Example Item Year 1 Year 2Year 3

Lecture 7

This lecture note was originally written by a student and partly modified and certified by the

course instructor.

Continuation of Indices

From last lecture, we saw that there are two ways to compute indices for more than one variable:

Mean Relative Index at a period n

=

sum of separate indices at period 𝑛_{number of indices}

=

∑𝐾𝑖=1_𝐾𝐼𝑛𝑖

Simple Aggregate Relative Index at a period n

=

_{sum of value in base period}sum of values in period 𝑛

=

∑_∑𝐾𝑖=1𝑉_𝑉𝑛𝑖

0𝑖 𝐾 𝑖=1

Where 𝐼

𝑛𝑖

is the individual index at period 𝑛 for commodity 𝑖, 𝑉

𝑛𝑖

and 𝑉

0𝑖

are the values for

commodity 𝑖 at period 𝑛 and base period respectively. 𝐾 is the number commodities present.

Disadvantages of the Mean/Simple Aggregate Index

1.

Suppose the price of a loaf of bread was GH¢2 in year 1 and GH¢2.40 in year 2; also the

price of a tonne of butter was GH¢2800 in year 1 and GH¢3000 in year 2,

Simple Aggregate Index for year 2 =

3000+2.4₂₈₀₀₊₂

× 100% = 107.15%

Suppose we were to ignore the value for bread, then

Simple Aggregate Index for year 2 =

3000₂₈₀₀

× 100% = 107.14%

The two index values are almost the same, implying butter has virtually no influence on the

composite index. This is one of the shortcomings of the simple aggregate index technique.

The mean relative index however does not suffer from this shortcoming and must be used

in such circumstances (i.e. when there are huge differences between the values for the

various commodities).

For example, using the mean relative index, we have:

Mean Relative Index at a period n

=

sum of separate indices at period 𝑛_{number of indices}

=

∑𝐾𝑖=1𝐼𝑛𝑖

Index for butter at year 2 =

3000₂₈₀₀

∗ 100% = 107.14

𝐾 = 2

Mean Index for year 2=

120+107.14₂

= 113.57

2.

Mixing of Units

Sometimes, two or more commodities might be in different units which might result in a

meaningless/un-interpretable composite index.

Example: Suppose in 2014, 5000kg of steel and 1000 gallons of diesel were sold, and in

2015, 6000kg of steel and 1200 gallons of diesel were sold. According to simple

aggregate index,

Simple Aggregate Index for year 2015 =

5000𝑘𝑔+1000𝑔_{6000𝑘𝑔+1200𝑔}

× 100%

The value obtain from the resulting composite index is meaningless as we are mixing Apples

and Oranges.

WEIGHTED INDICES

Importance Weight

Most times, a commodity or an aspect of a commodity could be more important compared to

another commodity. In this case, the manufacturer may want to place more emphasis (or

weight) on the most important commodity. This weight must be taken into account when

calculating the indices.

Take the example in the table below. Commodity A has a price of 2 but sold 20,000 units

whereas commodity B’s price is 1000 but sold only 2. Basing a simple aggregate index on

price/quantity will yield result that ignores the size of quantity or price. When emphasis is on

price, commodity B is more important than commodity A. The opposite is true when emphasis

is on quantity. Thus, when one is after price index, the quantity values could be used as

weights. Similarly, when quantity index is desired, the price values could be used as weights.

Price

Quantity

A

2

20,000

Weighted Indices

The above reasoning leads to a modification of the previous indices for more than one variable:

a.

Weighted Mean (Average) Relative Index

𝐼

𝑊𝑀

=

∑ 𝑊𝑖𝐼𝑛𝑖

𝑛 𝑖=1

∑𝑛𝑖=1𝑊𝑖

where, W

i

= weight factor for commodity i

I

ni

= index for commodity i at period n

b.

Weighted Aggregate Index

𝐼

𝑊𝐴

=

∑ 𝑊𝑖𝑉𝑛𝑖

𝑛 𝑖=1

∑𝑛𝑖=1𝑊𝑖𝑉_0𝑖

× 100%

where, W

i

= weight factor for commodity i

V

ni

= value for commodity i at time n

V

0i

= value for commodity i at base time period

Example

Item Year 1 Year 2 Year 3

Price (P0) Price (Pn) Weight (Wi)

A 20 24 6

B 55 51 4

C 63 84 1

D 28 34 8

a.

