### Chapter 13

### Annuities and

### Annuities and

### Sinking Funds

### Sinking Funds

### 1.

### Differentiate between contingent

### annuities and annuities certain

### 2.

### Calculate the future value of an

### ordinary annuity and an annuity

### due manually and by table lookup

### Annuities and Sinking Funds

**#13**

**#13**

**Learning Unit Objectives**

**Annuities: Ordinary Annuity and **

**Annuity Due (Find Future Value)**

**LU13.1**

**LU13.1**

### 1.

### Calculate the present value of an ordinary

### annuity by table lookup and manually

### check the calculation

### 2.

### Compare the calculation of the present

### value of one lump sum versus the present

### value of an ordinary annuity

### Annuities and Sinking Funds

**#13**

**#13**

**Learning Unit Objectives**

**Present Value of an Ordinary Annuity **

**(Find Present Value)**

**LU13.2**

**LU13.2**

### 1.

### Calculate the payment made at the end

### of each period by table lookup

### 2.

### Check table lookup by using ordinary

### annuity table

### Annuities and Sinking Funds

**#13**

**#13**

**Learning Unit Objectives**

**Sinking Funds (Find Periodic Payments**

**LU13.3**

**LU13.3**

### 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.**

**Future value of annuity **
**-the future dollar amount of **
**a series of payments plus **
**interest**

**Present value of an annuity - the **
**amount of money needed to **

**invest today in order to receive a **
**stream of payments for a given **
**number of years in the future**

**Annuity - A series of **
**payments**

**$0.00**
**$0.50**
**$1.00**
**$1.50**
**$2.00**
**$2.50**
**$3.00**
**$3.50**
**1** **2** **3**
**End of period**
$1.00
$2.08
$3.2464

### Figure 13.1 Future value of an

### annuity of $1 at 8%

### 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**

### 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**

**Step 1. For period 1, no interest calculation is **
**necessary, since money is invested at the end of **
**period **

**Step 3. Add the additional investment at the **
**end of period 2 to the new balance.**

### Calculating Future Value of an Ordinary

### Annuity Manually

**Step 4. Repeat steps 2 and 3 until the end**
**of the desired period is reached.**

**Step 2. For period 2, calculate interest **
**on the balance and add the interest to **
**the previous balance. **

### 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%**
*Manual Calculation*
3,000.00
$ End of Yr 1
240.00
3,240.00
3,000.00
6,240.00
End of Yr 2
499.20
6,739.20
3,000.00
9,739.20
End of Yr 3

**Step 1. Calculate the number of periods and **
**rate per period **

**Step 2. Lookup 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 pymt. x Ordinary annuity****ordinary annuity each period table factor **

### Calculating Future Value of an Ordinary

### Annuity by Table Lookup

Period 2% 3% 4% 5% 6% 7% 8% 9% 10% 1 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 2 2.0200 2.0300 2.0400 2.0500 2.0600 2.0700 2.0800 2.0900 2.1000 3 3.0604 3.0909 3.1216 3.1525 3.1836 3.2149 3.2464 1.0000 3.3100 4 4.1216 4.1836 4.2465 4.3101 4.3746 4.4399 4.5061 4.5731 4.6410 5 5.2040 5.3091 5.4163 5.5256 5.6371 5.7507 5.8666 5.9847 6.1051 6 6.3081 6.4684 6.6330 6.8019 6.9753 7.1533 7.3359 7.5233 7.7156 7 7.4343 7.6625 7.8983 8.1420 8.3938 8.6540 8.9228 9.2004 9.4872 8 8.5829 8.8923 9.2142 9.5491 9.8975 10.2598 10.6366 11.0285 11.4359 9 9.7546 10.1591 10.5828 11.0265 11.4913 11.9780 12.4876 13.0210 13.5795 10 10.9497 11.4639 12.0061 12.5779 13.1808 13.8164 14.4866 15.1929 15.9374 11 12.1687 12.8078 13.4863 14.2068 14.9716 15.7836 16.6455 17.5603 18.5312 12 13.4120 14.1920 15.0258 15.9171 16.8699 17.8884 18.9771 20.1407 21.3843 13 14.6803 15.6178 16.6268 17.7129 18.8821 20.1406 21.4953 22.9534 24.5227 14 15.9739 17.0863 18.2919 19.5986 21.0150 22.5505 24.2149 26.0192 27.9750 15 17.2934 18.5989 20.0236 21.5785 23.2759 25.1290 27.1521 29.3609 31.7725

