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**PSU Study Session Fall 2010 Dan Sprik**

Exam FM/2 PSU Study Session Fall 2010 Dan Sprik

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**Lesson 2: Annuities Why is it important?**

This lesson is mostly deriving closed mathematical formulas for a series of payments. Knowing these formulas is an essential component of working with Loans (lesson 3) and Bonds (lesson 4) How many questions should I expect on this? Probably of my 35 questions when I took the exam boiled down to knowing an annuity formula (this includes annuities in the context of loans and bonds) How should I learn this material? Learn how to derive the geometric series formula (on the next slide). If you know this formula, you can derive ANY of the annuity formulas you want. It helps to have the formulas memorized, too!

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**THE MOST IMPORTANT FORMULA ON THIS EXAM!!**

Geometric series: 1+𝑟+ 𝑟 2 +…+ 𝑟 𝑛 = 1− 𝑟 𝑛+1 1−𝑟 , 𝑟≠1 Proof: Let 𝑆=1+𝑟+ 𝑟 2 +…+ 𝑟 𝑛 𝑟𝑆=𝑟+ 𝑟 2 + 𝑟 3 +…+ 𝑟 𝑛+1 𝑆−𝑟𝑆=1− 𝑟 𝑛+1 𝑆= 1− 𝑟 𝑛+1 1−𝑟 If 𝑟 <1 then lim 𝑛→∞ 1+𝑟+…+ 𝑟 𝑛 = 1 1−𝑟

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**Basic Annuities Level annual payments at the end of the year.**

Annuity Immediate

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Accumulated Value Accumulated value of level annual payments at the time of final payment. AV= sn┐i% = (1+i)n an┐i% = (1+i)n – 1 i

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**Annuity Due Level annual payments at the beginning of the year.**

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Accumulated Value Accumulated value of level annual payments one year after the time of final payment. AV= 𝑠 n┐i% = (1+i)n än┐i% = (1+i)n – 1 d

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**Relationships än┐i% = (1+i) an┐i% än┐i% = 1 + an-1┐i%**

𝑠 n┐i% = (1+i) sn┐i% 𝑠 n┐i% = sn+1┐i% - 1

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Perpetuities Perpetuity immediate ä ∞ ┐i% = 1 d a ∞ ┐i% = 1 i

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**Useful Variations ä2n┐i% / än┐i% = a2n┐i% /an┐i%**

= ä (m)2m ┐i% / ä (m)n ┐i% = a(m) 2m ┐i% / a(m) n ┐i% = 1 + vn a3n┐i% = (1 + vn + v2n) an┐i%

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Mthly Annuities Remember an m-thly annuity pays 1/m every mth

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**Avoid the Trap!! Mthly perpetuity due ä (m)∞ ┐i% = 1/ d(m)**

Mthly perpetuity immediate a(m) ∞ ┐i% = 1/ i(m)

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**Continuous Annuity Continuous payment at a rate of one per year.**

=( i/ δ) an┐i% = ∫vt dt

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**Increasing Annuities (Ia) n┐i% = (än┐i% - nvn )/ i (immediate)**

(Iä) n┐i% = (än┐i% - nvn )/ d (due) (Is) n┐i% = ( 𝑠 n┐i% - n ) / i (immediate) (I 𝑠 ) n┐i% = ( 𝑠 n┐i% - n ) / d (due)

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**Increasing Annuities (Ia)∞ ┐i% = 1/ i + 1 / i2 (immediate)**

(Iä) ∞ ┐i% = 1 / d2 (due)

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**Decreasing Annuities (Da) n┐i% = ( n - an┐i% ) / i (immediate)**

(Dä) n┐i% = ( n - an┐i% ) / d (due)

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**Geometric Payments (VERY IMPORTANT!!)**

Present value of geometric payment pattern with the first payment of $1 one year from now. PV = ( 1 – {(1 + g )/ (1 + i )}n ) i - g

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SOA May 1988 John took out a 2,000,000 construction loan, disbursed to him in three installments. The first installment of 1,000,000 is disbursed immediately and this is followed by two 500,000 installments in six months intervals. The interest on the loan is calculated at a rate of 15% convertible semiannually and accumulated to the end of the second year. At that time, the loan and the accumulated interest will be replaced by a 30 year mortgage at 12% convertible monthly. The amount for the monthly mortgage payment for the first five years will be one-half of the payment of the sixth and later years. The first monthly mortgage payment is due exactly two years after the initial disbursement of the construction loan. Calculate the amount of the 12th mortgage payment. A) 13, B) 13, C) 16, D) 16,955 E) 25,811

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SOA November 1993 Mark receives 500,000 at his retirement. He invest 500,000 – X in an annual payment 10 year annuity immediate and X in an annual payment perpetuity immediate. His total annual payments received during the first 10 years are twice as large as those received thereafter. The annual effective rate of interest is 6%. Calculate X. A) 345, B) 346, C) 346,835 D) 347,335 E) 348,835

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SOA May 2003 At an annual effective interest rate of i, i>0, both of the following annuities have a present value of X: a 20 year annuity-immediate with annual payments of 55. a 30 year annuity immediate with annual payments that pays 30 per year for the first 10 years, 60 per year for the second 10 years, and 90 per year for the final 10 years. Calculate X. A) B) 585 C) 595 D) E) 615

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SOA May 1998 Jeff deposit 100 at the end of each year for 13 years into Fund X. Antoinette deposit 100 at the end of each year for 13 years into Fund Y. Fund X earns an annual effective interest rate of 15% for the first 5 years and an annual effective rate of 6% thereafter. Fund Y earns an annual effective rate of i. At the end of 13 years, the accumulated value of Fund X equals the accumulated value of Fund Y. Calculate i. A) 6.4% B) 6.7% C). 7.0% D) 7.4% E). 7.8%

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SOA May 1990 A 20 year annuity pays 100 every other year beginning at the end of the second year, with additional payments of 300 each at the end of years 3, 9, and 15. The effective annual interest rate is 4%. Calculate the present value of the annuity. A) B) C) D) E) 1430

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SOA May 1998 Francois purchases a 10 year annuity immediate with annual payments of 10X. Jacques purchases a 10 year decreasing annuity immediate which also makes annual payments. The payment at the end of year one is equal to 50. At the end of year 2, and at the end of each year through year 10, each subsequent payment is reduced over what was paid in the previous year by an amount equal to X. At an annual effective rate of 7.072%, both annuities have the same present value. Calculate X, where X<5. A) B) C) D) E) 3.69

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CAS May 1999 An annuity provides for 12 annual payments. The first payment is 100, paid at the end of the first year, and each subsequent payment is 5% more than the one preceding it. Calculate the present value of this annuity if i=7%.

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ANNUITIES Sequences and Series

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