Introduction to Actuarial Science Week 3 PDF

INTRODUCTION TO ACTUARIAL SCIENCE WEEK 3 1  Recap Annuities Types Ordinary Annuity 2  Find the future value of the cash flow of $600 a month for 5 years at 9% interest compounded monthly. If Sandra contributes GH...

INTRODUCTION TO ACTUARIAL SCIENCE WEEK 3 1  Recap Annuities Types Ordinary Annuity 2  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? 3  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? 4  What is the current value of an annuity of $7,500 paid at the end of each half-year for 10 years in an account bearing 11% compounded annually? Discussion Jenny needs $20,000 in the near future. She is saving up $875 at the end of each month in an account bearing 15% interest compounded annually. When will the $20,000 be available? 5 Focus Annuity due Deferred annuity Perpetuity 6 Annuity due: future and current values Annuity due is just like an ordinary annuity except that the payments occur at the beginning of each payment term instead of at the end. Insurance premiums and property rentals are typical examples of annuities due. 7 Annuity due: future values 1+𝑟 𝑛 −1 𝐹𝑉𝑑 = 𝐴 (1 + 𝑟) 𝑟 Example If Simon deposits $200 at the beginning of each week for 2 years in an account that pays 2.5% compounded weekly, how much would he have at the end of the year? 8 Annuity due: future values Solution The weekly rate = 0.025/52 = 0.00096 n = 52 × 2 = 104. 9 Annuity due: current values 1− 1+𝑟 −𝑛 𝐶𝑉𝑑 = 𝐴 (1 + 𝑟) 𝑟 Example What is the current value of an annuity due for an investment fund paying 5% interest compounded monthly if $750 is deposited at the beginning of each month for 3 years? 10 Annuity due: future and current values (Table Method) 𝐹𝑉𝑑 = 𝐴. 𝑆𝑛 ¬𝑟. (1 + 𝑟) 𝐶𝑉𝑑 = 𝐴. 𝑎𝑛 ¬𝑟. (1 + r) 11 Annuity due: future and current values (Table Method) If Robert and his wife deposit $6,600 at the beginning of each year for 5 years at an interest rate of 7% annually, how much will they receive at the end of the 5 years using the table method. What is the current value of an annuity due for an investment fund if the interest rate is 6% compounded semiannually and the deposit is $4,500 twice a year for 5 years. 12 Try Jack and his wife will deposit $550 in their savings account at the first day of each month for the next 5 years in order to have a large enough down payment to buy a larger house. If their savings account pays 7% interest compounded monthly, how much down payment will they collect? 13  A deferred annuity is an ordinary annuity except that the payoff starts at a later time. The first payment is made after a certain time has been passed according to the financial Deferred contract. Annuity This time is called the deferment period. For example, if the first payment is made at the fifth payment period, the annuity would be deferred four payment periods. 14 Future And Current Values Of A Deferred Annuity Future Value Using the table value, the future value of a deferred annuity is given by 𝐹𝑉𝑑𝑒𝑓 = 𝐴. 𝑆𝑛 ¬𝑟 15 Future And Current Values Of A Deferred Annuity Example Find the future value of an annuity of $500 payable quarterly for 2 years where the first payment is made at the end of 6 months given that the interest rate is 8% compounded quarterly. 16 Future And Current Values Of A Deferred Annuity Solution A = $500; r = 0.08/4 = 0.02; n = 8 𝐹𝑉𝑑𝑒𝑓 = 500. 𝑆8 ¬0.02 17 Future And Current Values Of A Deferred Annuity Current Value 𝑪𝑽𝒅𝒆𝒇 = 𝑨. 𝒂𝒏 ¬𝒓 (𝟏 + 𝒓)−𝒅 where d is the deferment period Example Find the current value of an annuity of $800 payable quarterly for 2 years where the first payment is made at the end of 6 months given that the interest rate is 8% and the deferment period is 1. 18  Future And Current Values Of A Deferred Annuity Solution A = $800; r = 0.08/4 = 0.02; n = 8, d = 1 𝐶𝑉𝑑𝑒𝑓 = 800. 𝑎8 ¬0.02 (1 + 0.02)−1 19  A perpetuity is an annuity with a payment that goes on forever. Endowment accounts and philanthropic funds are all examples of perpetuities. Perpetuities Thus, the formula is given by 𝐴 = 𝑟. 𝐶𝑉∞ 𝐴 𝑟 = 𝐶𝑉∞ 𝐴 𝐶𝑉∞ = 𝑟 20 Perpetuities Example An amount of $750,000 was given by a former professor’s wife to his department for setting up a new scholarship in his name. The department invested the gift at 10% interest compounded annually. Find the size of the annual scholarship. 21 Perpetuities Solution 𝐴 = 𝑟. 𝐶𝑉∞ r= 0.1, 𝐶𝑉∞ = 750000 𝐴 = 0.1. 750000 = 75000 annual scholarship fund 22  Try A family left a portion of its fortune to a local church, which invested the fund at 8% interest compounded semiannually. If the church receives $2,650 a month as payment from that fund, Find the original family gift to the church. 23 Solution 𝐴 𝐶𝑉∞ = 𝑟 A = 2,650 *6 = 15900 r = 0.08/2 = 0.04 15900 𝐶𝑉∞ = 0.04 ◦ = 397500 24  Find the current value of an annuity due of $900 each week for 1.5years at 8% interest compounded weekly (use the table) Try A man wishes to give his choice of charities $150 each quarter for an indefinite time. He has $5,000 to invest in a perpetuity for this purpose. What should the interest rate be for him to make his wish come true? 25