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Homework 1 — Actuarial Foundations

This assignment reinforces the foundational concepts introduced in Class 1. Show all formulas used. Numerical answers should be clearly labeled.

Unless otherwise stated:

  • Time is measured in discrete annual periods
  • Payments occur at the end of each period
  • Interest rates are annual effective rates

Problem 1 — Probability (Indicator Variables)

Let \(I\) be an indicator variable equal to \(1\) if a policyholder dies during the year and \(0\) otherwise.

  1. Write the definition of \(I\) using a piecewise function.
  2. If \(P(\text{death}) = 0.02\), compute \(E[I]\).

Problem 2 — Probability (Expectation)

Let \(X\) represent the benefit payment in a one-year term life insurance contract:

  • \(X = 100{,}000\) if the insured dies during the year
  • \(X = 0\) otherwise

If the probability of death during the year is \(q = 0.01\):

  1. Write the probability distribution of \(X\)
  2. Compute \(E[X]\)

Problem 3 — Mortality and Survival (Single Period)

At age \(x\), the probability of death during the year is \(q_x = 0.015\).

  1. Compute the one-year survival probability \(p_x\)
  2. Interpret \(p_x\) in words

Problem 4 — Mortality and Survival (Multiple Periods)

Suppose the following mortality rates apply:

  • \(q_9 = 0.10\)
  • \(q_{10} = 0.20\)

An individual is alive at the beginning of age 9.

  1. Compute the probability that the individual survives to the end of age 10
  2. Clearly show the formula used for \({}_2 p_9\)

Problem 5 — Time Value of Money (Present Value)

Assume an annual effective interest rate of \(i = 5\%\).

A payment of \(1{,}000\) will be made at the end of each of the next 3 years.

  1. Write the present value formula
  2. Compute the numerical present value

Problem 6 — Time Value of Money (Changing Interest Rates)

Suppose interest rates vary by year:

  • Year 1: \(i_1 = 4\%\)
  • Year 2: \(i_2 = 5\%\)
  • Year 3: \(i_3 = 6\%\)

A payment of \(500\) is made at the end of each year.

  1. Write the present value formula using year-specific discount factors
  2. Compute the present value

Problem 7 — Life Insurance (Expected Present Value)

Consider a one-year term life insurance contract that pays a benefit \(B\) at the end of the year of death.

Assume: - \(q_x = 0.01\) - \(i = 5\%\)

  1. Write the expected present value (EPV) of the benefit
  2. Express the EPV in terms of \(B\)

(No need to solve for a numerical value of \(B\).)

Problem 8 — Life Insurance (Premium Calculation)

Using the setup from Problem 7, suppose the level premium \(P\) is paid at the end of the year only if the insured is alive.

Under the equivalence principle:

  1. Write the equation that equates the EPV of premiums to the EPV of benefits
  2. Solve for the premium \(P\) in terms of \(B\)

Problem 9 — Annuity (Expected Present Value)

Consider a life annuity-immediate that pays \(A\) at the end of the year if the annuitant survives the year.

Assume:

  • \(p_x = 0.98\)

  • \(i = 5\%\)

  • Write the expected present value of the annuity payment

  • Express the EPV in terms of \(A\)

Problem 10 — Life Insurance vs Annuity (Conceptual)

Answer the following in words, no calculations required.

  1. Which probability is central to valuing life insurance contracts?
  2. Which probability is central to valuing annuity contracts?
  3. Explain why life insurance and annuities hedge opposite risks.

Problem 11 — Python Exercise (Required)

In this problem, you will write a small, reusable Python function that computes the present value (PV) of a sequence of deterministic cash flows under a constant annual effective discount rate.

This function will serve as a foundational building block for later topics, including annuities, bonds, and insurance cash-flow modeling.

11.1 Business Requirement (BRD)

Objective

Design a Python function that converts a sequence of future cash flows into a single present value using standard time value of money principles.

The function should be generic, readable, and reusable.

11.2 Function Specification

Item Description
Function name present_value
Purpose Compute the present value of a list of future cash flows
Discounting Annual effective interest rate
Timing Cash flows occur at the end of each year

11.3 Inputs

Parameter Type Description
cash_flows list[float] Cash flows paid at the end of each year
rate float Annual effective discount rate (e.g., 0.05)

Assumptions

  • The first cash flow occurs at the end of year 1
  • The discount rate is constant across all years
  • Cash flows may be positive or zero

11.4 Output

Output Type Description
Present Value float Discounted value of all cash flows at time 0

11.5 Mathematical Definition

Let:

  • \( CF_t \) = cash flow at the end of year \( t \)
  • \( i \) = annual effective interest rate

The present value is defined as:

\[ PV = \sum_{t=1}^{n} \frac{CF_t}{(1 + i)^t} \]

11.6 Implementation Requirements

Your Python function must:

  • Use a loop or list comprehension
  • Apply the correct discount factor for each year
  • Return a single numeric value
  • Be clearly written and commented

11.7 Test Case (Required)

Use the following inputs to test your function:

  • Cash flows: [100, 100, 100]
  • Discount rate: 0.05

You should:

  1. Call your function using these inputs
  2. Print or display the resulting present value

11.8 Deliverables

Include all of the following in your submission:

  1. The Python function definition
  2. The test code that calls the function
  3. The numerical output of the test case

11.9 Example Output (for reference)

Your numerical result should be close to (but not necessarily exactly): \(272.32\)

Submission Notes

  • Show formulas clearly before substituting values
  • Clearly label all answers
  • Python code should be readable and commented

This assignment prepares you for valuation techniques used in later classes.