Business Mathematics - Chapter 13

Janine invests $1,000 a month for 1 year into an ordinary annuity earning a 2% rate per period. How much will Janine have at the end of Year 1?

1 year x 12 months = 12 Interest = 2% Table Factor: 13.4120 1,000 x 13.4120 = $13,412

order the steps for calculating future value of an annuity by Table lookup.

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

Order the steps for manually calculating future value of an ordinary annuity.

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

Annuity

A series of payments

After paying off a car loan or credit card, don't remove this amount from your budget. Instead, invest in your future by applying some of it to your retirement account. How much would $450 invested at the end of each quarter be worth in 10 years at 4% interest? (Please use the following provided Table.)

Amount after 10 years: Principal: $450 Quarterly 10 Years 4% Interest 10 x 4 = 40 4% / 4 = 1% Table Factor: 48.8864 $450 x 48.8864 = $21,998.88

Match the annuity type to the respective definition.

Annuities Certain: have a fixed (or specified) number of payments (such as a home loan) Contingent Annuities: have no fixed number of payments, and instead depend on an uncertain event (such as death of the policy holder)

Given an investment of $20,000 after 2 years made quarterly at 12%, match the future value to the annuity type.

Annuity Due: 2 years x 4 Quarters = 8 + 1 = 9 periods 12% / 4 Quarters = 3% Table Factor = 10.1591 $20,000 x 10.1591 = $203,182 $203,182 - $20,000 = $183,182 Annuity Due: $183,182 Ordinary Annuity: 2 years x 4 quarters = 8 periods 12% / 4 quarters = 3% Table Factor: 8.8923 $20,000 x 8.8923 = $177,846 Ordinary Annuity: $177,846

Match the annuity group with the timing of payments.

Annuity Due: has payments made at the beginning of the period (such as live insurance premiums) Ordinary Annuity: made at the end of the period (such as stock dividends)

Present Value of ordinary Annuity Payment

Annuity Payment x Present value of ordinary annuity table

A sinking fund accumulates interest using the __________ interest method.

compound

Given the annuity due of $10,000 per year at 10% for 2 years, match the value to the time frame.

Beginning of Year 1: $10,000 End of Year 1: $11,000 Beginning of Year 2: $21,000 End of Year 2: $23,100

Annuity Due

Regular deposits/payments made at the beginning of the period Monthly: January 1 Quarterly: April 1 Semiannually: July 1 Annually: January 1

Discharge bonded indebtedness, replacing worn-out equipment, or purchasing plant expansion are some of the uses of:

Sinking Fund

Abby Mia wants to know how much must be deposited in her local bank today so that she will receive yearly payments of $18,000 for 20 years at a current rate of 9% compounded annually. (Use the tables in the handbook.)

Principal: $18,000 Time: 20 Years Interest = 9%, Compounded Annually Present Value of Annuity Table Factor: 9.1285 18,000 x 9.1285 = $164,313

An annuity is a:

Stream of payments

Term of the annuity

The time from the beginning of the first payment period to the end of the last payment period

Carlyle plans to invest $12,000 per year for the next 5 years at 10% interest. How much will he have at the end of year 1?

$12,000 - The investment is made at the end of the period; therefore it has not yet earned interest. At the end of Year 1 Carlyle's investment totals $12,000.

order the steps for calculating future value of an ordinary annuity by Table lookup.

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

Select the table factor for a quarterly annuity at 8% for 2 years.

2 years x 4 quarters = 8 Periods 8% / 4 = 2% Table factor: 8.5829

You are earning an average of $46,500 and will retire in 10 years. If you put 20% of your gross average income in an ordinary annuity compounded at 7% annually, what will be the value of the annuity when you retire? (Please use the following provided Table.)

20% x $46500 =$9,300 7% annually 10 years Table Factor: 13.8164 $9,300 x 13.8164 = $128,492.52

Given the ordinary annuity formula, A = PMT x (1 + i)^n - 1) [-----------------] i match the term to the definition.

A = Future value or an ordinary annuity I = Interest N = Number of Periods Pmt = Annuity Payment

True of False: Compound interest and annuities cannot be combined.

False: Investments can include both compound interest and annuity components.