Ordinary annuity table: Compound sum of an annuity of $1 (Partial)

### Table 13.1 Ordinary annuity table:

### Compound sum of an annuity of $1

**N = 3 x 1 = 3**
**R = 8%/1 = 8%**
**3.2464 x $3,000**
**$9,739.20**

### Future Value of an Ordinary Annuity

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

### 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.

### 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%**
*Manual Calculation*
3,000.00
$ Beginning Yr 1
240.00
3,240.00
3,000.00
Beginning Yr 2
6,240.00
499.20
6,739.20
3,000.00
Beginning Yr 3
9,739.20
779.14
10,518.34
End of Yr. 3

### 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 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.

Step 4. Subtract 1 payment from Step 3.

### Future Value of an Annuity Due

**Find the value of an **
**investment after 3 **
**years for a $3,000 **
**annuity due at 8%** **N = 3 x 1 = 3 + 1 = 4**
**R = 8%/1 = 8%**
**4.5061 x $3,000**
**$13,518.30 - $3,000**
**$10,518.30**

**$0.00**
**$0.50**
**$1.00**
**$1.50**
**$2.00**
**$2.50**
**$3.00**
**$3.50**
**1** **2** **3**
**End of period**
$.93
$1.78
$2.5771

### Figure 13.2 - Present value of

### an annuity of $1 at 8%

### 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 an ordinary 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 x Present value of****ordinary annuity pymt. Pymt. ordinary annuity table**

Period 2% 3% 4% 5% 6% 7% 8% 9% 10% 1 0.9804 0.9709 0.9615 0.9524 0.9434 0.9346 0.9259 0.9174 0.9091 2 1.9416 1.9135 1.8861 1.8594 1.8334 1.8080 1.7833 1.7591 1.7355 3 2.8839 2.8286 2.7751 2.7232 2.6730 2.6243 2.5771 2.5313 2.4869 4 3.8077 3.7171 3.6299 3.5459 3.4651 3.3872 3.3121 3.2397 3.1699 5 4.7134 4.5797 4.4518 4.3295 4.2124 4.1002 3.9927 3.8897 3.7908 6 5.6014 5.4172 5.2421 5.0757 4.9173 4.7665 4.6229 4.4859 4.3553 7 6.4720 6.2303 6.0021 5.7864 5.5824 5.3893 5.2064 5.0330 4.8684 8 7.3255 7.0197 6.7327 6.4632 6.2098 5.9713 5.7466 5.5348 5.3349 9 8.1622 7.7861 7.4353 7.1078 6.8017 6.5152 6.2469 5.9952 5.7590 10 8.9826 8.5302 8.1109 7.7217 7.3601 7.0236 6.7101 6.4177 6.1446 11 9.7868 9.2526 8.7605 8.3064 7.8869 7.4987 7.1390 6.8052 6.4951 12 10.5753 9.9540 9.3851 8.8632 8.3838 7.9427 7.5361 7.1607 6.8137 13 11.3483 10.6350 9.9856 9.3936 8.8527 8.3576 7.9038 7.4869 7.1034 14 12.1062 11.2961 10.5631 9.8986 9.2950 8.7455 8.2442 7.7862 7.3667 15 12.8492 11.9379 11.1184 10.3796 9.7122 9.1079 8.5595 8.0607 7.6061

Present value of an annuity of $1 (Partial)

### Table 13.2 - Present Value

### of an Annuity of $1

### 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.**

**N = 3 x 1 = 3**
**R = 8%/1 = 8%**
**2.5771 x $8,000**
**$20,616.80**
*Manual Calculation*
20,616.80
$
1,649.34
22,266.14
(8,000.00)
14,266.14
1,141.29
15,407.43
(8,000.00)
7,407.43
592.59
8,000.02
(8,000.00)
0.02
Interest ==>
Payment ==>
End of Year 3 ==>
Interest ==>
Interest ==>
Payment ==>
Payment ==>

