Constant-Force Conversion Between Continuous and Discrete Values

Why the Mortality Assumption Matters

A unit life insurance benefit can be paid immediately when death occurs or at the end of the policy year of death. Both benefits depend on the same survival probabilities, but the payment dates differ. Converting between their expected present values therefore requires an assumption about when deaths occur within each year, as well as an interest assumption.

Under a constant force of mortality, the time to death has an exponential distribution. This makes the continuous value an integral and the year-end value a geometric series. Their ratio is not generally the same as under the uniform distribution of deaths (UDD). The distinction matters even when both mortality assumptions have the same probability of death in the first year.

Definition: Suppose the force of mortality is constant at \(\mu>0\) from age \(x\) onward, so \(\mu_{x+t}=\mu\) for all \(t\geq 0\). Then \({}_tp_x=e^{-\mu t}\), and the probability of death within one year is \(q=1-e^{-\mu}\). A continuous unit whole-life insurance pays 1 at the moment of death; a discrete unit whole-life insurance pays 1 at the end of the year of death.

Let \(i\) be the effective annual interest rate, \(v=1/(1+i)\), and \(\delta=\ln(1+i)\) the equivalent force of interest. Thus a payment at time \(t\) has present value \(e^{-\delta t}\). The formulas below assume \(i>-1\), and the mortality model is used for the full duration being valued.

Whole-Life Conversion Under Constant Force

For a continuous benefit, death at time \(t\) has density \(\mu e^{-\mu t}\), and its discount factor is \(e^{-\delta t}\). For the discrete benefit, death in policy year \(k+1\) has probability \(e^{-k\mu}(1-e^{-\mu})\), and the benefit is discounted for \(k+1\) years. These two approaches give the whole-life values directly.

\bar{A}_x =\int_0^\infty e^{-\delta t}\mu e^{-\mu t}\,dt =\frac{\mu}{\mu+\delta}, \qquad A_x =\sum_{k=0}^\infty v^{k+1}e^{-k\mu}(1-e^{-\mu}) =\frac{v(1-e^{-\mu})}{1-ve^{-\mu}}.

The geometric series for \(A_x\) converges when \(ve^{-\mu}<1\), which holds for \(i>-1\) and \(\mu>0\). Dividing the two expressions gives a conversion factor that depends on both the interest and mortality forces.

\frac{\bar{A}_x}{A_x} = \frac{\mu(1-ve^{-\mu})} {(\mu+\delta)v(1-e^{-\mu})}.

This ratio can also be justified policy year by policy year. In year \(k+1\), the continuous expected present value contribution is the probability of surviving to time \(k\), multiplied by the integral of the discounted death density over the next year. The discrete contribution is the probability of surviving to time \(k\), multiplied by the probability of dying in that year and the end-of-year discount factor. Their common survival-to-year-\(k\) factor cancels, leaving the same ratio in every year when \(\mu\) is constant. Summing those contributions therefore preserves that ratio for whole life.

\frac{\displaystyle e^{-k(\mu+\delta)} \int_0^1 e^{-\delta s}\mu e^{-\mu s}\,ds} {\displaystyle e^{-k\mu}(1-e^{-\mu})e^{-\delta(k+1)}} = \frac{\mu(1-e^{-(\mu+\delta)})} {(\mu+\delta)e^{-\delta}(1-e^{-\mu})} = \frac{\bar{A}_x}{A_x}.

The equality of the year-specific and whole-life ratios depends on the same constant \(\mu\) applying in every policy year. If the force is constant only within each year but changes from one year to another, each year can have a different ratio. In that case, the whole-life ratio is determined by the weighted sum of the year-specific contributions; the formula above cannot be used as a single conversion factor without further justification.

Worked Example: Constant-Force Whole-Life Values

A hypothetical policy pays 1 at death if the benefit is continuous, or 1 at the end of the year of death if it is discrete. Mortality follows a constant force \(\mu=\ln 2\) for life, and the effective annual interest rate is \(i=10\%\). Find both expected present values and their ratio. Use four decimal places for final values.

Method. Use the constant-force whole-life formulas. First, \(v=1/1.10=0.9090909\), \(\delta=\ln(1.10)=0.0953102\), and \(e^{-\mu}=1/2\). Thus the annual probability of death is \(q=1-1/2=0.5\).

