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 UNIT 11 : MATHEMATICS OF INVESTMENT

 LESSON 5: GENERAL ANNUITIES HOMEWORK QUESTIONS

 

Homework Questions: (Solutions below)

Note: Answers to # 1 4 should be to the nearest hundredth.

1. Find the nominal (annual) interest rate, compounded annually, equivalent to 7.6%/a, compounded semi-annually.

2. Find the nominal (annual) interest rate, compounded semi-annually, equivalent to 6.6%/a, compounded quarterly.

 

3. Find the nominal (annual) interest rate, compounded monthly, equivalent to 4.2%/a, compounded semi-annually.

 

4. Find the nominal (annual) interest rate, compounded quarterly, equivalent to 6%/a, compounded monthly.

 

 

5. Evaluate each of the following general annuities. Include a complete time line diagram for each.

 

a) $500 per month for 60 months at 8.4%/a, compounded annually.

 

b) $700 per year for 12 years at 6.8%/a, compounded quarterly.

 

c) $750 every 3 months for 12 years at 8.4%/a, compounded semi-annually.

 

d) $50 per week for 6 years at 7%/a, compounded quarterly.

 

 

Solutions:

1. Find the nominal (annual) interest rate, compounded annually, equivalent to 7.6%/a, compounded semi-annually.

 

Solution:

Ie. We must find the annual rate that is equivalent to 7.6%/a, compounded semi-annually.

 

Step 1: Using the formula A = P(1 + i)n, find the value of $1 invested at 7.6%/a, compounded semi-annually after 1 year.

Step 2: Let the equivalent annual rate be i %.

Now find the value of $1 invested at i % per year after 1 year.

A = 1(1 + i)1 ** n = 1, the number of times interest is compounded per year.

Step 3: These two amounts must be equal. Hence

 

2. Find the nominal (annual) interest rate, compounded semi-annually, equivalent to 6.6%/a, compounded quarterly.

 

Solution:

Ie. We must find the annual rate, compounded semi-annually that is equivalent to 6.6%/a, compounded quarterly.

 

Step 1: Using the formula A = P(1 + i)n, find the value of $1 invested at 6.6%/a, quarterly after 1 year.

Step 2: Let the equivalent year rate be i %. (Note the equivalent yearly rate would be 2i %.)

 

Now find the value of $1 invested at i % per year after 1 year.

A = 1(1 + i)2 ** n = 2, the number of times interest is compounded per year.

Step 3: These two amounts must be equal. Hence

 

3. Find the nominal (annual) interest rate, compounded monthly, equivalent to 4.2%/a, compounded semi-annually.

Solution:

Ie. We must find the annual rate, compounded monthly that is equivalent to 4.2%/a, compounded semi-annually.

 

Step 1: Using the formula A = P(1 + i)n, find the value of $1 invested at 4.2%/a, semi-annually after 1 year.

Step 2: Let the equivalent monthly rate be i %. (Note the equivalent yearly rate would be 12i %.)

 

Now find the value of $1 invested at i % per month after 1 year.

A = 1(1 + i)12 ** n = 12, the number of times interest is compounded per year.

Step 3: These two amounts must be equal. Hence

 

4. Find the nominal (annual) interest rate, compounded quarterly, equivalent to 6%/a, compounded monthly.

Solution:

Ie. We must find the annual rate, compounded quarterly that is equivalent to 6%/a, compounded monthly.

 

Step 1: Using the formula A = P(1 + i)n, find the value of $1 invested at 6%/a, monthly after 1 year.

Step 2: Let the equivalent year rate be i %. (Note the equivalent yearly rate would be 4i %.)

 

Now find the value of $1 invested at i % per year after 1 year.

A = 1(1 + i)4 ** n = 4, the number of times interest is compounded per year.

Step 3: These two amounts must be equal. Hence

 

5. Evaluate each of the following general annuities. Include a complete time line diagram for each.

 

a) $500 per month for 60 months at 8.4%/a, compounded annually.

 

Solution:

Here the payment interval( monthly ) is different than the interest period (annual). This is a general annuity.

We must match the interest period to the payment interval.

Ie. We must find the monthly rate that is equivalent to 8.4%/a, compounded annually.

 

Step 1: Using the formula A = P(1 + i)n, find the value of $1 invested at 8%/a, compounded annually after 1 year.

A = 1(1.084)1 = 1.084

Step 2: Let the equivalent monthly rate be i %. (Note the equivalent yearly rate would be 12i %.)

Now find the value of $1 invested at i % per month after 1 year.

A = 1(1 + i)12 ** n = 12, the number of times interest is compounded per year.

Step 3: These two amounts must be equal. Hence

 

Now find the amount of the annuity using the annuity formula.

Interest per. 0 1 2 . . . 58 59 60 Accumulated value

Payment 500 500 500 500 500

 


500

 

500(1.006744132)1

 

500(1.006744132)2

 

.

 

.

 

.

