Fixed Indexed Annuity
Insurance companies offer sophisticated retirement planning products to their clients.
One of these products is a “Fixed Indexed Annuity.” In this contract, the investor’s
account has a credited return that is a function of stock index returns.
In many of these contracts the investor’s credited return might be “floored” at zero
and perhaps “capped” at some amount. For example, if the stock market returned
5.00% in a year, the contract makes 5.00% but if the market returns a negative
amount, the credit would be zero.
There is also the concept of a “participation rate.” This means that the investor’s
crediting rate might be some percentage of stock returns. Consider the following
example:
Market Return Uncapped
Crediting Rate
Uncapped
Crediting Rate with
a 75%
Participation Rate
Crediting Rate with
a 5.00% Cap and
75% Participation
Rate
-10.00% 0.00% 0.00% 0.00%
-5.00% 0.00% 0.00% 0.00%
+5.00% +5.00% +3.75% +3.75%
+10.00% +10.00% +7.50% +5.00%
Note that the investor’s money does not actually go into the stock market, their money
stays on deposit at the Insurance Company. Any returns made by the investor are a
liability for the company; hence, the company must hedge this risk.
The various terms of these contracts can be quite complex. In our case, we will keep
it as simple as possible, and then investigate the hedging aspects in this study.
About the Project
This project is an Excel spreadsheet based project. You are welcome to work together
in small groups; however you must turn in your own version of your workbook. Please
don’t collaborate with students from past semesters.
We will be tying together a number of relevant financial topics while investigating the
“replication of risk” concept.
1. Simulate an arbitrary number of Geometric Brownian Motion pathways. You’ll
probably want to simulate at least 1,000 paths. Each path depends on a set of random
numbers, a constant volatility and default-free rates. This is a simulation for the
underlying stock index, S(t) up until a terminal time T. (No dividend yield.)
2. Along each pathway, model the value of a theoretical (Black-Scholes) call option on
S, with a strike K. Also model the delta of this option at each time step. This will
represent the embedded option in the FIA contract. We will be simulating the
performance of uncapped returns with a participation rate.
3. Now replicate the value of the long call by simulated trading the underlying. Also
make sure you keep track of cash. In other words, construct a replicating portfolio.
This is driven by the equation 𝑐 = Δ𝑆 + 𝐵. This is equivalent to:
𝑐𝑎𝑙𝑙 𝑜𝑝𝑡𝑖𝑜𝑛 = (#𝑜𝑓 𝑠ℎ𝑎𝑟𝑒𝑠)($ 𝑝𝑒𝑟 𝑠ℎ𝑎𝑟𝑒) + (𝑐𝑎𝑠ℎ 𝑎𝑐𝑐𝑜𝑢𝑛𝑡)
At each time step, make an appropriate trade that serves to adjust your stock position,
all the while earning interest on credits or paying interest on debits.
4. For each simulated pathway, set aside the terminal value of the replicating portfolio,
the terminal value of the underlying and the terminal value of the theoretical call
option. The difference of the portfolio and the call option is “replicating error.”
• Describe (in words) the distribution of replicating errors.
• Make a plot of this distribution.
• Describe (in words) the value of the replicating portfolio and the value of the
theoretical call as a function of the underlying at time T.
• Make a plot of the value of the replicating portfolio and the value of the
theoretical call as a function of the underlying at time T.
• Investigate the effect of changing the crediting rate, interest rates, volatility,
etc.
