2.5 Annuities

Page 1

● Unknown Time ● Unknown Rate of Interest ● Varying Interest ● Annuities not Involving Compound Interest

Example:

An investment of $1000 is to be used to make payments of $100 at the end of every year for as long as possible. If the fund earns an annual effective rate of interest of 5%, find how many regular payments can be made and find the amount of smaller payment: To be paid on the date of the last regular payment. To be paid one year after the last regular payment. To be paid during the year following the last regular payment.

0

1000

100

100

1

2

... ... ... ...

100

100

... ... ... ...

13

14

14+k

15

1

3

2

The equation value; đ?‘›|

=

To find the value of n; By using financial calculator, set, I = 100 (0.5) = 5

PV = 1000 PMT = -100 And compute N. OR

By using formula, = .

− − đ?‘›+

+

+

=

.

=

.

.

=

ln ln .

− đ?‘›+

− .

≈

.

We will get 14 < n < 15. Thus, 14 regular payments ca be made plus a smaller final payment. The equation value at the end of 14th year: |

+

=

=

.

.

−

|

=$

.

The equation value at the end of the 15th year: |

.

.

=

.

+

=

=

.

. =$

.

=

or in general,

−

|

+ .

In this case the equation of value becomes: + |

+

=

.

n = 14 so,

+

Thus,

= =

.

= . = . .

.

−

=$

.

−

A fund of $25000 is to be accumulated by means of deposits of $1000 made at the end of every year as long as necessary. If the fund earns an effective rate of interest of 8%, find how many regular deposits will be necessary and the size of a final deposit to be made one year after the last regular deposit.

0

1000

1000

... ... ... ...

25000

1

2

... ... ... ...

n

The equation value; đ?‘›|

=

To find the value of n; By using financial calculator, set, I=8 FV = 25000 PMT = -1000 And compute N. OR

By using formula, = . + +

=

.

. đ?‘›+

=

.

đ?‘›+

=

ln ln .

− +

≈

.

, 14 < n < 15

The equation value at the end of 15th year: |

+

=

= = −$

−

|−

The last regular payment deposit brings the fund close to enough to the $25000 that interest alone over the last year is sufficient to cause the fund to exceed $25000. The balance in the fund at the end of the 14th year:

So,

| |

=$

=

.

=$

.20

Unknown Rate Of Interest 1.

The best way to determine an unknown rate of interest for a basic annuity is to use a financial calculator.

2.

Despite the widespread availability of financial calculators, it is nevertheless instructive to note that a simple approximation formula to determine an unknown rate of interest exists.

3.

This simple approximation formula can even be computed by hand, yet it often produces reasonable answers if high levels of accuracy are not required.

4.

5.

Consider the situation in which

đ?‘›|

= đ?‘” . An approximation for is given by :

≈

đ?‘”

−đ?‘” +

The comparable formula that could be used to find an đ?’ˆâˆ’đ?’? đ?’Šâ‰ˆ đ?’ˆ đ?’?−

approximate answer for

đ?‘›|

= đ?‘” is

Example : At what rate of interest, convertible quarterly, is $16,000 the present value of $1000 paid at the end of every quarter for five years? Let

=

/ , so that the equation of value becomes |

|

=

=

Method 1 (Financial calculator): N

=

20

PV

=

16

PMT

=

-1

And compute I obtaining I

=

2.2262

J

=

0.022262

Thus, we have

So that = . = .

Method 2 (Approximation formula): We have n = 20 and g = 16, so that formula (3.21) gives − ≈

= 0.0238

Method 3 (Approximation): |

.

<

. |

.

>

|

.

=

.

− . − .

=

−

=

+

+ +

=

= − . − .

=

|

= .

. = .

= . = .

=

−

+ +

−

=

.

−

.

Example : Find the rate of interest at which �

|

= . .

In this example, we can use a simple algebraic approach. �

|

=

=

= +

=

+ + + = . + + + = . + + = . + − . = = =− .

− Âą

− Âą =

−

−

− .

− Âą

=− . = .

.

Varying Interest In this section we will consider the situation in which the rate of interest can vary each period, but compound interest is still in effect. The first pattern would be for to be the applicable rate for period k regardless of when the payment is made. This method could be called portfolio rate method since one rate applies to the e tire a uity alue portfolio at a y poi t i ti e. Present value becomes,

đ?‘›|

=

+

−

+

+

−

đ?’‚đ?’?| =

đ?’? =

+ =

−

+. . . +

+đ?’Š

−

+

−

...