Financial analysts recommend investing 15% to 20% of your annual income in your retirement fund to reach a replacement rate of 70% of your income by age 65. This recommendation increases to almost 30% if you start investing at 45 years old. Mallori Rouse is 25 years old and has started investing $3,000 at the end of each year in her retirement account. How much will her account be worth in 20 years at 8% interest compounded annually? How much will it be worth in 30 years? What about at 40 years? How much will it be worth in 50 years? (Please use the following provided Table 13.1)

Future Value after 20 Years: P = $3,000 R = 8% T = 20 Years Table Factor: 45.7620 $3,000 x 45.7620 = $137,286 Future Value after 30 Years: P = $3,000 R = 8% T = 30 Years Table Factor: 113.2833 $3,000 x 113.2833 = $339,849.90 rounded to nearest whole dollar amount: $339,850 Future Value after 40 Years: P = $3,000 R= 8% T = 40 Years Table Factor: 259.0569 $3,000 x 259.0569 = 777,171 Future Value after 50 Years: P = $3,000 R= 8% T = 50 Years Table Factor:573.7711 $3,000 x 573.7711 = $1,721,313

Rob Herndon, an accountant with Southwest Airlines, wants to retire 50% of Southwest Airlines bonds by 2035. Calculate the payment Rob needs to make at the end of each year at 6% compounded annually to reach his goal of paying off $300,000 in 20 years. (Use Table 13.3.)

Future Value after 20 years: P: $300,000 N= 20 Years I = 6% compounded annually Annuity Table Factor: 0.0272 $300,000 x 0.0272 = $8,160

Sinking Fund Payment

Future Value x Sinking Fund Table Factor

Alice Longtree has decided to invest $400 quarterly for 4 years in an ordinary annuity at 8%. As her financial adviser, calculate for Alice the total cash value of the annuity at the end of year 4. (Use Table 13.1.)

Future Value: n = 4 x 4 = 16 I = 8% / 4 = 2% TF: 18.6392 $18.6392 x 400 = 7455.68

Future Value of Annuity

Future dollar amount of a series of payments plus interest

Josef Company borrowed money that must be repaid in 20 years. The company wants to make sure the loan will be repaid at the end of year 20. So it invests $12,500 at the end of each year at 12% interest compounded annually. What was the amount of the original loan? (Use Table 13.1.)

Future value of an ordinary annuity: N = 20 I = 12% Table Factor: 72.0524 $12,500 x 72.0524 = $900,655

Match the future values of a lump-sum and an annuity

Lump-Sum: Principal plus interest Annuity: Future dollar amount of a series of payments plus interest

Find the value of an investment after 3 years for a $30,000 ordinary annuity at 8% (Manual Calculation)

Manual Calculation: $3,000.00 (End of year 1) $ 240.00 (Plus interest) --------------------------- $3,240.00 $3,000.00 (Year 2 investment) --------------------------- $6,240.00 (End of year 2) 499.20 --------------------------- $6,739.20 $3,000.00 (Year 3 Investment) --------------------------- $9,739.20 (End of Year 3)

Thomas wants to retire in 6 years. What amount should Tom invest now to be able to withdraw $30,000 at the end of each year for 20 years after retirement if he can earn 8%.

N = 20 I = 8% Present Value of an annuity table factor: 9.8181 9.8181 x $30,000 = $294,543 N = 6 I = 8% Present Value of $1 at end of period = .6302 .6302 x $294,543 = $185,621

Perez wants to withdraw $10,000 at the end of each of 4 years. Interest is 6% annually. How much must he invest today to receive this stream of payments.

N = 4 Years I = 6% Table Factor: 3.4651 3.4651 x $10,000 = $34,651

Greg invests $1,000 at the end of each year for 5 years at 5% annually, then leaves the investment to grow (without additional payments) for an additional 5 years. What is the future value of this investment after 10 years.

N = 5 I = 5% Annuity Table Factor = 5.5256 5.5256 x $1,000 = $5,525.60 N = 5 I = 5% Future Value of $1 at compound interest table factor: 1.2763 1.2763 x $5,525.60 = $7,052.32

Carlyle plans to invest $12,000 per year for the next 5 years at 10% interest. How much will he have at the end of Year 5?

