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Topics Covered

Discount Factor v Present Value Z Whole-Life Insurance Term Insurance Endowment Insurance Pure Endowment Moment of Death (continuous) End of Year of Death (discrete) Net Single Premium
On this page
  1. 1. Interest, Discount & the Discount Factor v
  2. 2. Two Modes of Benefit Payment
  3. 3. Whole-Life Insurance
  4. 4. n-Year Term Insurance
  5. 5. n-Year Pure Endowment
  6. 6. n-Year Endowment Insurance
  7. 7. Deferred Insurance
  8. 8. Varying Benefits
  9. 9. Relationships Between Insurance Models
  10. 10. Net Single Premium
  11. Summary Table — Insurance Present Values
  12. Key Take-aways

1. Interest, Discount & the Discount Factor v

Throughout this unit we assume a constant annual effective rate of interest \(i > 0\). Key quantities:

SymbolMeaningFormula
\(i\)Effective annual interest rateGiven
\(v\)Discount factor\(v = 1/(1 + i)\)
\(d\)Effective discount rate\(d = 1 - v = i/(1+i)\)
\(\delta\)Force of interest (continuous)\(\delta = \ln(1+i),\;\; v = e^{-\delta}\)
\(i^{(m)}\)Nominal rate, compounded m times/yr\((1 + i^{(m)}/m)^m = 1 + i\)

A payment of 1 at time \(t\) (years from now) is worth \(v^t = e^{-\delta t}\) today.

2. Two Modes of Benefit Payment

Life-insurance benefits can be modelled in two ways:

  1. Continuous — benefit paid at the exact moment of death. Uses continuous future lifetime \(T = T(x)\).
  2. Discrete — benefit paid at the end of the year of death. Uses curtate future lifetime \(K = K(x) = \lfloor T \rfloor\).

The discrete form is what classical life tables produce; the continuous form is mathematically cleaner.

3. Whole-Life Insurance

DEFINITION

A whole-life insurance pays a benefit of ₹1 (or any face amount, by linearity) on the death of (\(x\)) whenever it occurs — there is no expiration. Premiums may be paid for a defined period.

3.1 Continuous Case — \(\bar A_x\)

Present value at time 0 (age \(x\)) of the benefit:

\[ Z \;=\; v^T, \quad T = T(x). \]

The actuarial present value (APV) = net single premium is:

\[ \bar A_x \;=\; E(Z) \;=\; E(v^T) \;=\; \int_0^{\omega - x} v^t \cdot {}_tp_x\, \mu(x+t)\, dt. \]

For constant force of mortality \(\mu(x+t) = \mu\) and \(v = e^{-\delta}\):

\[ \bar A_x \;=\; \int_0^{\infty} e^{-\delta t} e^{-\mu t} \mu\, dt \;=\; \dfrac{\mu}{\mu + \delta}. \]

3.2 Discrete Case — \(A_x\)

Benefit ₹1 paid at the end of the year of death:

\[ Z \;=\; v^{K+1}, \quad K = K(x). \] \[ A_x \;=\; E(Z) \;=\; \sum_{k=0}^{\infty} v^{k+1} \cdot {}_kp_x \cdot q_{x+k} \;=\; \sum_{k=0}^{\infty} v^{k+1} \cdot {}_{k|}q_x. \]

3.3 Variance of Z

For the continuous whole-life:

\[ E(Z^2) \;=\; \int_0^{\infty} (v^2)^t \cdot {}_tp_x\, \mu(x+t)\, dt, \]

Note: this equals \({}^2\bar A_x\) — the same formula computed with force of interest doubled (\(\delta \to 2\delta\), or \(v \to v^2\)).

\[ \text{Var}(Z) \;=\; {}^2\bar A_x - (\bar A_x)^2. \]

3.4 Relationship Continuous ↔ Discrete

Under UDD (uniform distribution of deaths within each year):

\[ \bar A_x \;=\; \dfrac{i}{\delta}\, A_x. \]

The factor \(i/\delta > 1\) accounts for paying the benefit on average half a year earlier in the continuous version.

3.5 Recursion for \(A_x\)

\[ A_x \;=\; v\, q_x + v\, p_x \, A_{x+1}. \]

Either (\(x\)) dies in year 1 (probability \(q_x\), present value \(v\)) or survives to age \(x+1\) (probability \(p_x\), present value \(v \cdot A_{x+1}\)).