Calculate the Weighted Mean Price Index

b.

Calculate the Weighted Aggregate Price Index

Solution:

a.

𝑰𝒏𝒊 𝑾𝒊𝑰𝒏𝒊

A ₂₄

20

� × 100% = 120% 6 × 120 = 720

B ₅₁

55

� × 100% = 92.73 4 × 92.73 = 370.92

C ₈₄

63

� × 100% = 133.33 1 × 133.33 = 133.33

D ₃₄

28

� × 100% = 121.43 8 × 121.43 = 971.44

Total 2195.69

𝐼𝑊𝑀 =2195.69₁₉ = 115.56%

b. .

𝑾𝒊𝑷𝒏𝒊 𝑾𝒊𝑷𝟎𝒊

A _{6 × 24 = 144 6 × 20 = 120}

B _{4 × 51 = 204}

4 × 55 = 220

C _{1 × 84 = 84 1 × 63 = 63}

D _{8 × 34 = 272 8 × 28 = 224}

Total _{704 627}

𝐼𝑆𝐴=704_{627 × 100% = 122.20%}

SPECIAL INDICES

Laspeyres Indices (Base-weight Index)

This is a special case of a weighted aggregate index which uses base time period value as weight. It is commonly associated with price and quantity.

Laspeyres Price Index: This uses base period quantity as the weight. Laspeyres Quantity Index: This uses base period price as the weight. Illustration

Item P0 Pn Q0 Qn

A 10 15 95 100

B 20 25 4 2

Finding Laspeyres Price Index: 𝐿𝑝=∑ 𝑞0𝑖𝑝𝑛𝑖

𝑛 𝑖=1

∑𝑛_𝑖=1𝑞_0𝑖𝑝_0𝑖

Finding Laspeyres Quantity Index:

𝐿𝑞 =

∑𝑛_𝑖=1𝑝_0𝑖𝑞_𝑛𝑖 ∑𝑛 𝑝_0𝑖

Paasche (Current-weighted) Indices

This is a special case of the weighted aggregate indices which uses current time period values as the weight for the index. It is commonly associated with price and quantity.

Paasche Price Index uses current time period quantities as weight. Paasche Quantity Index uses current time period price as weight. Finding Paasche Price Index:

𝑃𝑝=∑ 𝑞𝑛𝑖𝑝𝑛𝑖

𝑛 𝑖=1

∑𝑛_𝑖=1𝑞_𝑛𝑖𝑝_0𝑖

Finding Paasche Quantity Index: 𝑄𝑞=∑ 𝑝𝑛𝑖𝑞𝑛𝑖

𝑛 𝑖=1

∑𝑛_𝑖=1𝑝_𝑛𝑖𝑞_0𝑖

Example

A company buys four products with the following features;

Number of units Price paid per unit

Item Year 1 Year 2 Year 1 Year 2

A 20 24 10 11

B 55 51 23 25

C 63 84 17 17

D 28 34 19 20

a. Calculate the base-weighted price and quantity index.

b. Calculate the current-weighted price and quantity index.

Solution:

a.

𝐿𝑝=(20 × 11) + (55 × 25) + (63 × 17) + (28 × 20)_{(20 × 10) + (23 × 55) + (63 × 17) + (28 × 19) × 100% = 105.15%}

𝐿𝑞 =(10 × 24) + (23 × 51) + (17 × 84) + (19 × 34)_{(10 × 20) + (23 × 55) + (17 × 63) + (19 × 28) × 100% = 113.66%}

b.

𝑃𝑝=(24 × 11) + (51 × 25) + (84 × 17) + (34 × 20)_{(24 × 10) + (51 × 23) + (84 × 17) + (34 × 19) × 100% = 104.59%}

INTRODUCTION TO FINANCIAL MATHEMATICS

A.

CASHFLOW

It is the inflow and outflow of money over a period of time.

Tools used to represent Cash flow

Cash flow diagram

Cash flow table

A cash flow table lists the period and the corresponding cash in a table. However, unlike a cash flow diagram, it doesn’t show whether cash is received at the beginning or at the end of the period.

Example: Depict the following cash flows of a company on a cash flow diagram.

a. When money is paid out/received at the beginning of the year? b. When money is paid out/received at the end of the year?

Solution

a.

b.