+

đ?‘›

−

Example : Find the present value of an annuity-immediate which pays 1 at the end of each half-year for 5 years, if the rate of interest is 8% convertible semiannually for the first 3 years and 7% convertible semiannually for the last 2 years.

1

0

PV

1

1

1

1

2

= %

1

1

3

1

1

4

= %

1

1

5

=

=

𝑥 | %/ | .

+

+

− + . .

= . =$ .

𝑥 | %/ | .

−

+

+ .

𝑣

+

2

− + . .

− 𝑥 −

+ .

−

The second pattern would be to compute present values using rate for the payment made at time k over all k periods. This method could be called the yield curve method.

Present value becomes,

đ?‘›|

=

+

−

+

đ?’‚đ?’?| =

đ?’? =

+

−

+đ?’Š

+. . . + −

+

đ?‘›

−đ?‘›

The payment for the first 3 years are discounted at 8% convertible semiannually and the payments for the last 2 years are discounted at 7% convertible semiannually.

1

0

PV

1

1

1

2

1

= %

1

1

3

1

1

4

= %

1

1

5

𝑥 | %/

=

=

| .

+

+

− + . .

= . =$ .

𝑥 | %/ | .

−

+

+ .

𝑣

+

2

− + . .

− 𝑥 −

+ .

−

We now turn to accumulated values. We will consider values of ��| rather than ��| so that all values of for k = 1, 2, . . . . ., n will enter the formula. If the applicable rate for period k is regardless of he the pay e t is ade portfolio rate ethod

Accumulated values become ��| =

đ?‘› =

=

+

đ?‘›âˆ’ +

Example : Find the accumulated value of 10-year annuity-immediate of $100 per year if the effective rate of interest is 5% for the first 6 years and 4% for the last 4 years.

0

100

100

100

100

100

100

100

100

100

100

1

2

3

4

5

6

7

8

9

10

= %

= %

=

=

=$

+ .

.

.

.

−

| %

+ %

+ . .

+

+

| %

+ .

+

−

.

.

If the payment made at time k earns at rate cur e ethod

o er the rest of the accu ulatio period yield

Accumulated values become

��| =

đ?‘› =

+

đ?‘›âˆ’ +

The first 6 payments are invested at an effective rate of interest of 5% and if the final 4 payments are invested at 4%

0

100

100

100

100

100

100

100

100

100

100

1

2

3

4

5

6

7

8

9

10

= %

= %

+ %

| %

+ .

= =

.

.

−

+ .

=$

.

+

| %

+ .

+

.

+

−

.

.

Annuities not involving compound interest The present value of an n-period annuity-immediate is equal to the sum of the present values of the individual payments. đ?‘›|

=

đ?‘› =

We turn next to finding the value for ��| . If we assume that 1 invested at time t, where t = 1, 2, . . . , n

-1 will accumulate to

đ?‘Ž đ?‘› đ?‘Ž

at time n, then we would have

��| =

đ?‘› =

=

đ?‘› =

This would seem to be inappropriate procedure in certain cases, e.g. a problem involving varying force of interest over the n periods. The accumulated value of an n-period annuity-immediate in which each payment is invested at simple interest from the date of payment until the end of the n periods. + + + + + â‹Ż+ [ + − ]

��| =

Another formula ( symmetry and completeness )

đ?‘›|

=

đ?‘›âˆ’ =

đ?‘›âˆ’ =

=

đ?‘›âˆ’ =

Example : Compare the value of � | at 10% interest: Method 1 (Assuming Compound Interest)

�

��| = | .

=

= .

đ?‘›

+

+ . .

−

−

Method 2 (Using Formula) ��| = �

| .

đ?‘›

=

=

= .

= . = .

.

.

đ?‘›

+

.

=

+

.

+

.

+

.

+

Method 3 (Using Formula)

�

��| = | .

đ?‘›âˆ’ =

= + . + . + . + . + . = .

.

Find � | if � t = 0.02t for 0 ≤ t ≤ 5

� |=

=

đ?‘’

đ?‘Ą

�r �

=đ?‘’

.

= . = .

Alternatively, formula is applicable,

=đ?‘’

��| =

=đ?‘’ =đ?‘’

đ?‘› =

=

+đ?‘’

=đ?‘’ . =đ?‘’ . = .

.

+đ?‘’

+ .

2

.

đ?‘Ą

+đ?‘’

+ .

.

+

+ .

+

� 0 �r �

2 đ?‘Ą 0

.

2

.

=

� |=

= .

đ?‘’

đ?‘› =

=

đ?‘’− . + đ?‘’− . + đ?‘’− . + đ?‘’− +đ?‘’ . +đ?‘’ . +đ?‘’ . +

.

+ đ?‘’−

.

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