EXAMPLE 1 — Constant force, continuous

\(\mu = 0.02,\; \delta = 0.05\). Then \(\bar A_x = 0.02/(0.02 + 0.05) = 0.286\).

So for a benefit of ₹1 lakh, the APV is ₹28 600 — the single premium an insurer would charge for full life cover under these assumptions.

\(_2 \bar A_x = 0.02/(0.02 + 0.10) = 0.167\) (using \(2\delta = 0.10\)).

\(\text{Var}(Z) = 0.167 - 0.286^2 = 0.167 - 0.0818 = 0.0852\); SD \(\approx 0.292\).

EXAMPLE 2 — Discrete, age 30, i = 5 %

From a life table, \(q_{30} = 0.001\). Using the recursion (and assuming \(A_{31}\) is known from the table, say 0.105):

\(A_{30} = v(0.001) + v(0.999)(0.105)\)
where \(v = 1/1.05 = 0.9524\)
\(= 0.9524 \cdot 0.001 + 0.9524 \cdot 0.999 \cdot 0.105 = 0.000952 + 0.0999 = 0.1009\).

Net single premium for ₹10 lakh whole-life = 10,00,000 × 0.1009 = ₹1 00 900.

4. n-Year Term Insurance

DEFINITION

n-year term insurance pays ₹1 on death only if death occurs within \(n\) years from now. No benefit if (\(x\)) survives \(n\) years.

4.1 Continuous Case

Present value: \(Z = v^T\) if \(T \le n\), else \(Z = 0\).

\[ \bar A^{\,1}_{x:\overline{n}|} \;=\; \int_0^n v^t\, {}_tp_x\, \mu(x+t)\, dt. \]

4.2 Discrete Case

\[ A^{\,1}_{x:\overline{n}|} \;=\; \sum_{k=0}^{n-1} v^{k+1}\, {}_{k|}q_x. \]

The superscript "1" over \(x\) (the life) indicates that the benefit is contingent on death of \(x\) before time \(n\).

4.3 Connection

Under UDD: \(\bar A^{\,1}_{x:\overline{n}|} = (i/\delta)\, A^{\,1}_{x:\overline{n}|}\).

5. n-Year Pure Endowment

DEFINITION

A pure endowment pays ₹1 if (\(x\)) is alive at time \(n\); 0 otherwise. No payment on death within \(n\) years.

\[ A^{\;\;\;\;1}_{x:\overline{n}|} \;=\; v^n\, {}_np_x \;=\; {}_nE_x. \]

The "1" is above \(\overline{n}\), indicating the benefit is contingent on survival, not death. Often written \({}_nE_x\).

6. n-Year Endowment Insurance

DEFINITION

Endowment insurance is the combination: pays ₹1 on death within \(n\) years (term part) or ₹1 on survival to time \(n\) (pure endowment part). Either way, ₹1 is paid at most once.

6.1 Decomposition

CONTINUOUS \[ \bar A_{x:\overline{n}|} \;=\; \bar A^{\,1}_{x:\overline{n}|} + {}_nE_x \;=\; \text{(term)} + \text{(pure endowment)}. \] DISCRETE \[ A_{x:\overline{n}|} \;=\; A^{\,1}_{x:\overline{n}|} + {}_nE_x. \]

It guarantees a payment at either the policyholder's death (if before maturity) or maturity. From a customer perspective it functions as both protection and savings.

EXAMPLE 1 — Term vs Endowment

\(\delta = 0.05\), \(\mu = 0.02\), n = 20 years, age \(x = 40\). Constant force model.

\(_{20}p_x = e^{-0.02 \cdot 20} = e^{-0.4} = 0.6703\).

Term insurance APV:

\(\bar A^{\,1}_{x:\overline{20}|} = \dfrac{\mu}{\mu+\delta}(1 - e^{-(\mu+\delta) n}) = \dfrac{0.02}{0.07}(1 - e^{-0.07 \cdot 20}) = 0.286 \cdot (1 - 0.2466) = 0.286 \cdot 0.7534 = 0.2155\).

Pure endowment APV: \(v^{20}\, _{20}p_x = e^{-0.05 \cdot 20}\, e^{-0.02 \cdot 20} = e^{-1.4} = 0.2466\).

