Actuarial Notes

Profit Margin

Study guide: Exam ALTAM

The Profit Margin of a policy is its net present value divided by the expected present value of its premiums, both calculated at the risk discount rate. It expresses the profit from a profit test per unit of premium income, so that policies of different sizes, terms and premium patterns can be compared on one scale.

\text{Profit margin} = \frac{\text{NPV}}{\text{EPV of premiums}}

\text{EPV of premiums} = \sum_{t=0}^{n-1} P\,{}_tp_x\,(1+r)^{-t}

  • Symbols: Following Dickson, Hardy & Waters, the NPV is the present value of the profit signature at the risk discount rate r, P is the premium paid at the start of each year while the policy is in force, and {}_tp_x is the probability that the policy is in force at time t. For a level premium, the denominator is P\,\ddot{a}_{x:\overline{n}|} calculated at rate r, and with lapses the survival factor allows for every decrement.
  • What it says: A profit margin of 5% means that the shareholders expect to gain 5 cents of present value for every dollar of premium the policy brings in. The margin has the same sign as the NPV, but unlike the NPV it does not grow with the size of the policy.
  • Against the other measures: The margin always exists, while an internal rate of return may not, but its value depends on the choice of r. The NPV, IRR, margin and discounted payback period are usually reported together.
  • Setting a premium: A target margin m fixes the premium through \text{NPV}(P) = m\,P\,\ddot{a}_{x:\overline{n}|} at rate r. When the reserves do not depend on the premium, both sides are linear in P and the equation has a single solution.

Profit Margin of a 5-Year Term Insurance A 5-year term insurance on (60) has a premium of $500 a year and an NPV of 93.00 at a risk discount rate of 10%, as found in Profit Testing. Mortality follows the SULT at 5%, so {}_tp_{60} for t = 1 to 4 is 0.996602, 0.992823, 0.988619 and 0.983943. Find the profit margin.

Answer The premium paid at time t is discounted at 10% and weighted by the probability that the policy is still in force to pay it.

\begin{align*}
\text{EPV} &= 500\Big(1 + \tfrac{0.996602}{1.1} + \tfrac{0.992823}{1.1^2} \\
&\qquad + \tfrac{0.988619}{1.1^3} + \tfrac{0.983943}{1.1^4}\Big) \\
&= 500(1 + 0.906002 + 0.820515 + 0.742764 + 0.672046) \\
&= 500(4.141327) \\
&= 2{,}070.66 \\
\text{Profit margin} &= \frac{93.00}{2{,}070.66} \\
&= 4.49\%
\end{align*}

The policy earns about 4.5 cents of present value for every dollar of premium, measured at the shareholders' required rate of 10%.

The Premium That Gives a 10% Profit Margin For the same policy, each extra dollar of premium raises the NPV at 10% by 3.8710, because each year's profit rises by 0.97(1.06) = 1.0282 in proportion to the policies in force. The reserves do not depend on the premium. Find the premium that gives a profit margin of 10%.

Answer The NPV is 93.00 + 3.8710(P - 500), and the EPV of premiums is 4.1413\,P. Setting the NPV equal to 10% of the EPV gives a linear equation in P.

\begin{align*}
93.00 + 3.8710(P - 500) &= 0.41413\,P \\
3.8710\,P - 0.41413\,P &= 1{,}935.50 - 93.00 \\
3.45687\,P &= 1{,}842.50 \\
P &= 533.00
\end{align*}

Check: at $533.00, the NPV is 93.00 + 3.8710(33.00) = 220.74 and the EPV of premiums is 533.00(4.1413) = 2{,}207.31, a ratio of 10.0%. Raising the premium by $33 more than doubles the NPV, because almost all of the extra premium is profit.

How it appears on the exam

  • ALTAM asks candidates to calculate the profit margin alongside the NPV, IRR and discounted payback period of a profit test, for traditional, universal life and equity-linked policies.
  • Premium from a target: The Profit Analysis objectives also ask for premiums based on a specified profit objective, and a target profit margin is one such objective.
  • Comparisons: A question may compare the margin under best-estimate and stress-test assumptions, or for two premium levels.

Exam traps

  • The rate in the denominator: The EPV of premiums is taken at the risk discount rate r, the same rate as the NPV. Valuing the premiums at the earned rate or the reserve rate mixes two bases in one ratio.
  • Premiums in advance: Premiums are paid at times 0 to n-1, so the first premium is not discounted and the last is weighted by {}_{n-1}p_x. Shifting them to the ends of the years understates the denominator.
  • Survival in the denominator: A premium is received only from a policy still in force. Using P\,\ddot{a}_{\overline{n}|} without the survival factor overstates the premiums and understates the margin.
  • The right numerator: The numerator is the NPV of the profit signature, not the NPV of premiums less benefits. Expenses and reserves must already be in it.