The continuous value is

\bar{A}_x =\frac{\ln 2}{\ln 2+\ln(1.10)} =\frac{0.6931472}{0.7884574} =0.8791.

The discrete value is

A_x =\frac{(0.9090909)(0.5)} {1-(0.9090909)(0.5)} =\frac{0.4545455}{0.5454545} =0.8333.

Therefore \(\bar{A}_x/A_x=0.8791/0.8333=1.0549\), to four decimal places. The continuous benefit has the larger value because it is paid earlier than the year-end benefit. The result is checked by the ratio formula: substituting \(v=0.9090909\), \(e^{-\mu}=0.5\), and \(\delta=0.0953102\) also gives approximately \(1.0549\). The common mistake avoided here is treating the annual death probability \(0.5\) as if it were the force \(\mu\); they are different quantities, related by \(q=1-e^{-\mu}\).

Finite-Term Benefits

For an \(n\)-year term insurance, the benefit is paid only if death occurs during the first \(n\) years. Under the same constant-force model, the continuous value integrates the death density over \([0,n]\). The discrete value includes only the first \(n\) terms of the year-end series.

\bar{A}^{\,1}_{x:\overline{n}|} =\int_0^n e^{-\delta t}\mu e^{-\mu t}\,dt =\frac{\mu}{\mu+\delta}\left(1-e^{-n(\mu+\delta)}\right),

Here \(\bar{A}^{\,1}_{x:\overline{n}|}\) denotes the expected present value of a continuous unit \(n\)-year term benefit. For the discrete term benefit, the sum of the first \(n\) terms is

A^{\,1}_{x:\overline{n}|} =\sum_{k=0}^{n-1} v^{k+1}e^{-k\mu}(1-e^{-\mu}) =\frac{v(1-e^{-\mu})\left(1-(ve^{-\mu})^n\right)} {1-ve^{-\mu}}.

The same per-year ratio applies under a constant force, so the ratio of the continuous and discrete \(n\)-year term values is the whole-life ratio. This holds because both values sum exactly the same \(n\) policy-year contributions, each with that same ratio. It would not follow merely from the two benefits having the same term if the force varied by policy year.

Worked Example: Converting a Three-Year Term Value

A hypothetical three-year term contract has unit benefits, with continuous payment at death or discrete payment at the end of the year of death. Assume \(\mu=\ln 2\) throughout the term and \(i=10\%\). Calculate both values and check their ratio. Round final values to four decimal places.

Method. From the preceding example’s stated assumptions, \(v=10/11\), \(e^{-\mu}=1/2\), and \(\delta=\ln(1.10)\). In the discrete calculation, \(ve^{-\mu}=(10/11)(1/2)=5/11\). Therefore

A^{\,1}_{x:\overline{3}|} =\frac{(5/11)\left(1-(5/11)^3\right)}{1-5/11} =\frac{(5/11)(1206/1331)}{6/11} =\frac{1005}{1331} =0.7551.

For the continuous value, \(\mu+\delta=0.7884574\), and \(e^{-3(\mu+\delta)}=(e^{-\mu}e^{-\delta})^3=(5/11)^3=125/1331\). Hence

\bar{A}^{\,1}_{x:\overline{3}|} =\frac{0.6931472}{0.7884574} \left(1-\frac{125}{1331}\right) =(0.8791)(0.9061) =0.7966.

The ratio is \(0.7966/0.7551\approx1.0549\), matching the whole-life conversion factor. This is a useful check: with constant mortality force, truncating both values at the same term duration does not change their ratio. A common error is to include the survival payment at year 3 in the term value; a term insurance pays only for deaths during the term, so there is no maturity benefit in this calculation.

Comparison with UDD

Under UDD, the time of death conditional on dying during a policy year is uniform within that year. When the annual death probability is \(q\), the continuous contribution from a policy year is therefore related to its year-end discrete contribution by the factor \(i/\delta\). This gives the familiar UDD conversion \(\bar{A}_x=(i/\delta)A_x\), provided the same annual mortality probabilities are used in both values.

Under constant force, the conditional timing of death within a year is not uniform: deaths are more concentrated earlier in the year than under UDD. Consequently, the UDD factor should not be applied to a constant-force model. The next example holds the first-year death probability fixed to isolate the effect of the within-year assumption.