 

500(1.006744132)58

 

500(1.006744132)59

 

Hence the amount of the annuity is $36 827.58.

 

 

b) $700 per year for 12 years at 6.8%/a, compounded quarterly.

Solution:

Here the payment interval( yearly ) is different than the interest period (quarterly). This is a general annuity.

We must match the interest period to the payment interval.

Ie. We must find the yearly rate that is equivalent to 6.8%/a, compounded quarterly.

 

Step 1: Using the formula A = P(1 + i)n, find the value of $1 invested at 6.8%/a, compounded quarterly after 1 year.

 

Step 2: Let the equivalent yearly rate be i %.

Now find the value of $1 invested at i % per year after 1 year.

A = 1(1 + i)1 ** n = 1, the number of times interest is compounded per year.

Step 3: These two amounts must be equal. Hence

 

Now find the amount of the annuity using the annuity formula.

Interest period0 1 2 . . . 10 11 12 Accumulated value

Payment 700 700 700 700 700

 


700

 

700(1.069753736)1

 

700(1.069753736)2

 

.

 

.

 

.

 

700(1.069753736)10

 

700(1.069753736)11

 

Hence the amount of the annuity is $12 503.78.

 

c) $750 every 3 months for 12 years at 8.4%/a, compounded semi-annually.

 

Solution:

Here the payment interval( year ) is different than the interest period ( semi-annual). This is a general annuity.

We must match the interest period to the payment interval.

Ie. We must find the quarterly rate that is equivalent to 8.4%/a, compounded semi-annually.

 

Step 1: Using the formula A = P(1 + i)n, find the value of $1 invested at 8.4%/a, compounded semi-annually after 1 year.

 

 

Step 2: Let the equivalent quarterly rate be i %. (Note the equivalent yearly rate would be 4i %.)

Now find the value of $1 invested at i % per year after 1 year.

A = 1(1 + i)4 ** n = 4, the number of times interest is compounded per year.

Step 3: These two amounts must be equal. Hence

 

Now find the amount of the annuity using the annuity formula with (1 + i) = 1.011230201.

Interest per. 0 1 2 . . . 46 47 48 Accumulated value

Payment 750 750 750 750 750

 


750

 

750(1.011230201)1

 

750(1.011230201)2

 

.

 

.

 

.

 

750(1.011230201)46

 

750(1.011230201)47

 

Hence the amount of the annuity is $47 365.62.

 

 

d) $50 per week for 6 years at 7%/a, compounded quarterly.

 

Solution:

Here the payment interval( weekly ) is different than the interest period ( quarterly). This is a general annuity.

We must match the interest period to the payment interval.

Ie. We must find the weekly rate that is equivalent to 7%/a, compounded quarterly.

 

Step 1: Using the formula A = P(1 + i)n, find the value of $1 invested at 7%/a, compounded quarterly after 1 year.

 

Step 2: Let the equivalent weekly rate be i %. (Note the equivalent yearly rate would be 52i %.)

Now find the value of $1 invested at i % per week after 1 year.

A = 1(1 + i)52 ** n = 52, the number of times interest is compounded per year.

Step 3: These two amounts must be equal. Hence

 

Now find the amount of the annuity using the annuity formula with (1 + i) = 1.001335401.

Interest period0 1 2 . . . 310 311 312 Accumulated value

Payment 50 50 50 50 50

 


50

 

50(1.001335401)1

 

50(1.001335401)2

 

.

 

.

 

.

 

50(1.001335401)310

 

50(1.001335401)311

 

 

Hence the amount of the annuity is $19 336.61.

 

 

 

6. Rahim has just purchased a new car for 34 500. He borrows the money and will repay it in monthly instalments over 6 years. The interest rate is 7.8%/a, compounded quarterly. Find the monthly payment.

 

Solution:

Here the payment interval( monthly ) is different than the interest period ( quarterly). This is a general annuity.

We must match the interest period to the payment interval.

Ie. We must find the monthly rate that is equivalent to 7.8%/a, compounded quarterly.

 

Step 1: Using the formula A = P(1 + i)n, find the value of $1 invested at 7.8%/a, compounded quarterly after 1 year.

Step 2: Let the equivalent monthly rate be i %. (Note the equivalent yearly rate would be 12i %.)

Now find the value of $1 invested at i % per month after 1 year.

A = 1(1 + i)12 ** n = 12, the number of times interest is compounded per year.

Step 3: These two amounts must be equal. Hence

 

The money in question is borrowed now at point 0 on the time line. Hence this is a PV general annuity question

 

Interest Period 0 1 2 3 70 71 72

Payment R R R R R R

R(1.006458202)-1

R(1.006458202)-2

.

.

R(1.006458202)-70

R(1.006458202)-71

 

R(1.006458202)-72

 

 

This forms the following geometric series:

R(1.006458202)-72 + R(1.006458202)-71 + . . . + R(1.006458202)-2 + R(1.006458202)-1

 

 

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