N = 5 Years I = 10% Annuity Table Factor: 6.1051 6.1051 x $12,000 = $73,261.20

Janine invests $1,000 a month for 1 year into an ordinary annuity earning a 2% rate per period. How much will Janine have at the end of Year 1.

N =1 year I = 2% 1 year x 12 Months = 12 $1,000 per year x 13.4120 = $13,412

Bankrate.com reported on a shocking statistic: only 54% of workers participate in their company's retirement plan. This means that 46% do not. With such an uncertain future for Social Security, this can leave almost 1 in 2 individuals without proper income during retirement. Jill Collins, 20, decided she needs to have $250,000 in her retirement account upon retiring at 60. How much does she need to invest each year at 5% compounded annually to meet her goal? (Please use the following provided Table.)

P = $250,000 T = 60 - 20 = 40 Years I = 5% compounded annually TF = 0.0083 Future Value x Sinking Fund Factor = Sinking Fund Payment $250,000 x 0.0083 = $2,075 (Sinking fund table reports the reciprocal of the future value of an annuity)

The goal of a sinking fund is to determine the amount of the:

Periodic Payment: A sinking fund determines the amount of the periodic payment.

Find the value of an investment after 3 years for a $30,000 ordinary annuity at 8% (Table Lookup)

Periods (N) = 3 x 1 = 3 + 1 = 4 Rate (R) = 8% / 1 = 8% Table Factor = 4.5061 4.5061 x $3,000 = $13,518.30 $13,518.30 - $3,000 = $10,518.30

Patricia and Joe Payne are divorced. The divorce settlement stipulated that Joe pay $525 a month for their daughter Suzanne until she turns 18 in 4 years. Interest is 6% a year compounded monthly. How much must Joe set aside today to meet the settlement? (Please use the following provided Table.)

Present Value of Ordinary Annuity: Principal: $525 per month Time: 4 Years Interest: 6% Compounded monthly N = 4 x 12 = 48 I = 6% / 12 = 50% Table Factor: 42.5803 $525 x 42.5803 = $22,354.66

Scott deposits $5,000 at the end of each year into an account for five years. Assuming 6% interest annually, what is the value of his account in five years?

Principal: $5,000 Time: 5 Years Interest: 6%, compounded annually Future Value Table Factor: 5.6371 $5,000 x 5.6371 = $28,185.50

Janet Woo decided to retire to Florida in 6 years. What amount should Janet invest today so she can withdraw $50,000 at the end of each year for 20 years after she retires? Assume Janet can invest money at 6% compounded annually. (Use Table 13.2 and Table 12.3.)

Principal: $50,000 Time: 20 Years Interest: 6%, compounded annually Present Value of Annuity Table Factor: 11.4699 $50,000 x 11.4699 = $573,495 Principal: $573,495 Time: 6 Years Interest: 6% Present Value at end of period Table Factor: 0.7050 $573,495 x .7050 = $404,313.98

Ordinary Annuity

Regular deposits/payments made at the end of the period. Monthly: January 31 Quarterly: June 30 Semiannually: Dec. 31 Annually: Dec. 31

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.

An annuity terms is:

The time from the beginning of the first payment period to the end of the last payment period.

The present value of an ordinary annuity looks at how much needs to be invested ______ to recieve a stream of payments for a given number of years in the ______.

Today, Future

Sinking Fund

a financial arrangement that sets aside regular periodic payments of a particular amount of money. Compound interest accumulates on these payments to a specific sum at a predetermined future date.

Mike Macaro is selling a piece of land. Two offers are on the table. Morton Company offered a $40,000 down payment and $35,000 a year for the next 5 years. Flynn Company offered $25,000 down and $38,000 a year for the next 5 years. Assume money can be invested at 8% compounded annually. (Use Table 13.2.)

a. What is the value of the offers? Morton Company: $40,000 down payment $35,000 Annually 5 Periods 8% (Table 13.2) 3.9927 x $35,000 = $139,744.50 $139,744.50 + $40,000 = $179,744.50 Flynn Company: $25,000 down payment $38,000 annually 5 periods 8% 3.9927 x $38,000 = $151,722.60 $151,722.60 + $25,000 = $176,722.60 b. Which offer is better for Mike? Morton Company