Thursday, 30 April 2015

Compound Interest for annuities

Compound Interest for annuities

Some scheme requires regular deposit for higher interest. These regular deposits may be at start of year or end of year. Compound interest on this interest may be worked out by two method i.e. formula and details working.

Example of advance Deposit

Investment
3,000
Years
3
Deposit Start of the years

Interest Rate
6%

Calculate the future value

Solution

1.       First Method

Year 1
 3,000 x ( 1+.06)3
3573
Year 2
3,000 x ( 1+.06)2
3371
Year 3
3,000 x ( 1+.06)1
3180


10,124




2.       Alternate Methods

= P [ (1+i)n+1-1]-1
            I
= 3,000 [3.374]
= $ 10,124


Example of Arrears Deposit

Investment
3,000
Years
3
Deposit at year end

Interest Rate
6%

Calculate the future value

Solution

3.       First Method

Year 1
 3,000 x ( 1+.06)2
3,371
Year 2
3,000 x ( 1+.06)1
3,180
Year 3
3,000
3,000


9,550




4.       Alternate Method (by formula)

= P [ (1+i)n-1]
            I
= 3,000 [3.1836]
= $ 9,550



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Compound Interest Examples

Compound Interest Examples

Compound interest is calculated by the following formula
Future value = P (1+r) n
P= Present Value
r= rate of interest
n= number of period

Compound Interest Example # 1

Deposit
$ 300,000
Interest Rate
15%
Interest Nature
Compound
Calculate interest for
4 years



Solution
Future value = P (1+r) n
= $ 300,000(1.15)4
=$ 524,702

Interest = future value – present value
= 524,702-300,000
=$ 224,702

Compound Interest Example # 2

Year 1
100,000

Year 2
150,000

Year 3
120,000

Year 4
110,000

Interest Rate 15%
480,000


Calculate the future value?



Solution

Year 1
100,000
(1+i)4
174,900
Year 2
150,000
(1+i)3
228,131
Year 3
120,000
(1+i)2
158,700
Year 4
110,000
(1+i)1
156,500



718,231



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Compound Interest Formula

Compound Interest Formula

In compound interest the interest is reinvested each year and therefore each interest amount for each year is different i.e. interest increase with passage of time. Future value in case of compound interest is calculated by the following formula

Future value = P (1+r) n

P= Present Value
r= rate of interest
n= number of period

Compound Interest Formula Example

Mr, Khalil invested $ 50,000 in a bank for three year a@ 6% per annum. What amount Mr. Khalil would receive in three year assuming the compound interest rate?
Future value = P (1+r)n
= $ 50,000(1.06)3
=$ 59,550

Interest = future value – present value
= 59,550-50,000

=9,550
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Compound Interest Formula

Compound Interest Formula

In compound interest the interest is reinvested each year and therefore each interest amount for each year is different i.e. interest increase with passage of time. Future value in case of compound interest is calculated by the following formula

Future value = P (1+r) n
P= Present Value
r= rate of interest
n= number of period

Compound Interest Formula Example

Mr, Khalil invested $ 50,000 in a bank for three year a@ 6% per annum. What amount Mr. Khalil would receive in three year assuming the compound interest rate?
Future value = P (1+r)n
= $ 50,000(1.06)3
=$ 59,550

Interest = future value – present value
= 59,550-50,000

=9,550
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Simple Interest Example

Simple Interest Example

Mr. Ali has deposited an amount of $ 150,000 with a bank offer a 13% interest. Bank will make equal installment of interest each year. How much will Mr. Ali receive each year and total amount?

Simple Interest Formula

In simple interest equal amount of interest is paid each year. Therefore bank is offering a simple interest rate and following formula may be used for calculating total interest.
Future value = P ( 1+ Rn)
= $ 150,000 (1+ .13x5)
=$ 150,000 x 1.65
=247,500 (Total amount)
= 247,500 – 150,000 = $ 97,500 (Interest)
=$ 97,500/5 year = $ 19,500 (Interest per year)

R= rate of interest
N = number of period
P = Present value

Alternate Method annual Interest

Interest = P x r
= 150,000 x .13
= $ 19,500 (annual interest)
= $ 19,500 x 5 = $ 97,500 (Total Interest = Annual interest x No of years)
= $ 97,500 + 150,000 = 247,500 (total Future Value= Deposit amount + Total interest)


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Simple Interest Formula

Simple Interest Formula

Future value is calculated by the following simple formula
Future value = P ( 1+ Rn)
R= rate of interest
N = number of period
P = Present value

Simple Interest Formula Example

Bank is offering a simple rate of interest 12% (p.a) . Mr. A wants to deposit $ 25,000 today. How much amount Mr. A will received in 4 years?

= $ 25,000 (1+ .12x4)
=$ 25,000 x 1.48
=37,000


In simple interest the interest in not re invested therefore the interest amount remains same for each year.
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Future Annuity Formula

Future Annuity Formula

Present value of Future annuity may be calculated by the following formula
C x (Annuity Factor) x Discount Factor

Future Annuity Formula Example

Mr. Saleem Khan was offered an annuity amount of $ 25,000 for 3 years. Discount rate for annuity is 5%. Annuity will start after 5 years. Calculate the present value of the annuity.

