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Module 3 ANNUITY Engr. Gerard Ang School of EECE

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Annuity Annuity – it is a series of equal cash flows occurring each period over a range of periods. Types of Annuity: 1. Ordinary Annuity 2. Deferred Annuity 3. Annuity Due 4. Perpetuity

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Ordinary Annuity Ordinary Annuity – is a series of equal payments or receipts occurring over a specified number of periods with the payments or receipts occurring at the end of each period. It is also referred as annuity-immediate. 𝐏=𝐀 𝟏− (𝟏+𝐢) −𝐧 𝐢 𝐅=𝐀 (𝟏+𝐢) 𝐧 −𝟏 𝐢 Where: P = present worth A = series of periodic equal payments n = number of interest period i = interest rate per interest period

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**P/A and A/P Factors: Notation and Equations**

Name Find/Given Factor Formula Standard Notation Equation Excel Functions (P/A,i,n) Uniform-series present worth P/A 1− (1+𝑖) −𝑛 𝑖 P = A(P/A,i,n) PV(i%,n,A) (A/P,i,n) Capital recovery A/P 𝑖 1− (1+𝑖) −𝑛 A = P(A/P,i,n) PMT(i%,n,P)

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**F/A and A/F Factors: Notation and Equations**

Name Find/Given Factor Formula Standard Notation Equation Excel Functions (F/A,i,n) Uniform-series compound amount F/A (1+𝑖) 𝑛 −1 𝑖 F = A(F/A,i,n) FV(i%,n,A) (A/F,i,n) Sinking fund A/F 𝑖 (1+𝑖) 𝑛 −1 A = F(A/F,i,n) PMT(i%,n,,F)

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**Sample Problems on Ordinary Annuity**

1. What is the current value of a $50 payment to be made at the end of each of the next three years if the prevailing rate of interest is 7% compounded annually? 2. An obligation of Php20,000 is to be repaid in uniform annual amounts each of which included repayment of the debt and interest over a period of 5 years. If interest is 10% per year, what is the annual payment? 3. Maintenance cost for a small bridge expected to last for 60 years is estimated to be Php1,000 each for the first 5 years, followed by a Php10,000 expenditure in the 15th year and Php10,000 in the 30th year. If interest is 10% per year, what is the equivalent uniform annual cost over the 60 year period?

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**Sample Problems on Ordinary Annuity**

4. What is the equivalent previous worth of Php500 annuity to be paid constantly in 60 years 72 years ago, if annual interest is 1%? 5. Find the annual payment to extinguish a debt of Php10,000 payable for 5 years at 12% interest. 6. A savings loan is made between a man and banker. What should be the uniform monthly payment that the man should make if he is to borrow Php50,000 and he is to pay in 10 years? Interest is taken as 6% compounded quarterly. 7. What annuity is required over 10 years to equate with the future amount of Php15,000. Assume i = 5%.

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Deferred Annuity Deferred Annuity – are annuities that are computed on different present year and/or future year. It is annuity where the first payment is made several periods after the beginning of the annuity. Where: k = number of deferred periods

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**Methods of Solving Deferred Annuity Problems**

1. Draw the cash flow diagram. 2. Select any convenient focal date. Temporary focal date is used to convert deferred annuity to ordinary annuity Final focal date is used to obtained the required value. 3. Project all values to temporary focal date. 4. Obtain the final value. 𝐏′=𝐀 𝟏− (𝟏+𝐢) −𝐧 𝐢 Where: k = number of deferred periods 𝐏=𝐏′ (𝟏+𝒊) −𝒌

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**Sample Problems on Deferred Annuity**

1. Find the value of x in the cash flow diagram, given that would make the equivalent present worth of the cash flow diagram to Php22,000 and interest rate is at 13% per year.

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**Sample Problems on Deferred Annuity**

2. Determine the uniform annual payments which would be equivalent to the cash flow diagram given. Interest rate of 12% per year.

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Annuity Due Annuity Due – is a series of equal payments or receipts occurring over a specified number of periods with the payments or receipts occurring at the beginning of each period. 𝐏=𝐀 𝟏− 𝟏+𝐢 −𝐧 𝐢 (𝟏+𝐢) 𝐅=𝐀 𝟏+𝐢 𝐧 −𝟏 𝐢 (𝟏+𝐢) Where: P = present worth A = series of periodic equal payments n = number of interest period i = interest rate per interest period

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**Sample Problems on Annuity Due**

1. What is the current value of a $50 payment to be made at the beginning of each year, for three years if the prevailing rate of interest is 7% compounded annually? 2. What is the accumulated value of a $25 payment to be made at the beginning of each of the next three years if the prevailing rate of interest is 9% compounded annually?

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Perpetuity Perpetuity – are uniform payments which are done infinitely. It is also called as perpetual annuity.

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Types of Perpetuity 1. Ordinary Perpetuity – first payment is done one period after the focal date. 2. Deferred Perpetuity – first payment is done several periods after the focal date. 𝐏= 𝐀 𝐢 𝐏= 𝐀 𝐢 (𝟏+𝒊) −𝒌 Where: k = number of deferred periods

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**Sample Problems on Perpetuity**

1. How much should Mr. Sy invest on a bank that offers 10% interest so that he would earn Php1,000 each year in perpetuity. 2. Don Jose deposited Php5,000,000 on a bank that earns 10% compounded annually. Five years later he died. His will states that his beneficiary is an orphanage which will be receiving the money in perpetuity a year after he died. How much is the yearly fund the orphanage will be receiving? 3. If money is worth 8% compounded quarterly, compute the present value of the perpetuity of Php1,000 payable quarterly.

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