Expected Cash Flows Versus Individual Outcomes

Introduction

A state-contingent benefit is paid according to the policyholder’s state or history under the contract. Before the future path is known, an actuary can calculate the expected present value of those payments by weighting each possible outcome by its probability. That calculation describes an average across possible outcomes under the assumed model. It does not describe the cash-flow path that one policy will actually produce.

This distinction matters when interpreting valuations. An expected value can be useful for pricing and comparison, but the policyholder’s path may produce no payment, one payment, or several payments. The expected amount is not a promise that the policy will pay that amount. This tutorial develops the distinction using discrete state and path outcomes.

Definition: A realized cash-flow path is the sequence of states actually experienced by a policy and the payments generated by those states and the contract rules. The expected present value is the probability-weighted average of the present values of all possible cash-flow paths, under the stated model and discount basis.

Weighting Possible Outcomes

Suppose a contract has possible outcomes indexed by $j$, with probabilities $p_j$. Let $Z_j$ be the present value of the payments if outcome $j$ occurs. The probabilities must satisfy $p_j\geq 0$ and $\sum_j p_j=1$. The expected present value is

E[Z]=\sum_j p_j Z_j.

For payments at several dates, let $C_t$ denote the random payment at time $t$, and let $v=1/(1+i)$, where $i$ is the effective annual interest rate. If payment dates are annual and each payment is made at year-end, then the present value on a realized path is $Z=\sum_{t=1}^{n}v^tC_t$. Its expected present value is

E[Z]=E\left[\sum_{t=1}^{n}v^tC_t\right] =\sum_{t=1}^{n}v^t E[C_t].

This equality follows from the linearity of expectation. It does not require the cash flows at different dates to be independent. But the result is still an expectation: the sequence $E[C_1],\ldots,E[C_n]$ is generally not the sequence of payments made on any individual path.

Key distinction: A state probability is a weight assigned to a possible outcome when calculating an expectation. It does not mean that a fraction of the corresponding benefit is paid on each policy, nor that the expected payment is guaranteed.

Worked Example: A Benefit at One Future Date

A hypothetical contract pays $4,000 at the end of one year if the covered person is in state $C$, and pays nothing otherwise. At the payment date, the stipulated distribution is $P(C)=0.25$ and $P(H)=0.75$. Assume an effective annual interest rate of 10%, no mortality during the one-year term, and no other cash flows. Find the expected present value and describe the possible realized values.

Method and assumptions. The outcome is a two-point discrete distribution: the payment is $4,000 with probability 0.25 and $0 with probability 0.75. The discount factor is $v=1/1.10=0.9090909\ldots$.

If state $C$ occurs, the present value is $4{,}000v=4{,}000/1.10=\$3{,}636.36$ to the nearest cent. If state $H$ occurs, the present value is $0. Weighting these outcomes gives

E[Z]=0.25(3{,}636.36\ldots)+0.75(0) =909.09\ldots.

The expected present value is therefore $909.09, rounded to two decimals. On an individual policy, however, the present value is either $3,636.36… or $0 under these assumptions. Neither possible path produces the expected value of $909.09.

Check. The same result follows by first finding the expected payment at year-end: $E[C_1]=0.25(4{,}000)+0.75(0)=\$1{,}000$. Discounting gives $1{,}000/1.10=\$909.09$ to two decimals. A common mistake this example avoids is interpreting that $1,000 expected payment as a guaranteed year-end payment.

Expected Payments Across Several Dates

When payments are made at multiple dates, the future path determines the full stream. State probabilities at a particular date can weight that date’s payment if the payment rule depends only on the state at that date. If eligibility also depends on duration, prior transitions, or another history condition, the outcome categories must distinguish the relevant possibilities. A probability for a broad state alone cannot replace a more specific contract rule.

The following example uses a finite set of complete paths, so each path’s payment stream and present value can be calculated directly. The path probabilities are supplied assumptions; they are not inferred from transition intensities or a mortality table.

