Rate of Interest

Rate by which Principal amount increase with the time period. There are three types of interest

1-      Nominal rate of interest

2-      Effective rate of interest

3-      Equivalent rate of interest

1)      Nominal rate of interest:

When rate of interest is apply for one year or the rate which is written on the document is called nominal rate of interest.

2)      Effective rate of interest:

When nominal rate of interest is apply on quarterly basis or monthly basis is called effective rate of interest. Because effectiveness of interest increase due to shorter period compounding.

Effective rate of interest = [1+r/m]^n*m100

m = no. of compounding periods

r = nominal rate of interest

3)      Equivalent rate of interest:

Whenever two different rate of interest for different time periods give the same future value is called equal lent rate of interest

For example 18 % per annum = 1.38888843 % per month will give us the same future value

Equal lent rate of interest = [(1+r_y)^1/m-1]100

m  = no. of periods

Q no 10:

Suppose you were to receive $ 1000 at the end of 10 year. If your opportunity rate is 10 percent, what is the present value of this if interest is compounded a) annually? (b)quarterly?(c)continuously ?

Solution:

When annually interest rate:

F.V = $ 1000

n = 10 year

r = 10 %

P.V =F.V (1+r)^-n

P.V = 1000(1+0.10)^-10

P.V = $ 385.55

When interest rate is quarterly:

P.V = F.V (1+r/m)^-n*m

P.V = 1000(1+0.10/4)^-10*4

P.V = $ 372.43

When interest rate continuously:

P.V = F. V (e)^-r*n

Where

e = 2.71828

P.V = $ 367.89

Q no 6:

Vernal Equinox wishes to borrow $ 10000 for three years. A group of individuals agree to lend him this amount if he contracts to pay them $ 16000 at the end of the three years. What is the implicit compound annual interest rate implied by this contract ( to the nearest whole present)?

Solution:

P.V = 10000

F.V = 16000

n = 3 years

r =?

F.V = P.V(1+r)ⁿ

16000 = 10000(1+r)³

1.6^1/3 = (1+r)

r = 16.96%

Annuity

Annuity is equal instalments for payments or receipts for specific period of time.

Or

Annuity is a series of equal payments or receipts occurring over a specific period of time.

Ist of annuity:

·         Ordinary annuity:

In which instalments paid or receive at the end of the each period.                                                                                                                                                                                                                   

                       

          0                     1^st year           2^nd year            3^rd year            4rd year             5^th year               

                               100                     100                   100                    100                      100                      

We can find

Future value annuity =  P[(1+r)ⁿ-1/r]

Present value annuity = P[1-(1+r)^-n/r]

Q # 3(Ordinary annuity)

Joe Hernandez has inherited $ 25000 and wishes to purchase an annuity that will provide him with a steady income over the next 12 years. He has heard that the local saving and loan association is currently paying 6 percent compounded interest on an annual basis. If he were to deposit his funds, what year -end equal dollar amount ( to the nearest dollar ) would  he  able to withdraw annually such that he would have  a zero valance after  his last withdrawal 12 years from now?

Solution:

 Rate of interest = 6 % p.a

No of year = 12

P.V = $ 25000

Installments (p) = ?

We know the formula of P.V annuity

P.V = P[ 1-(1+r)^-n/r]

25000 = P[1-(1+0.06)^-12/0.06]
P = 2982 will withdraw annuity 


Q # 4 ( Ordinary annuity)
You need to have $ 50000 at the end of 10 year. To accumulate this sum, you have decided to save a certain amount at the end of each year of next 10 years and deposit it in the bank. The bank pay 8 % interest compounded annually for long-term deposits. How much will you have to save each year (to the nearest dollar)?

Solution:

 F.V = $ 50000

n = 10 year

r = 8 % p.a

Installments (p) = ?

By using F.V annuity formula

F. V a = P[(1+r)ⁿ-1/r]

50000 = P[(1+0.08)¹⁰-1/0.08]

P = 3453.04

Amortizing Example

S.C Problem No 9:

Solution:

Year	Installments	Principal	Interest	Balance
0	-	-	-	50000
1	7451.47	3451.47	4000	46548.53
2	7451.47	3727.588	3723.882	42820.94
3	7451.47	4025.795	3425.675	38795.15
4	7451.47	4347.858	3103.612	34447.29
5	7451.47	4695.687	2755.783	29751.6
6	7451.47	5071.342	2380.128	24680.26
7	7451.47	5477.049	1974.421	19203.21
8	7451.47	5915.213	1536.257	13288
9	7451.47	6388.43	1063.04	6899.568
10	7451.47	6899.505	551.9654	0

B:

Total interest paid = 24514.76

Simple interest

There are three factors which affect the amount of interest principal, rate and time so we can say that

“Whenever interest is paid or earn on principal amount is called simple interest.”

year		Interest	Balance
1	100000@10%p.a	10000	110000
2	100000@10%p.a	10000	120000
3	100000@10%p.a	10000	130000
4	100000@10%p.a	10000	140000

Discounts offer during the sale or purchase of commodity is the example of simple interest.

Formula:

Simple Interest = rate * time * principal

Compound Interest

“Whenever interest is earn/paid on interested amount or on principle plus interest amount is called compound interest”

Year		Interest	Balance
1	100000*10%p.a	10000	110000
2	110000*10%p.a	11000	121000
3	121000*10%p.a	12100	133100
4	133100*10%p.a	13310	146410

Examples:

In banks loans or lease of assets e.t.c is the example of compound interest.