Year 1 2 3 4 5 6 7

B.

INTEREST AND INTEREST RATES

Suppose the principal (P) is the current amount of money and I is the interest earned over a

time period n. If Pn is the amount of money attained at period n, then:

Interest (I) at period 𝑛= 𝑃_𝑛− 𝑃₀

The business world however, prefers the use of ‘interest rate’ which is the percentage of the interest earned over the principal than ‘interest’.

Let 𝑖 be interest rate. Then: 𝑖=_𝑃𝐼

0× 100%

Thus, Interest (I) = 𝑃₀× 𝑖

Types of Interest

- Simple Interest (for very short-life projects and usually smaller interest rate) - Compound Interest

Simple Interest Compound Interest

Year Beginning _amount Interest Ending _amount Beginnin_{g amount Interest} Ending amount 0

P 0 P P 0 P

1 𝑃 P × 𝑖 P + P𝑖 𝑃 P × 𝑖 P + P𝑖 = 𝑃(1 + 𝑖)

2 P + P𝑖 P × 𝑖 P + 2P𝑖 𝑃(1 + 𝑖) 𝑃(1 + 𝑖) ∗ 𝑖 𝑃(1 + 𝑖) + 𝑃(1 + 𝑖) ∗ 𝑖_{= 𝑃(1 + 𝑖)}2

3 P + 2P𝑖 P + 3P𝑖 𝑃(1 + 𝑖)2 _{𝑃(1 + 𝑖)}2_{∗ 𝑖} 𝑃(1 + 𝑖)2+ 𝑃(1 + 𝑖)2

∗ 𝑖 = 𝑃(1 + 𝑖)3 4 P + 3P𝑖 P + 4P𝑖 𝑃(1 + 𝑖)3 _{𝑃(1 + 𝑖)}3_{∗ 𝑖 𝑃(1 + 𝑖)}3_{+ 𝑃(1 + 𝑖)}3

∗ 𝑖 = 𝑃(1 + 𝑖)4

… … … …

n P + (n − 1)P𝑖 P + nP𝑖 𝑃(1 + 𝑖)𝑛−1 _{𝑃(1 + 𝑖)}𝑛−1_{∗ 𝑖} 𝑃(1 + 𝑖) 𝑛−1

+ 𝑃(1 + 𝑖)𝑛−1_{∗ 𝑖}

= 𝑃(1 + 𝑖)𝑛

For ‘Simple’: Total Interest =𝑛𝑃𝑖 𝑖= interest rate

𝑃= Beginning amount 𝑛= time period

For compound: Total Interest =𝑃(1 + 𝑖)𝑛_{− 𝑃}

𝑃=Present value 𝑖=interest rate 𝑛= future value

Total amount after period 𝑛 = 𝑃(1 + 𝑖)𝑛

Thus compound interest is the interest earned not only on the original principal, but also on all interests earned previously. In other words, at the end of each year, the interest earned is added to the original amount (principal) and the money is reinvested.

The formula, 𝐹 = 𝑃(1 + 𝑖)𝑛 _{is at the heart of many calculations in finance.}

Simple vs Compound Interest

First, we observe that simple interest is easy to calculate. However, it does not tell the true picture of how interest is calculated in practice. So when would one apply Simple Interest?

Let’s compare the two formulas:

For simple, the future amount is: 𝑃 + 𝑛𝑃𝑖 = 𝑃[1 + 𝑛𝑖]

For compound, the future amount is: 𝑃(1 + 𝑖)𝑛

For one to prefer ‘simple’ (since it is easier to calculate) over ‘compounding’, we must have the following:

𝑃[1 + 𝑛𝑖] ≈ 𝑃(1 + 𝑖)𝑛

This can occur only when 𝑛 and 𝑖 are really small (and to some extent not very large 𝑃)

There is a special case when 𝑛 = 1. Then, irrespective of the size of 𝑖, both simple and compound yield the same amount since 𝑃[1 + 𝑛𝑖] = 𝑃(1 + 𝑖)𝑛_{= 𝑃(1 + 𝑖) when 𝑛 = 1.}

Example: When 𝑖 is really small

Suppose 𝑛 is 10 and 𝑖 = 1%. Then

For simple, [1 + 10 ∗ 0.01] = 1.1

For compound, (1 + 0.01)10_{= 1.1046}