Endowment insurance APV: \(0.2155 + 0.2466 = 0.4621\) — higher than term alone because of the survival benefit.

EXAMPLE 2 — Single premium pricing

An insurer issues an endowment policy with face ₹10 lakh, \(n = 25\), \(i = 6\%\), age 35. Using a life table, suppose \(A^{\,1}_{35:\overline{25}|} = 0.072\) and \(_{25}E_{35} = 0.225\). Then \(A_{35:\overline{25}|} = 0.297\).

Net single premium = ₹10,00,000 × 0.297 = ₹2 97 000.

7. Deferred Insurance

DEFINITION

An m-year deferred whole-life insurance pays ₹1 at death only if death occurs after \(m\) years. The first \(m\) years are a "no benefit" deferment period.

CONTINUOUS \[ {}_{m|}\bar A_x \;=\; \int_m^{\infty} v^t\, {}_tp_x\, \mu(x+t)\, dt \;=\; v^m\, {}_mp_x\, \bar A_{x+m}. \] DISCRETE \[ {}_{m|}A_x \;=\; \sum_{k=m}^{\infty} v^{k+1}\, {}_{k|}q_x \;=\; v^m\, {}_mp_x\, A_{x+m}. \]

Useful identity: \(A_x = A^{\,1}_{x:\overline{m}|} + {}_{m|}A_x\) — whole-life splits into "die within m years" + "die after m years".

8. Varying Benefits

8.1 Increasing Insurance (IA)x

Benefit increases each year. If benefit = (k + 1) when death occurs in year \(k + 1\) (so K = k):

\[ (IA)_x \;=\; \sum_{k=0}^{\infty}(k + 1) v^{k+1}\, {}_{k|}q_x. \]

8.2 Decreasing Term (DA)1x:n|

\[ (DA)^{\,1}_{x:\overline{n}|} \;=\; \sum_{k=0}^{n-1}(n - k) v^{k+1}\, {}_{k|}q_x. \]

Often used in mortgage-protection insurance.

9. Relationships Between Insurance Models

Under uniform distribution of deaths (UDD) within each year, the continuous and discrete forms are linked by simple multiplicative factors.

Continuous= Constant ×Discrete
\(\bar A_x\)\(i/\delta\)\(A_x\)
\(\bar A^{\,1}_{x:\overline{n}|}\)\(i/\delta\)\(A^{\,1}_{x:\overline{n}|}\)
\(\bar A_{x:\overline{n}|}\)—\((i/\delta)A^{\,1}_{x:\overline{n}|} + {}_nE_x\) (pure endowment factor 1)

The factor \(i/\delta\) appears because under UDD each death within a year occurs on average half a year earlier than the year-end, so a continuous benefit is paid earlier and is therefore more valuable.

10. Net Single Premium

The net single premium (NSP) of an insurance benefit is the actuarial present value of the benefit, computed assuming the equivalence principle:

\[ \text{NSP} \;=\; E\bigl[\text{PV of future benefits}\bigr]. \]

Multiplying NSP by the face amount gives the lump-sum premium that would fund the future benefits exactly on average.

For benefit \(b\) face amount:

NSP is the input for computing periodic premiums (Unit 5).

Summary Table — Insurance Present Values

TypePV ZAPV / NSP (per ₹1 benefit)
Whole-life (cts)\(v^T\)\(\bar A_x\)
Whole-life (disc)\(v^{K+1}\)\(A_x\)
n-year Term (cts)\(v^T \cdot \mathbb{1}_{T \le n}\)\(\bar A^{\,1}_{x:\overline{n}|}\)
n-year Term (disc)\(v^{K+1} \cdot \mathbb{1}_{K < n}\)\(A^{\,1}_{x:\overline{n}|}\)
Pure Endowment\(v^n \cdot \mathbb{1}_{T > n}\)\({}_nE_x = v^n\, _np_x\)
Endowment (disc)\(v^{\min(K+1, n)}\)\(A_{x:\overline{n}|}\)
m-deferred Whole-life\(v^T \cdot \mathbb{1}_{T > m}\)\({}_{m|}A_x = v^m\, _mp_x\, A_{x+m}\)

Key Take-aways

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