Introduction to Actuarial Science Week 2 PDF

INTRODUCTION TO ACTUARIAL SCIENCE WEEK 2 1  Core areas Definition of and scope application Recap Simple and Role of Compound Actuaries Inter...

INTRODUCTION TO ACTUARIAL SCIENCE WEEK 2 1  Core areas Definition of and scope application Recap Simple and Role of Compound Actuaries Interest 2  Recap If you invest $1,000 at 7% interest for 3 years, how much would you accumulate if the compounding occurs annually; quarterly; monthly and weekly? Hint 𝑟 𝑛𝑡 𝐴 = 𝑃 1 + 𝑛 3 Focus Annuities Types Ordinary Annuity 4 Annuity Any set of equal payments made at equal intervals of time. Example periodic savings or investment deposits purchases on installment mortgages life insurance premiums Social Security deductions 5 Types of Annuities 6  Term of annuity: the time between the beginning of the first interval and the end of the last interval. For example, a 3-year term would be between January 1, 2009 and December 31, 2011. Annuity certain: an annuity for which the beginning and end of the term are designated and recognised in Types of advance. Example a mortgage or car loan. Contingent annuity: an annuity for which the beginning Annuities of the term is known but the end is contingent upon a certain event. An example is life insurance. Payment of the premium starts at the time of purchase and continues while the insured is alive, and the end occurs at the time of the insured’s death. Perpetuity: an annuity for which the beginning of the term is known but the end is infinite. 7  Future Value Of An Ordinary Annuity Recall from previous week; 𝐹𝑉 𝑃𝑉 = (1+𝑟)𝑡 Thus 𝐹𝑉 = 𝑃𝑉 (1 + 𝑟)𝑡 Let a payment of a quarterly annuity be A and assume a term of annuity (n) of 1 year. The first payment would be due at the end of the first quarter and the last would be due at the end of the last quarter. 8 Future Value Of An Ordinary Annuity 1st Quarter: 𝐹𝑉 = 𝐴(1 + 𝑟)3 2nd = 𝐴(1 + 𝑟)2 3rd = 𝐴(1 + 𝑟)1 = 𝐴(1 + 𝑟) 4th = 𝐴(1 + 𝑟)0 = 𝐴 Therefore, FV= 1ST + 2nd+ 3rd+ 4th Quarter 𝐹𝑉 = 𝐴(1 + 𝑟)3 + 𝐴(1 + 𝑟)2 +𝐴 1 + 𝑟 + 𝐴 9  Future Value Of An Ordinary Annuity Thus, the future value of an ordinary annuity with n terms of payment is given by 𝑨 (𝟏+𝒓)𝒏 −𝟏 𝑭𝑽 = 𝒓 Example At age 25 Adam started to contribute to his retirement account by making yearly contributions of $1200. If his employer pays 6% interest compounded annually, how much will he collect when he retires at age 65, and how much will he make on his investment? 10 Future Value Of An Ordinary Annuity 𝐴 (1+𝑟)𝑛 −1 𝐹𝑉 = 𝑟 1200 (1+0.06)40 −1 𝐹𝑉 = 0.06 ◦ = $185714. 36 11 Example Kelvin wanted to deposit $150 a month in an account bearing 4% interest compound quarterly. It is for his 6-year-old son, who would cash it when he starts his college education at age 18. How large will the son’s education fund be? Solution A is quarterly and is equal to 150 × 3 = 450 r is on a quarterly rate: 0.04/4 = 0.01 n is measured by quarters = (18−6) × 4 = 48 12  Try At age 25, Eben started to make payment for a mortgage by making monthly contributions of $200. If the interest rate is 3% compounded semiannually, how much will he have paid by the age of 50? 13 Solution A is semiannually and is equal to 200 × 6 = 1200 r is on a semiannual rate: 0.03/2 = 0.015 n is measured by semiannual = (50−25) × 2 = 50 Hence, 1200 (1+0.015)50 −1 𝐹𝑉 = 0.015 = $88419.39 14 FV Using Tables The future value formula can therefore be rewritten as 𝑭𝑽 = 𝑨. 