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**Annuities and Sinking Funds**

Chapter Thirteen Annuities and Sinking Funds McGraw-Hill/Irwin Copyright © 2014 by The McGraw-Hill Companies, Inc. All rights reserved. 13-

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**Learning unit objectives**

Differentiate between contingent annuities and annuities certain. Calculate the future value of an ordinary annuity and an annuity due manually and by table lookup. LU13-1: Annuities: Ordinary Annuity and Annuity Due (Find Future Value) LU 13-2: Present Value of an Ordinary Annuity (Find Present Value) Calculate the present value of an ordinary annuity by table lookup and manually check the calculation. Compare the calculation of the present value of one lump sum versus the present value of an ordinary annuity. LU 13-3: Sinking Funds (Find Periodic Payments) Calculate the payment made at the end of each period by table lookup. Check table lookup by using ordinary annuity table. 13-

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**Compounding Interest (Future Value)**

Term of the annuity – The time from the beginning of the first payment period to the end of the last payment period Annuity – A series of payments Present value of an annuity – Tthe amount of money needed to invest today in order to receive a stream of payments for a given number of years in the future Future value of annuity – The future dollar amount of a series of payments plus interest

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**Future value of an annuity of $1 at 8% (Figure 13.1)**

$3.2464 $2.0800 $1.00 End of period

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**Classification of Annuities**

Contingent annuities – have no fixed number of payments but depend on an uncertain event Annuities certain – have a specific stated number of payments Life Insurance payments Mortgage payments

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**Classification of Annuities**

Ordinary annuity – regular deposits/payments made at the end of the period Annuity due – regular deposits/payments made at the beginning of the period Jan. 31 Monthly Jan. 1 June 30 Quarterly April 1 Dec. 31 Semiannually July 1 Dec. 31 Annually Jan. 1

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**Calculating Future Value of an Ordinary Annuity Manually**

Step 1. For period 1, no interest calculation is necessary, since money is invested at the end of the period. Step 2. For period 2, calculate interest on the balance and add the interest to the previous balance. Step 3. Add the additional investment at the end of period 2 to the new balance. Step 4. Repeat Steps 2 and 3 until the end of the desired period is reached.

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**Calculating Future Value of an Ordinary Annuity Manually**

Find the value of an investment after 3 years for a $3,000 ordinary annuity at 8%.

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**Calculating Future Value of an Ordinary Annuity by Table Lookup**

Step 1. Calculate the number of periods and rate per period. Step 2. Look up the periods and rate in an ordinary annuity table. The intersection gives the table factor for the future value of $1. Step 3. Multiply the payment each period by the table factor. This gives the future value of the annuity. Future value of = Annuity payment x Ordinary annuity ordinary annuity each period table factor

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**Ordinary annuity table: Compound sum of an annuity of $1 (Table 13.1)**

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**Future Value of an Ordinary Annuity**

Find the value of an investment after 3 years for a $3,000 ordinary annuity at 8%. Periods (N) = 3 x 1 = 3 Rate (R) = 8%/1 = 8% (table factor) x $3,000 = $9,739.20

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**Calculating Future Value of an Annuity Due Manually**

Step 1. Calculate the interest on the balance for the period and add it to the previous balance. Step 2. Add additional investment at the beginning of the period to the new balance. Step 3. Repeat Steps 1 and 2 until the end of the desired period is reached.

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**Calculating Future Value of an Annuity Due Manually**

Find the value of an investment after 3 years for a $3,000 annuity due at 8%.

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**Calculating Future Value of an Annuity Due by Table Lookup**

Step 1. Calculate the number of periods and rate per period. Add one extra period. Step 2. Look up in an ordinary annuity table the periods and rate. The intersection gives the table factor for the future value of $1. Step 3. Multiply the payment each period by the table factor. Step 4. Subtract 1 payment from Step 3.

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**Future Value of an Annuity Due**

Find the value of an investment after 3 years for a $3,000 annuity due at 8%. Periods (N) = 3 x 1 = = 4 Rate (R) = 8%/1 = 8% (table factor) x $3,000 = $13,518.30 $13, $3,000 = $10,518.30

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**Present value of an annuity of $1 at 8% (Figure 13.2)**

$2.5771 $1.7833 $.9259 Number of periods

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**Calculating Present Value of an Ordinary Annuity by Table Lookup**

Step 1. Calculate the number of periods and rate per period. Step 2. Look up the periods and rate in the present value of an annuity table. The intersection gives the table factor for the present value of $1. Step 3. Multiply the withdrawal for each period by the table factor. This gives the present value of an ordinary annuity . Present value of Annuity Present value of ordinary annuity payment payment ordinary annuity table = x

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**Present Value of an Annuity of $1 (Table 13.2)**

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**Present Value of an Annuity**

John Fitch wants to receive a $8,000 annuity in 3 years. Interest on the annuity is 8% semiannually. John will make withdrawals at the end of each year. How much must John invest today to receive a stream of payments for 3 years. Interest ==> Payment ==> End of Year 3 ==> N = 3 x 1 = 3 periods R = 8%/1 = 8% (table factor) x $8,000 = $20,616.80

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**Lump Sums versus Annuities**

John Sands made deposits of $200 to Floor Bank, which pays 8% interest compounded annually. After 5 years, John makes no more deposits. What will be the balance in the account 6 years after the last deposit? Step 1. Future value of an annuity N = 5 x 2 = 10 periods R = 8%/2 = 4% Step 2. Future value of a lump sum (table factor) x $200 = N = 6 x 2 = 12 periods $2,401.22 R = 8%/2 = 4% (table factor) x $2, = $3,844.35

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**Lump Sums versus Annuities**

Mel Rich decided to retire in 8 years to New Mexico. What amount must Mel invest today so he will be able to withdraw $40,000 at the end of each year 25 years after he retires? Assume Mel can invest money at 5% interest compounded annually. Step 1. Present value of an annuity Step 2. Present value of a lump sum N = 8 x 1 = 8 periods N = 25 x 1 = 25 periods R = 5%/1 = 5% R = 5%/1 = 5% .6768 x $563,756 = $381,550.06 x $40,000 = $563,756

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**Sinking Funds (Find Periodic Payments)**

financial arrangement that sets aside regular periodic payments of a particular amount of money Sinking fund = Future x Sinking fund payment value table factor

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**SINKING FUND TABLE BASED ON $1**

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Sinking Fund To retire a bond issue, Moore Company needs $60,000 in 18 years from today. The interest rate is 10% compounded annually. What payment must Moore make at the end of each year? Use Table 13.3. N = 18 x 1 = 18 periods R = 10%/1 = 10% x $60,000 = $1,314 Check Future Value of an annuity table N = 18, R= 10% $1,314 x = $59,917.35* * Off due to rounding

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