4{,}450/1.03+7{,}230(800/10609)=\$10{,}011.54\) to the nearest cent, agreeing with the cause-by-cause calculation. The calculation avoids carrying year-1 probabilities into year 2 when the year-2 assumptions have changed.

Worked Example: A Three-Year Horizon and the Active-Life Probability

For a hypothetical three-year valuation, the annual effective interest rate is \(4\%\). In each year, the probabilities of retirement, withdrawal, disability, and death are \(0.05, 0.08, 0.02,\) and \(0.01\), respectively. The member remains active with probability \(0.84\) in each year. The end-of-year benefit values are $50,000 for retirement, $9,000 for withdrawal, $42,000 for disability, and $18,000 for death. There are no benefits after year 3. Calculate the total EPV by cause.

Method. Since the probabilities are constant, the discounted probability weight for a cause-specific benefit in year \(k\) is \(v^k(0.84)^{k-1}\). With \(v=1/1.04=25/26\), the three weights are

$$ v=\frac{25}{26}, \qquad v^2(0.84)=\frac{525}{676}, \qquad v^3(0.84)^2=\frac{11025}{17576}. $$

Their sum is \(2.3654415\ldots\). For each cause, multiply that sum by its annual cause probability times its benefit value:

  • Retirement: \(0.05(50{,}000)(2.3654415\ldots)=\$5{,}913.60\).
  • Withdrawal: \(0.08(9{,}000)(2.3654415\ldots)=\$1{,}703.12\).
  • Disability: \(0.02(42{,}000)(2.3654415\ldots)=\$1{,}986.97\).
  • Death: \(0.01(18{,}000)(2.3654415\ldots)=\$425.78\).

The total EPV, rounded to the nearest cent, is

$$ \operatorname{EPV}_{\mathrm{total}} = 5{,}913.60+1{,}703.12+1{,}986.97+425.78 = \$10{,}029.47. $$

Check. The annual cause-weighted benefit is \(2{,}500+720+840+180=\$4{,}240\). Applying the same three discounted survival weights gives \(4{,}240(2.3654415\ldots)=\$10{,}029.47\) to the nearest cent, consistent with the separate cause values.

Interpretation and an Exam Trap

The cause-specific components answer different questions: how much EPV is attributable to each type of first exit, under the stated benefit and decrement assumptions? The decomposition also makes assumption changes easier to assess. For example, an increase in the withdrawal benefit affects the withdrawal component directly, while a change in the probability of remaining active can affect the exposure to benefits in every later year.

A frequent error is to mistake the \(q_{j,k}\) values for shares conditional on an exit. In the first worked example, the total probability of exit in a year is \(0.21\). The quantity

$$ \frac{0.10(30{,}000)+0.08(8{,}000)+0.02(50{,}000)+0.01(20{,}000)}{0.21} = \frac{4{,}840}{0.21} = \$23{,}047.62 $$

is the expected benefit conditional on an exit during that year, not the unconditional expected benefit for a member active at the start of the year. Using it directly as the year-1 expected payment overstates the year-1 value. The correction is to multiply by the probability of an exit, \(0.21\), recovering \(0.21(\$23{,}047.62\ldots)=\$4{,}840\), before discounting. Equivalently, use the original cause probabilities \(q_j\) directly in the cause-specific formula.

Key takeaway: For each cause and year, multiply the probability of reaching the year active by the conditional cause-specific exit probability, benefit value, and discount factor. Add the resulting cause-specific EPVs to obtain total EPV. Do not normalize cause probabilities unless the calculation explicitly asks for a quantity conditional on an exit.

Check Your Understanding

For each problem, use the stated benefit payment timing and report money to the nearest cent.

  1. A one-year hypothetical valuation uses \(i=4\%\), with \(q_R=0.06\), \(q_W=0.04\), \(q_D=0.01\), and \(q_M=0.02\). End-of-year benefits are $20,000, $5,000, $30,000, and $10,000, respectively. Find the total EPV. (Answer: $3,076.92)
  2. A two-year model has \(i=5\%\). In both years, \(q_R=0.04\), \(q_W=0.06\), \(q_D=0.01\), and \(q_M=0.02\). Benefits in each year are $20,000 for retirement, $4,000 for withdrawal, $30,000 for disability, and $10,000 for death. Find the total EPV. (Answer: $2,816.00)
  3. A two-year model has \(i=3\%\). Year-1 probabilities are \(q_R=0.02\), \(q_W=0.10\), \(q_D=0.01\), and \(q_M=0.02\); year-2 probabilities are \(q_R=0.04\), \(q_W=0.05\), \(q_D=0.02\), and \(q_M=0.01\). Year-1 benefits in the same cause order are $30,000, $5,000, $40,000, and $15,000; year-2 benefits are $32,000, $6,000, $44,000, and $16,000. Find the total EPV. (Answer: $3,960.19)
  4. A two-year model has \(i=4\%\), with probabilities \(q_R=0.10\), \(q_W=0.05\), \(q_D=0.02\), and \(q_M=0.03\) in each year. Benefits in each year are $25,000, $8,000, $50,000, and $20,000, respectively. Find the retirement and withdrawal EPV components. (Answer: Retirement $4,038.46; withdrawal $1,015.38)
  5. A three-year model has \(i=4\%\), annual probabilities \(q_R=0.05\), \(q_W=0.08\), \(q_D=0.02\), and \(q_M=0.01\), and benefits of $50,000, $9,000, $42,000, and $18,000, respectively. What is the probability of reaching the start of year 3 active, and what discount-survival weight applies to a year-3 benefit? (Answer: Active probability \(0.84^2=0.7056\); year-3 weight \(v^3(0.84)^2=0.6273\), to four decimal places)

Previous

Next

Back to Tutorials
Clinical Biostats