Worked Example: Comparing a Path Value With an Expected Value

A hypothetical two-year policy pays $550 at the end of year 1 on path $B$. It pays $1,210 at the end of year 2 on path $A$, but only if that path occurs. The possible paths and their probabilities are: $B$, probability 0.30, with payments $(550,0)$; $A$, probability 0.50, with payments $(0,1{,}210)$; and $N$, probability 0.20, with payments $(0,0)$. Assume an effective annual interest rate of 10%, no mortality, and no other payments. Find the expected present value and compare it with the present value on each path.

Method and assumptions. This is a three-outcome discrete path distribution. The probabilities sum to $0.30+0.50+0.20=1.00$. Discount factors are $v=1/1.10$ and $v^2=1/1.10^2=1/1.21$.

On path $B$, the present value is $550/1.10=\$500.00$. On path $A$, it is $1{,}210/1.21=\$1{,}000.00$. On path $N$, it is $0. Weighting the path present values gives

E[Z]=0.30(500)+0.50(1{,}000)+0.20(0) =150+500=\$650.

The expected present value is $650.00. The realized present value is $500.00, $1,000.00, or $0.00, depending on which path occurs. None of those outcomes is $650.00.

As a check, the expected year-1 payment is $E[C_1]=0.30(550)+0.50(0)+0.20(0)=\$165$. The expected year-2 payment is $E[C_2]=0.30(0)+0.50(1{,}210)+0.20(0)=\$605$. Thus $vE[C_1]=165/1.10=\$150$, and $v^2E[C_2]=605/1.21=\$500$. Their sum is again $650.00.

Common mistake avoided. The $165 and $605 are expected payments, not payments on the policy. Treating them as guaranteed would suggest that every policy pays $165 at year 1 and $605 at year 2. In fact, those amounts summarize the probability-weighted payments across the stated outcomes; each realized path has its own specified cash flows.

Justification by Indicators

A useful way to see why state probabilities weight cash flows is to represent a payment using an indicator variable. Let $I_C$ equal 1 if state $C$ occurs at the payment date and 0 otherwise. If the contract pays a fixed amount $b$ exactly in state $C$, then $C_t=bI_C$. Since $I_C$ is 1 with probability $P(C)$ and 0 otherwise,

E[C_t]=E[bI_C] =bE[I_C] =b\bigl(1\cdot P(C)+0\cdot(1-P(C))\bigr) =bP(C).

Discounting a payment due at time $t$ multiplies each possible payment by the same deterministic discount factor $v^t$, so $E[v^tC_t]=v^tE[C_t]=v^tbP(C)$. This derivation explains the probability-weighted expected value. It does not imply that a policy pays $bP(C)$: the indicator on a realized policy is either 0 or 1.

Worked Example: Payments While in a Care State

A hypothetical long-term care contract pays $1,200 at the end of each of two years in which the insured is in care state $C$. The insured begins in healthy state $H$. The stipulated two-year path distribution is $HH:0.40$, $HC:0.20$, $HD:0.10$, $CH:0.10$, $CC:0.10$, and $CD:0.10$, where the first letter is the year-1 state and the second is the year-2 state. State $D$ denotes death and is absorbing. These supplied path probabilities include the mortality assumption; no separate mortality rate is assumed. Interest is 5% effective annually. Find the expected present value and the present value for each path.

Method and assumptions. Payments are at the end of each year and depend only on whether the state at that year-end is $C$. The discrete path probabilities sum to $0.40+0.20+0.10+0.10+0.10+0.10=1.00$. The discount factors are $v=1/1.05=0.95238095\ldots$ and $v^2=1/1.1025=0.90702948\ldots$.

