**Pensions and Annuities** Joe Taylor

**Basic Outline** • Quick overview of pensions and how they came to be • Annuities defined and studied • How to calculate an annuity • Examples from Actuarial Tests for Exam FM/2 on annuity calculations

**What is a Pension?** • An arrangement to provide people with an income when they are no longer earning a regular income from employment • Often in form of life annuity • Employment based pensions • Social and State pensions • Disability pensions

**Defined Benefit Plans ** • Benefit is calculated by a predetermined formula • Dollars Times Service Plan • If the plan states that the person will get $100 per month for each year of service, a person with 30 years of service would get $3000 per month • Final Salary Plan • Final Average Pay • Interest Problems • Early Retirement provisions

**Funded vs. Unfunded** • Unfunded: no assets are set aside • Usually paid for out of current taxes • U.S. Social Security • Funded: contributions are regularly made by the employer or employee prior to the maturity of the pension • Most private benefit plans

**Defined Contribution Plans** • Contributions made into accounts for each individual • Usually reinvested into a stock market • Retirees usually cash out at retirement and purchase an annuity • Risk/Reward situation • IRAs 401(k)s

**Pensions in USA** • “promises” made to veterans of Revolutionary and Civil War • Eventually were offered by local governments in late 1800’s • Popular during World War II • Wage freezes prohibited increases in pay • World wide crisis: aging population + lower birth rates =big trouble

**Annuities** • A financial contract in the form of an insurance product according to which an issuer makes a series of future payments to a buyer in exchange for immediate payment of a lump sum (or a series of payments) prior to the onset of the annuity

**Annuity Immediate** • Payments made at end of year • v= present value of first payment • n= number of payments • i=interest rate • d= discount rate

**Example**

**Future Value Annuity Immediate** Perpetuities

**Actuarial Exam Question** • A perpetuity immediate pays X per year. Brian receives the first n payments, Colleen receives the next n payments, and Jeff receives the remaining payments. Brian’s share of the present value of the original perpetuity is 40%, and Jeff’s share is K. • Calculate K • Answer: 36%

**Annuity Due** • Very similar to annuity immediate, but payments are made at the beginning of each time period • Multiply annuity immediate by i/d=1+i

**Example** • Given i=5% and n=10, find annuity due present value. • Notice this is different than previous answer of 7.7217

**Sample Actuarial Question** • Kathryn deposits 100 into an account at the beginning of each 4-year period for 40 years. The account credits interest at an annual effective interest rate of i. • The accumulated amount in the account at the end of 40 years is X, which is 5 times the accumulated amount in the account after 20 years. • Find X • Answer: 6195

**Increasing Annuities with Terms in Arithmetic Progression** An annuity where payments increase by 1 unit(or more) after each payment If i= 5% and n= 4

**Decreasing Annuities with Terms in Arithmetic Progression** • This is an annuity where each payment decreases by an equal amount from the previous payment • All other formulas for the decreasing annuity and increasing annuity can be derived from these two equations

**Actuarial Example** • The present value of a 25-year annuity-immediate with a first payment of 2500 and decreasing by 100 each year thereafter is X. Assuming an annual effective interest rate of 10%, calculate X • Answer:

**Annuities with Terms in Geometric Progression** • An annuity where each payment increases by g% over the previous payment • Using geometric series(we all learned it but probably forget):

**One Last Example** • Given i=10%, find PV of sequence of payments of

**Works Cited** • }http://en.wikipedia.org/wiki/Pension • }http://en.wikipedia.org/wiki/Life_annuity • }Emms, Paul. "Income Drawdown Schemes for a Defined Contribution Pension Plan." The Journal of Risk and Insurance 75.3 (2008): 739-61. Print. • }Horneff, Wolfram. "Optimal Gradual Annuitization: Quantifying the Costs of Switching to Annuities." The Journal of Risk and Insurance75.4 (2008): 1019-037. Print. • }Antolin, Pablo. "Private Pensions and THe Financial Crisis: How to Ensure Adequate Retirement Income from DC Pension Plans."OECD Journal 2 (2010): 153-71. Print. • }Hassett, Matthew, Toni Garcia, and Amy Steeby.ACTEX Study Manual SOA Exam 2. United States: ACTEX Publications, 2010. Print.