1.       Calculate the annuity at year 5
= 1-(1.05)-3
        .05
=2.723
= $ 25,000 x 2.723
=68,081

2.       Discount the present value at year zero

$ 68,081 x ( 1+.05)-5
=$68,081 x .7835
=53,343

Tip of future annuity

Two present value are calculated

1.       Present value is calculated at future year by annuity factor (Single Value)
2.       Present value is calculated by discounting the value calculates by annuity factor.
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Annuity Factor Formula

Annuity Factor Formula

1-(1+i)-n
     I

I= Interest Rate
N= number of year of annuity

Annuity Factor Formula Example

Interest Rate 9% and annuity is period is 5 years. Calculate annuity factor?

= 1-(1.09)-5
         .09
=.35/.09
=3.889


Annuity factor may directly be seen in annuity table in year five row under the interest rate of 9%.

Annuity Factor use

Annuity factor is used to calculate the present value of annuity (equal cash flows). For example if the annuity is $ 10,000 (p.a) , then we can find present value of annuity .

= $ 10,000 x 3.889

=38,890
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Annuity Example

Annuity Example

Annuity is equal cash for a limited period of time. Present value of annuity may be either calculated by the following or may be obtained from the annuity table.
1-(1+i)-n
     i

I= Interest Rate
N= number of year of annuity

Annuity Example

Mr. Amin was offered a annuity of $ 20,000 per year. Interest rate for the annuity is 6% .term of the annuity is 4 years .Calculate the annuity factor and present value of annuity?

1.       Annuity Factor

= 1-(1.06)-4
         .06
=.207/.06
=3.45


Annuity of 3.45 may also be checked in annuity table in the year row of 4 under the interest rate of 6%.

2.       Present value of annuity

$ 20,000 x (3.25)
= $ 65,000


Present value of annuity is calculated by multiplying the annuity amount with annuity factor.
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NPV Formula

NPV Formula

NPV may be calculated in two different ways

1.       NPV Formula For Equal Cash Flows

NPV for even cash flow is calculated by multiplying the cash inflow with annuity factor and then deducting the initial investment. Annuity factor either can be calculated by formula or maybe get from the annuity table. The formula may be expressed as under

NPV = C x (Annuity Factor) – Initial investment

C= Cash inflows
Annuity Factor = 1- (1+i)-n
                                       I
NPV Formula Example for Equal Cash Flow

Company expects equal cash flow $ 20,000 for 5 years against an investment o $ 60,000. Discount rate for the investment is 13%. Calculate the net present value.

= 20,000 x 3.517
=$ 70,345
=$ 70,345- $ 60,000
= $ 10,345

2.       NPV Formula for un equal cash Flows

= [C (1+i)-n+ C(1+i)-n]-Initial Investment

C= Net cash flow during the year
I= Desired rate of return

NPV Formula Example for Equal Cash Flow

Company expects cash flow f $ 20,000 for first year and then increase $ 2000 per year for 3 years .Investment for the project $ 40,000.Desired rate of investment is 13%. What would be net present value of project?

Solution

Year
Inflows
Discount Factor
Present Value
0
$ 40,000
1
($ 40,000)
1
$ 20,000
(1+.13)-1 =.884
$ 17,680
2
$ 22,000
(1+.13)-2=.783
$ 17,226
3
$ 24,000
(1+.13)-3=.693
$ 16,632
NPV


$ 11,538


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Average Profit Example

Average Profit Example

Samina & Saima estimated profit for the future 5 years are as under. Calculate the average profit & average rate of return if investment is $ 500,000.

Year 1
$ 60,000
Year 2
$ 70,000
Year 3
$ 50,000
Year 4
$ 60,000
Year 5
$ 65,000

Solution

1.       Average Profit of Company

=   Sum of profit for n years/n years

Year 1
$ 60,000
Year 2
$ 70,000
Year 3
$ 50,000
Year 4
$ 60,000
Year 5
$ 65,000
Total Profit
$ 305,000

Average profit for 5 Years = $ 305,000/5 (Years)
= 61,000 –( Average Profit)

2.       Accounting Rate of Return

= Average profit/Average Investment
= $ 61,000/$500,000 x 100
= 12.2%


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Average Profit Formula

Average Profit Formula

Average profit concept is widely used in many calculations especially for calculating the accounting rate of return. Accounting rate of return is an effective decision making tool for the organization. Average profit provides the average expected performance.

=   Sum of profit for n years/Number of years

Average profit Formula Example

ABC & Company reported the following profit for the last five year .Calculate the average profit

Year 1
$ 50,000
Year 2
$ 55,000
Year 3
$ 60,000
Year 4
$ 70,000
Year 5
$ 45,000

Solution

Year 1
$ 50,000
Year 2
$ 55,000
Year 3
$ 60,000
Year 4
$ 70,000
Year 5
$ 45,000
Sum of profit
$ 280,000

Average profit = $ 280,000/5 (Years)
= 56,000 (average profit)

Average profit & Accounting Rate of Return

Average profit is important component for calculating the Average rate of return and average rate of return is calculated by dividing the average profit with average investment i.e. Average Profit/Average Investment.

If we assume the average investment for above example is $ 200,000, then accounting rate of return may be calculated as under:
= $ 56,000/$200,000 x 100

= 28%
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