2.6 More General Annuities

Page 1

In this topic, annuities for which payments are made more or less frequently than interest is convertible and annuities with varying payments will be considered.

1. Find the rate of interest,convertible at

the same frequency as payments are made,that is

equivalent to the given rate of interest.

2. Using this new rate of

interest,

fine

value of the annuity.

the

• Example 1 Find the accumulated value at the end of four years of an investment fund in which $100 is deposited at the beginning of each quarter for the first two years and $200 is

deposited at he beginning of each quarter for the second two years, of the fund earns 12% convertible monthly. $100

$200

0

2

AV

4

Solving Example 1

Step 1 : Find j. 12

 0.12  4 1    1  j  12  

3   j  1.03  1

 0.030301

Step 2 : Find accumulated value. AV  100s8  200s8      1.0303018  1   1.0303018  1  1.0303018  200   100  0.030301   0.030301       1.030301   1.030301   $2998.861484

• Example 2

A loan of $3000 is to be repaid with quarterly installment at the end of each quarter for five years. If the rate of interest charged on the loan is 10% convertible semiannually, find the amount of each quarterly payment

$3000

0

5

Solving Example 2

Step 2 : Find R.

Step 1: Find j. 2

 0.10  4 1    1  j  2   j  1.05  1 1 2

 0.024695

Ra 20  3000 R

3000

 1  1.02469520      0 . 024695    $191.8873849

•Example 3

At what annual effective rate of interest will payment of $100 at the end of every quarter accumulate to $2500 at the end of five years?

Solving Example 3 Step 1 : Find equation accumulated

100s20  2500 Step 2 : Use interpolation method to find j.

s 20

(1  i ) 20  1  i

Rate of interest

Symbol

Sum

0.0228

<

24.98648207

j

=

25

0.0229

>

25.01153035

j - 0.0228 25  24.98648207  0.0229 - 0.0228 25.01153035 j = 0.0228539675

Step 3 : Find i. (1  j ) 4  (1  i)

1.02285396754  1  i  i  (1.0228539675) 4  1 i  0.0945977126

Let • k – the number of interest conversion periods in one payment period • n – the term of annuity measured in interest conversion periods • i – the rate of interest per interest conversion period • R – a series of equal payments at the end of each period

Annuity Immediate The present value of an annuity which pays 1 at the end of each k interest conversion periods for a total of n interest conversion period

đ?‘Žđ?‘›| đ?‘ đ?‘˜|

The accumulated value of this annuity immediately after the last payment

đ?‘Žđ?‘›| đ?‘ đ?‘˜|

Annuity-Due The present value of an annuity which pays 1 at the beginning of each k interest conversion periods for a total of n interest conversion periods

đ?‘Žđ?‘›| đ?‘Žđ?‘˜|

The accumulated value of this annuity k interest conversion period after the last payment đ?‘Žđ?‘›| đ?‘Žđ?‘˜|

+đ?‘–

đ?‘›

đ?‘ đ?‘›| = đ?‘ đ?‘˜|

Other Considerations • Case I Present value of perpetuity-immediate payable less frequently than interest • Case II To find the value of a series of payments at a given force of interest, � Replace vtk with e-�tk and (1+i)tk with e�tk

• Example 4 An investment of $1000 is used to make payments of $100 at the end of each year for as long as possible with a smaller final payment to be made at the time of the last regular payment. If interest is 7% convertible semiannually, find the number of payments and the amount of the total final payment.

1000

100

100

0

1

2

……

100

r

n-1

n

𝑅=

𝑅+

𝑎𝑛|0.0 = 𝑠 |0.0 𝑎𝑛| = . <n< 𝑠 |0.0 = 𝑠 |0.0

.

𝑅=

−

. 9

.

.

• Example 4 Rework Example 1

• Let m be the number of payment periods in one interest conversion period

• Let n be the term of the annuity measured in interest conversion periods • Let i be the interest rate per interest conversion period.

Annuity Immediate

Annuity Due

• Example 5 At what annual effective rate of interest is the present value of a series of payments of $1 every six months forever, with the first payment made immediately, equal to $10?

Solving Example 5

Reference Kellison, S.G. (2009). The Theory of Interest. McGraw Hill.

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