Worked Example: Same Annual Mortality, Different Timing Assumption

A hypothetical whole-life policy has effective annual interest \(i=10\%\), and its annual probability of death in each policy year is \(q=0.5\). Compare the continuous value under UDD with the continuous value under a constant force model calibrated to this same \(q\). Use four decimal places for final values.

Method. The discrete whole-life value is the same under either timing assumption because both have the same annual death probabilities:

A_x=\frac{vq}{1-v(1-q)} =\frac{(10/11)(1/2)}{1-(10/11)(1/2)} =\frac{5}{6} =0.8333.

Under UDD, \(\delta=\ln(1.10)=0.0953102\), so

\bar{A}_x^{\mathrm{UDD}} =\frac{0.10}{0.0953102}(0.8333333) =0.8743.

For the constant-force model, calibrate \(\mu\) from \(q=1-e^{-\mu}=0.5\), which gives \(\mu=-\ln(0.5)=\ln 2\). The continuous value is then

\bar{A}_x^{\mathrm{CF}} =\frac{\ln 2}{\ln 2+\ln(1.10)} =0.8791.

The constant-force value exceeds the UDD value by \(0.8791-0.8743=0.0048\). Both assumptions produce the same discrete value and annual mortality probability, but the continuous values differ because they assign different timing to deaths within each year. The common mistake avoided is using \(i/\delta\) for both models: that factor is the UDD conversion, not the constant-force conversion.

Common Mistakes and Interpretation

Confusing \(q\) and \(\mu\). Under a constant force, \(q=1-e^{-\mu}\), not \(q=\mu\). The approximation \(q\approx\mu\) may be close for a small force, but it is not an identity.
Discounting a discrete benefit for the wrong time. A benefit paid at the end of the year of death is discounted by \(v^{k+1}\) when death occurs in year \(k+1\), not by \(v^k\).
Applying the UDD factor to a constant-force model. The conditional distribution of death time differs between the assumptions, so the continuous values need not agree even when the annual death probability agrees.
Using a whole-life ratio without checking the mortality model. A single ratio follows here because the force is constant across all policy years. If it changes by year, calculate the contributions under the specified model rather than assuming one universal factor.

Key takeaway: Under a constant force \(\mu\), \(\bar{A}_x=\mu/(\mu+\delta)\) and \(A_x=v(1-e^{-\mu})/(1-ve^{-\mu})\). Their ratio depends on both mortality and interest. For the same annual death probability, UDD instead gives the continuous-to-discrete factor \(i/\delta\).

Check Your Understanding

For each problem, state the model and use the payment timing specified. Probabilities and values are rounded to four decimal places where needed.

A whole-life policy has constant force of mortality \(\mu=\ln 2\) and effective annual interest \(i=5\%\). Find the discrete and continuous unit insurance values. (Answer: \(A_x=0.9091\), \(\bar{A}_x\approx0.9342\).)
A one-year term benefit pays 1 at the end of the year of death or continuously at death. Assume constant mortality force \(\mu=0.2\) and force of interest \(\delta=0.08\). Find both values. (Answer: discrete value \(\approx0.1673\); continuous value \(\approx0.1744\).)
Under UDD, a whole-life policy has annual death probability \(q=0.2\) and effective annual interest \(i=4\%\). Find the discrete and continuous values. (Answer: \(A_x=5/6=0.8333\), \(\bar{A}_x=(0.04/\ln(1.04))(5/6)\approx0.8499\).)
A two-year term benefit has constant force of mortality \(\mu=0.4\) and force of interest \(\delta=0.06\). Find the discrete year-end and continuous values, and verify that the continuous value is larger. (Answer: discrete value \(=e^{-0.06}(1-e^{-0.4})+e^{-0.52}(1-e^{-0.4})\approx0.5065\); continuous value \(=(0.4/0.46)(1-e^{-0.92})\approx0.5230\).)
A constant-force model has first-year death probability \(q=0.3\), and the effective annual interest rate is \(i=5\%\). Find \(\mu\), then calculate the whole-life discrete and continuous values. (Answer: \(\mu=-\ln(0.7)\approx0.3567\), \(A_x\approx0.7786\), and \(\bar{A}_x\approx0.8797\).)

What You'll Learn