Formulas:

Future value = Present value (1+ rate of interest)^ no. of periods

Present value = Future value / (1+ rate of interest)^ no. of periods

Q # 1

The following are exercises in future (terminal) values:

At The end of three years, how much is an initial deposit of $ 100 worth, assuming a compound annual interest rate of 10 percent?

Solution a:

P.V = $100

r = 10 % p.a

n = 3 years

F.V = P.V(1+r)ⁿ

F.V= 100(1+0.10)³

F.V = 133.1

Q no 2 (Present value example)

The following are exercise in present value:

(a):$ 100 at the end of three year is worth how much today , assuming a discount rate of 10 %?

Solution:

We know that

P.V = F.V (1+r)^-n

P.V = 100 (1+0.10)^-3

P.V = 75

(b):What is the aggregate present value of $ 500 receive at the of each of the next three year, assuming a discount of 25 %?

Solution:

P.V = 500(1+0.25)^-3

P.V = 256

Question no 8:

Sales of the P.J. Cramer company were $ 500000 this year, and they are expected to grow at a compound rate of 20 % for next 6 years. What will be the sales figure at the  end of each of the next six year?  

Solution:

P.V = 500000

r = 20 %

 We will find the F.V at the end of each next six years

F.V = p(1+r)ⁿ

F.V year 1 = 500000(1+0.20) = $ 600000

F.V year 2 = 500000(1+0.20)²= $ 720000

F.V year 3 = 500000(1+0.20)³=$ 864000

F.V year 4 = 500000(1+0.20)⁴=$ 1036800

F.V year 5 = 500000(1+0.20)⁵=$1244160

F.V year 6 = 500000(1+0.20)⁶=$1492992

Boston Cosulting Group Matrix

Stars:

 SBU's placed in this cell are highly attractive because the industry in which they are located is robust and the business has a strong competitive position in the industry. Stars generate large amounts of cash, but also require heavy investment to continue to grow and to maintain competitive positioning.e.g. Pepsi , Coka Cola ,USB Business,

Pepsi:

In Pakistan Market share of pepsi is approximately 47 % and company has a strong position in market .Pepsi every year invest large amount of money to grow and to maintain competitive positioning.

Cash Cows:

  These SBU's are the corporation's key source of cash, and are typically the core business. They possess a strong competitive position, but are located in an industry that is mature, not growing or declining. A Cash Cow generates more cash than it requires, providing funds to the corporation to invest in other ventures. E.g. Mobilink JAZZ, Automobile industry, Fuji Fertilizers,

Mobilink:Similarly Mobilink  Blackberry handsets which are used by the corporate customers generate heavy revenues. Jazz there is no heavy investment from the company on these brands because they are there maturity stages .

Question Marks

 When a business is located in a growing industry, but has not achieved a strong competitive position, an attempt to evaluate further investment is shrouded by ambiguity as to eventually becoming a "winner". These are termed "Question Marks" because they deserve attention to determine if the venture can be viable.E.g Zong , National Bank

Zong:

We can see that zong has much invest on there promotions but there market share is still there. Company management should find strategy problems.

Dogs :

 A business situated in a low growth or declining industry, such as at the decline stage, has a precarious future. If the business has not already developed a strong competitive advantage, it should be divested. E.g. Candia Milk, Pakistan Railways,

Pakistan Railways:

Pakistan Railways declining day by day A dog may not require substantial cash, but it ties up capital that could better be deployed elsewhere. Unless a dog has some other strategic purpose, it should be liquidated if there is little prospect for it to gain market share

Important Legal Documents

GPOA:

General power of attorney is a document which is write when a person agree to transfer all his rights or authority of property to other person. It is just like a blank cheque in which giver of this document did not specify any terms and conditions.

Elements of GPOA:

·        Write on 1000 Rs stamp paper.

·        It is sign , dated by giver  and taker of authority of  GPOA.

·        The place of sign must be mentioned(Because the law of jurisdiction).

·        It has to be noted by Notary Public.

·        It has two witnesses.

SPOA:

Specific power of attorney is a type of legal document in which giver specify his authority while giving to other person.

Elements Of SPOA:

·        Written on Rs 500 stamp paper.

·        It is for a specific cause only which is well defined and cleared.

·        Job and  time specific.

·        It is strongly recommended that duplicate power of attorney sign by both parties.

IPOA:

Irrevocable power of attorney is just like GPOSA and it can not be cancel.

Wakalat Nama:

It is letter which is given by client to lawyer so that he or she represent the case, legal issue  of client in court. It is important to ensure that  is completely fill out and accepted by lawyer . It should be explicitly agreed term and conditions.

Note: One wakalat nama for one case only and it nature is like SPOA.

Summan:

It is a legal formal notice of information to the respondent so that he can reply to case instituted by complain tent.

Types:

·        Simple notice

·        Registered post

·        Registered ad

·        Carrier

·        Past on wall 
-   Announcement

Difference between Notary Public and Oath commissioner:

Both are professional licence holder but oath commissioner only sign and test affidavit. Notary public can sign and test all documents except affidavit.

If There is a will there is a way.