𝑺𝒏 ¬𝒓 For example find; 1. 𝐹𝑉 = 1200. 𝑆30 ¬0.06 2. 𝐹𝑉 = 450. 𝑆28 ¬0.01 15 Current Value Of An Ordinary Annuity We can obtain the formula for the discounted future value of an ordinary annuity by reversing the amounts of the payments. 𝐹𝑉 Recall; 𝑃𝑉 = (1+𝑟)𝑡 Thus, For n number of payments, CV (PV) is given as 𝑨[𝟏−(𝟏+𝒓)−𝒏 ] 𝑪𝑽 = 𝒓 16 Current Value Of An Ordinary Annuity Example A man wants to cash in his trust fund, which pays him $500 a month for the next 10 years. The interest on the fund is 6.5% compounded monthly. How much would he receive? Solution The monthly interest rate is 0.065/12 = 0.0054; the monthly term (n) is 10 × 12 = 120. 500[1−(1+0.0054)−120 ] 𝐶𝑉 = 0.0054 17 Current Value Of An Ordinary Annuity Try What is the current value of an annuity involving $3,750 payable at the end of each quarter for 7 years at an interest rate of 8% compounded quarterly? 18  Current Value Of An Ordinary Annuity Solution The quarterly rate is 0.08/4 = 0.02 The term in quarters: 7 × 4 = 28 19 Current Value Of An Ordinary Annuity 𝑪𝑽 = 𝑨. 𝒂𝒏 ¬𝒓 For the previous example; 𝐶𝑉 = 𝐴. 𝑎28 ¬0.02 ◦ = 3750 * (21.281272) 20  Try Find the following; a) 750. 𝑎15 ¬0.08 b) 3900. 𝑎32 ¬0.015 c) 1300. 𝑎6 ¬0.1 21 Finding The Payment Of An Ordinary Annuity (A) Future Value (FV) 𝐴 (1+𝑟)𝑛 −1 𝐹𝑉 = 𝑟 With change of subject 𝑭𝑽∗𝒓 𝑨= (𝟏+𝒓)𝒏 −𝟏 22 Finding The Payment Of An Ordinary Annuity (A) Current Value 𝑪𝑽∗𝒓 𝑨= 𝟏− (𝟏+𝒓)−𝒏 23 Using the Table Method Future Value (FV) 𝟏 𝑨= 𝑭𝑽. 𝑺𝒏 ¬𝒓 Current Value (CV) 𝟏 𝑨= 𝑪𝑽. 𝒂𝒏 ¬𝒓 24  Jim deposited $18,000 in his savings account. He planned to set an automatic payment to his son. Determine how much Jim’s son would receive as a semiannual payment if the interest rate is 12% compounded semiannually for 6 years. Try Samantha is planning to buy a new car for $17,000 upon her graduation 5 years from now. How much will Samantha’s quarterly deposit be if the interest rate is 8% compounded quarterly for 5 years? (use the table method) 25 Finding The Term Of An Ordinary Annuity The term of annuity (n) can be found as Future Value (FV) 𝒓 𝒍𝒏 𝑭𝑽∗ +𝟏 𝑨 𝒏= 𝐥𝐧(𝟏+𝒓) Current Value (CV) 𝑪𝑽∗𝒓 𝒍𝒏 𝟏− 𝑨 𝒏= 𝐥𝐧(𝟏+𝒓) 26  Example A small business owner wants to buy equipment for GH60,000. He can save GH500 a month for his future purchase. Determine how long it will take him if his account is paying 8% interest compounded annually. 27 Solution The annual rate = 0.08 The deposit (A) is GH500 monthly = GH6,000 annually. 28  Find the future value of the cash flow of $600 a month for 5 years at 9% interest compounded monthly. If Sandra contributes GH630 at the end of each month Try to her retirement account that pays 8 % compounded semiannually, how much will she have when she retires 20 years from the start of contributions? Rosemary would like to buy a chalet overlooking the Alps for $300,000. She can save up to $10,000 a month in an account paying 15% interest compounded monthly. How long will it take her to wait? 29  Try (Use the table Method) Find the future fund for Kelly, who is saving $350 at the end of each month for the next 3 1 years, if her savings account bears 7 % interest 2 compounded quarterly. What is the current value of an annuity of $7,500 paid at the end of each half-year for 10 1 years in an account bearing 9 % compounded 2 annually? 30