Equity-Linked Insurance Contracts: Practice Problems and Solutions

This section will discuss how to approach problems dealing with equity-linked insurance contracts. There are few existing sample problems available on this topic, so this section will attempt to give some practice questions by which students might be able to learn the basic approach for it.

When working with any expression for a maximum, it may sometimes be necessary to transform that expression into an expression representing the payoff of an option. In that case, it is useful to keep in mind the identities:

Formula 80.1: max(AB, C) = B*max(A, C/B)

Formula 80.2: max(A, B) = A + max(0, B - A)

An equity-linked insurance contract will often charge a premium P at the start. In this case, in order for no arbitrage to be possible, the contract's value C(T) at time T = 0 must be equal to P. Remember to set P = C(0) (using whatever values, expressions, or variables for P and C(0) are given) when dealing with this kind of problem.

Problem ELIC1. The price of the stock of Treacherous Industries can be expressed as S(T), where T is some time period. The current (T = 0) price of Treacherous Industries stock is $50 per share. A contract expiring in one year has the following payoff:

S(0)*Max[S(1)/S(0), 1.2]. Which of these portfolios will have an equivalent payoff to that of the contract? All options are European, and more than one answer may be correct. Assume that the risk-free interest rate is zero.

(a) A $50-strike put option expiring in one year and $60.
(b) A $50-strike put option expiring in one year and one share of stock.
(c) A $60-strike put option expiring in one year and $60.
(d) A $60-strike call option expiring in one year and $60.
(e) A $60-strike put option expiring in one year and one share of stock.
(f) A $60-strike call option expiring in one year and one share of stock.

Solution ELIC1. S(0)*Max[S(1)/S(0), 1.2] = Max[S(1), 1.2S(0)]. We have essentially performed multiplication by S(0) on all terms within the Max expression and compensated for it by dividing by S(0) outside the Max expression.

Max[S(1), 1.2S(0)] = Max[S(1), 1.2*50] = Max[S(1), 60] = S(1) + Max[0, 60 - S(1)]. This is the same as the payoff on one share of stock plus one $60-strike put option. So (e) is a correct answer.

But it is also the case that Max[S(1), 60] = 60 + Max[S(1) - 60, 0]. This is the same as the payoff on $60 plus one $60-strike call option. So (d) is a correct answer.

Thus, (d) and (e) are correct answers.

Problem ELIC2. A particular equity-linked insurance contract is based on the price of a stock S(T) at time T and has a value of P*(1 - X)*F[S(T)], where P and (1 - X) are constant and F[S(T)] is some function of the stock price S(T). Here, F[S(T)] = S² + 3S, and S(T) = 50^(T/3). What is the ratio of the contract's value at T = 3 years to its value at T = 1 year?

Solution ELIC2. At T = 3, S(T) = 50^(3/3) = 50, and F[50] = 50² + 3*50 = F[50] = 2650.

At T = 1, S(T) = 50^(1/3) = 3.684031499, and

F[3.684031499] = 3.684031499² + 3*3.684031499 = F[3.684031499] = 24.62418258.

Since P and (1 - X) are constant and are present in the expression for the contract's value in both time periods, they simply cancel out in the division and can be ignored. The ratio we seek is thus F[50]/F[3.684031499] = 2650/24.62418258 = 107.6177855

Problem ELIC3. A particular equity-linked insurance contract is based on the price of a stock S(T) at time T and has a value of P*(1 - X)*F[S(T)], where P and (1 - X) are constant and F[S(T)] is some function of the stock price S(T). Here, F[S(T)] = S² + 3S, and S(T) = 50^(T/3). Now say that P is the premium that one must pay at time T = 0 to enter into the contract and that no arbitrage is possible. Find the value of X.

Solution ELIC3. If no arbitrage is to be possible, the value of the contract at T = 0 must be P. But our expression for the value of the contract also says that its value at T = 0 is P*(1 - X)*F[S(0)]. Thus, we set P = P*(1 - X)*F[S(0)], which means that

1 = (1 - X)*F[S(0)]. Now we find F[S(0)]. S(0) = 50^(0/3) = 1 and F[1] = 1² + 3*1 = 4. Thus, 1 = (1 - X)*4 and ¼ = 1 - X, so X = 0.75

Now we have the practice necessary to attempt an exam-style question.