### 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?**

**N = 5 x 2 = 10 **
**R = 8%/2 = 4%**
**12.0061 x $200**
**$2,401.22**
**N = 6 x 2 = 12**
**R = 8%/2 = 4%**
**1.6010 x $2,401.22**
**$3,844.35**
**Future **
**value of **
**an **
**annuity**
**Future value **
**of a lump **
**sum**
Step 1
Step 2

### 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.**
**N = 25 x 1 = 25 **
**R = 5%/1 = 5%**
**14.0939 x $40,000**
**$563,756**
**N = 8 x 1 = 8**
**R = 5%/1 = 5%**
**.6768 x $563,756**
**$381,550.06**
**Present **
**value of **
**an **
**annuity**
**Present **
**value of a **
**lump sum**
Step 2

### Sinking Funds (Find Periodic Payments)

Bonds

**Sinking Fund = Future x Sinking Fund **
**Payment Value Table Factor**

Period 2% 3% 4% 5% 6% 8% 10% 1 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 2 0.4951 0.4926 0.4902 0.4878 0.4854 0.4808 0.4762 3 0.3268 0.3235 0.3203 0.3172 0.3141 0.3080 0.3021 4 0.2426 0.2390 0.2355 0.2320 0.2286 0.2219 0.2155 5 0.1922 0.1884 0.1846 0.1810 0.1774 0.1705 0.1638 6 0.1585 0.1546 0.1508 0.1470 0.1434 0.1363 0.1296 7 0.1345 0.1305 0.1266 0.1228 0.1191 0.1121 0.1054 8 0.1165 0.1125 0.1085 0.1047 0.1010 0.0940 0.0874 9 0.1025 0.0984 0.0945 0.0907 0.0870 0.0801 0.0736 10 0.0913 0.0872 0.0833 0.0795 0.0759 0.0690 0.0627 11 0.0822 0.0781 0.0741 0.0704 0.0668 0.0601 0.0540 12 0.0746 0.0705 0.0666 0.0628 0.0593 0.0527 0.0468 13 0.0681 0.0640 0.0601 0.0565 0.0530 0.0465 0.0408 14 0.0626 0.0585 0.0547 0.0510 0.0476 0.0413 0.0357 15 0.0578 0.0538 0.0499 0.0463 0.0430 0.0368 0.0315 16 0.0537 0.0496 0.0458 0.0423 0.0390 0.0330 0.0278 17 0.0500 0.0460 0.0422 0.0387 0.0354 0.0296 0.0247

### Table 13.3 - Sinking Fund Table

### Based on $1

### 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**
**R = 10%/1 = 10%**
**0.0219 x $60,000**
**$1,314**
**Check**
**$1,314 x 45.5992**
**59,917.35***

*** Off due to rounding**

**N = 18, R= 10%**
**Future Value of **
**an annuity table**

**Problem 13-13:**

18 periods + 1 = 19, 5%
30.5389
X $2,000
$61,077.80
-$ 2,000.00
-$59,077.80
**Solution:**

**Solution:**

**Problem 13-17:**

20 periods, 12% (Table 13.1) $12,500 x 72.0524 = $900,655

**Problem 13-18:**

10 periods, 11% (Table 13.2) $15,000 x 5.8892 = $88,338

**Problem 13-23:**

16 periods, = 2%8%
4
$900 x 13.577 = $12,219.93
OR
$900 x 18.6392 = $16,775.28
x .7284 **(Table 12.3)**$12,219.11 2% 16 periods

**Solution:**

**Solution:**

**Problem 13-25:**

20 periods, 2% (Table 13.3)

.0412 x $88,000 = $3,625.60 quarterly payment

**Problem 13-26:**

Morton: 5 periods, 8%

3.9927 x $35,000 = $139,744.50 + $40,000 = $179,744.50 Flynn: 5 periods, 8%

3.9927 x $38,000 = $151,722.60 + $25,000 = $176,722.60 Morton offered a better.

**Problem 13-27:**

PV annuity table: 15 periods, 8% 8.5595 x $28,000 = $239,666 PV table: 10 years, 8% .4632 x $239,666 = $111,013.29