Calculate the path present values. Path $HH$ has no care-state payment, so its value is $0. Path $HC$ has only a year-2 payment: $1{,}200/1.1025=\$1{,}088.435\ldots$. Path $HD$ has no payment, so its value is $0. Path $CH$ has only a year-1 payment: $1{,}200/1.05=\$1{,}142.857\ldots$. Path $CC$ has both payments: $1{,}200/1.05+1{,}200/1.1025 =\$1{,}142.857\ldots+\$1{,}088.435\ldots =\$2{,}231.292\ldots$. Path $CD$ has only a year-1 payment, so its value is also $1,142.857….

Weighting the path values gives

\begin{aligned} E[Z] &=0.40(0)+0.20(1{,}088.435\ldots)+0.10(0)\\ &\quad+0.10(1{,}142.857\ldots)+0.10(2{,}231.292\ldots) +0.10(1{,}142.857\ldots)\\ &=217.687\ldots+114.286\ldots+223.129\ldots+114.286\ldots\\ &=\$669.39. \end{aligned}

The answer is $669.39, rounded to two decimals. Check by date: the probability of being in $C$ at year 1 is $0.10+0.10+0.10=0.30$, from paths $CH,CC,CD$. The probability of being in $C$ at year 2 is $0.20+0.10=0.30$, from $HC,CC$. Therefore the expected payment at each date is $0.30(1{,}200)=\$360$. Discounting gives $360/1.05=\$342.857\ldots$ and $360/1.1025=\$326.531\ldots$, totaling $669.39.

Common mistake avoided. The two $360 expected payments do not mean every policy receives $360 in each year. For example, path $HH$ receives nothing, while path $CC$ receives $1,200 at both dates. The supplied state and path probabilities weight these possibilities; they do not turn contingent payments into guaranteed partial benefits.

Interpreting the Result

Expected present value is a summary of the outcomes allowed by the model, the contract terms, and the discount assumptions. Changing any of these inputs can change the expected value. In particular, the probabilities must correspond to the payment condition being valued. When a benefit depends on more than the current state, the outcome description must preserve the information required by that rule.

The expected value is useful because it compresses many possible paths into one probability-weighted amount. That compression loses information about the range of realized values and the path-specific timing of payments. In an actuarial calculation, it is therefore important to state both what the expected value measures and what it does not measure.

Expected value: a probability-weighted average under specified assumptions.
Realized value: the present value of payments on the path that actually occurs.
State probability: a weight for an outcome, not a fractional payment or guarantee.

Key takeaway: First identify the possible payment paths and their probabilities. Then calculate and weight their present values. Report the result as an expected present value, not as the payment an individual policy is certain to receive.

Check Your Understanding

For each problem, distinguish the weighted expected value from the possible realized payments. Amounts are hypothetical.

A contract pays $2,200 at the end of one year if a specified state occurs, with probability 0.40, and otherwise pays nothing. Interest is 10% effective annually, and there is no mortality during the term. Find the expected present value. (Answer: $800.00)
At the end of year 1, a hypothetical policy pays $1,000 in state $C$ and nothing in states $H$ or $D$. The probabilities are $P(H)=0.50$, $P(C)=0.35$, and $P(D)=0.15$. Assume 5% effective annual interest and that these probabilities already account for mortality. Find the expected present value and state the possible realized present values. (Answer: Expected present value $333.33; realized values $952.38 or $0.00.)
A two-year contract has two possible paths: probability 0.60 gives a $500 payment at the end of year 1 and nothing at year 2; probability 0.40 gives nothing at year 1 and $1,100 at the end of year 2. Interest is 10% effective annually, with no mortality. Calculate the expected present value both by weighting path present values and by discounting expected payments by date. (Answer: $636.36.)
A hypothetical care benefit pays $600 at each year-end when the insured is in state $C$. The two-year paths are $HH:0.50$, $HC:0.20$, $CH:0.20$, and $CC:0.10$, beginning in $H$. There is no mortality in this model, and interest is 10% effective annually. Find the expected present value and explain whether any one path has that value. (Answer: $312.40; no path has that present value.)

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