Problem ELIC4.

Similar to Question 3 from the Society of Actuaries' Sample MFE Questions and Solutions:

A particular equity-linked insurance contract is based on the price of a stock S(T) at time T and has a value of P*(1 - X)*Max[S(T)/S(0), (1 + r)^T]. A premium P is paid for this contract when it is entered into (i.e., now). This contract will reach maturity in 3 years. The minimum guaranteed rate of return r is 0.1. The stock pays no dividend and has a price of $500 per share at time T = 0. Also at time T = 0, the price of one three-year, $665.5-strike European put option on the stock is $55.55. Find the value of X.

Solution ELIC4. First we work with the expression Max[S(T)/S(0), (1 + r)^T], which, for T = 3 is Max[S(3)/S(0), (1.1)³] = Max[S(3)/S(0), 1.331] = [1/S(0)]Max[S(3), 1.331S(0)] = [1/500]Max[S(3), 665.5], since S(0) = 500. We recall the formula

max(A, B) = A + max(0, B - A).

Thus, Max(S(3), 665.5) = S(3) + Max(0, 665.5 - S(3)), from which it follows that

[1/500]Max[S(3), 665.5] = (1/500)(S(3) + Max(0, 665.5 - S(3))).

The premium P will be equal to the present value of the equity-linked insurance contract, so

P = PV(P*(1 - X)*(1/500)(S(3) + Max(0, 665.5 - S(3))) →

The expression (P/500)*(1 - X)*(S(3) + Max[0, 665.5 - S(3)]) applies to time t = 3. How do we get its present value?

The present value of Max(0, 665.5 - S(3)) is just the price of the put option, 55.55.

The present value of S(3) is the current stock price, S(0). The other terms are not affected by present value considerations.

Thus, PV(P*(1 - X)*(1/500)(S(3) + Max(0, 665.5 - S(3))) = (P/500)*(1 - X)*(S(0) + 55.55), and so

P = (P/500)*(1 - X)*(S(0) + 55.55) = (P/500)*(1 - X)*(500 + 55.55).

Thus, 500 = (1 - X)*555.55, implying that

X = 1 - 500/555.55 = X = 0.0999909999.

Problem ELIC5.

Similar to Question 3 from the Society of Actuaries' Sample MFE Questions and Solutions:

A particular equity-linked insurance contract is based on the price of a stock S(T) at time T and has a value of P*(1 - X)*Max[S(T)/S(0), (1 + r)^T]. A premium P is paid for this contract when it is entered into (i.e., now). This contract will reach maturity in 4 years. The minimum guaranteed rate of return r is 0.0064844317. The stock pays no dividend and has a price of $700 per share at time T = 0. Also at time T = 0, the price of one four-year, $900-strike European call option on the stock is $91. Find the value of X.

Solution ELIC5.

This problem is somewhat of a twist on Problem ELIC4and the sample MFE question. Here, instead of being given the value of a put option, we are given the value of a call option, so we will need to transform the Max expression differently.

Max[S(T)/S(0), (1 + r)^T] = Max[S(4)/700, (1.064844317)⁴] = Max[S(4)/700, 9/7] =

(1/700)Max[S(4), 900] = (1/700)(900 + Max[S(4) - 900, 0]) (by Formula 80.2, taking advantage of the fact that we are given a call price rather than a put and trying to anticipate the use of that call price.)

We know that the premium is equal to the price of the contract, which is the present value of its value in four years.

Thus, P = PV(P*(1 - X)*(1/700)(900 + Max[S(4) - 900, 0]))

The present value of 900 in 4 years is 900/(1.064844317)⁴= 700.

The present value of Max[S(4) - 900, 0] is the current price of the call option, or 91. None of the other items in the expression above are affected by present value considerations.

Thus, PV(P*(1 - X)*(1/700)(900 + Max[S(4) - 900, 0])) = P*(1 - X)*(1/700)(700 + 91).

P*(1 - X)*(1/700)(700 + 91) = P*(1 - X)*(791/700).

Hence, P = P*(1 - X)*(791/700), and 1 - X = 700/791, so X = 0.115044248.

See other sections of The Actuary's Free Study Guide for Exam 3F / Exam MFE.