By the end of this chapter, you will be able to:
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This chapter introduces index numbers, essential tools in banking and finance for measuring changes in economic variables such as prices, quantities, or values over time. Index numbers enable professionals to monitor inflation, cost of living, and investment performance, crucial for decision-making in Kenya’s dynamic financial environment. Understanding the formulae and computation methods of index numbers equips finance practitioners with analytical skills to interpret market trends and economic data accurately.
Index numbers quantify relative changes between a base period and a comparison period. The formulas differ depending on the type of index number and the weighting scheme used. This section presents the main formulae used in banking and finance to compute index numbers, focusing on price and quantity indices relevant to Kenya’s economic data analysis.
A price index number measures the relative change in the price of a basket of goods or services over time. The general formula is:
$$ \text{Price Index} = \frac{\text{Price in current period}}{\text{Price in base period}} \times 100 $$
This formula is modified to incorporate quantities or weights depending on the index type.
Example 1: A bank tracks the price of a financial service that cost Ksh 200 in the base year and Ksh 250 in the current year. Calculate the price index.
Given: Base price = Ksh 200, Current price = Ksh 250
$$ \text{Price Index} = \frac{250}{200} \times 100 $$
$$ = 1.25 \times 100 $$
Answer: 125
Example 2: The price of a share was Ksh 150 in 2019 and Ksh 180 in 2020. Compute the price index for 2020 using 2019 as the base.
Given: Base price = Ksh 150, Current price = Ksh 180
$$ \text{Price Index} = \frac{180}{150} \times 100 $$
$$ = 1.2 \times 100 $$
Answer: 120
Example 3: A SACCO’s loan interest rate rose from 12% to 15% in one year. Find the price index for interest rates.
Given: Base rate = 12%, Current rate = 15%
$$ \text{Price Index} = \frac{15}{12} \times 100 $$
$$ = 1.25 \times 100 $$
Answer: 125
A quantity index number measures the relative change in quantities of goods or services over time, often used for volume analysis in banking transactions or commodity trading.
$$ \text{Quantity Index} = \frac{\text{Quantity in current period}}{\text{Quantity in base period}} \times 100 $$
Example 1: A bank processed 500 loan applications in 2018 and 600 in 2019. Calculate the quantity index for 2019.
Given: Base quantity = 500, Current quantity = 600
$$ \text{Quantity Index} = \frac{600}{500} \times 100 $$
$$ = 1.2 \times 100 $$
Answer: 120
Example 2: The number of mobile money transactions increased from 2,000,000 in 2020 to 2,400,000 in 2021. Find the quantity index.
Given: Base quantity = 2,000,000, Current quantity = 2,400,000
$$ \text{Quantity Index} = \frac{2,400,000}{2,000,000} \times 100 $$
$$ = 1.2 \times 100 $$
Answer: 120
Example 3: A retail bank’s deposits increased from Ksh 1 billion to Ksh 1.2 billion in a year. Compute the quantity index.
Given: Base quantity = 1,000,000,000, Current quantity = 1,200,000,000
$$ \text{Quantity Index} = \frac{1,200,000,000}{1,000,000,000} \times 100 $$
$$ = 1.2 \times 100 $$
Answer: 120
Weighted indices assign importance to items based on quantities or values, reflecting their relative significance in the basket.
$$ \text{Weighted Price Index} = \frac{\Sigma (p_1 \times q_0)}{\Sigma (p_0 \times q_0)} \times 100 $$
Where \(p_1\) = price in current period, \(p_0\) = price in base period, and \(q_0\) = quantity in base period.
Example 1: A bank’s investment portfolio has 3 assets with base prices \(p_0 = [100, 150, 200]\) and quantities \(q_0 = [10, 5, 8]\). Current prices \(p_1 = [110, 160, 210]\). Calculate the weighted price index.
Given: \(p_0 = [100, 150, 200]\), \(q_0 = [10, 5, 8]\), \(p_1 = [110, 160, 210]\)
$$ \text{Weighted Price Index} = \frac{(110 \times 10) + (160 \times 5) + (210 \times 8)}{(100 \times 10) + (150 \times 5) + (200 \times 8)} \times 100 $$
$$ = \frac{1100 + 800 + 1680}{1000 + 750 + 1600} \times 100 $$
$$ = \frac{3580}{3350} \times 100 = 1.06716 \times 100 $$
Answer: 106.72
Example 2: A bank’s loan portfolio includes 2 types: personal loans with quantity 1000 at base price 10% interest and mortgage loans with quantity 500 at base 7% interest. Current rates are 11% and 8%. Find the weighted price index.
Given: \(p_0 = [10, 7]\), \(q_0 = [1000, 500]\), \(p_1 = [11, 8]\)
$$ \text{Weighted Price Index} = \frac{(11 \times 1000) + (8 \times 500)}{(10 \times 1000) + (7 \times 500)} \times 100 $$
$$ = \frac{11000 + 4000}{10000 + 3500} \times 100 = \frac{15000}{13500} \times 100 = 1.1111 \times 100 $$
Answer: 111.11
Example 3: A cooperative society has 3 products with prices and quantities as follows: base prices [Ksh 50, 80, 120], quantities [20, 15, 10], current prices [55, 85, 130]. Calculate weighted price index.
Given: \(p_0 = [50, 80, 120]\), \(q_0 = [20, 15, 10]\), \(p_1 = [55, 85, 130]\)
$$ \text{Weighted Price Index} = \frac{(55 \times 20) + (85 \times 15) + (130 \times 10)}{(50 \times 20) + (80 \times 15) + (120 \times 10)} \times 100 $$
$$ = \frac{1100 + 1275 + 1300}{1000 + 1200 + 1200} \times 100 = \frac{3675}{3400} \times 100 = 1.0809 \times 100 $$
Answer: 108.09
Price relatives compare individual item prices between two periods and are the basis for many index numbers.
$$ \text{Price Relative} = \frac{p_1}{p_0} \times 100 $$
Example 1: The price of maize was Ksh 40/kg last year and Ksh 45/kg this year. Find the price relative.
Given: \(p_0 = 40\), \(p_1 = 45\)
$$ \text{Price Relative} = \frac{45}{40} \times 100 = 1.125 \times 100 $$
Answer: 112.5
Example 2: A bank’s service fee increased from Ksh 500 to Ksh 600. Calculate the price relative.
Given: \(p_0 = 500\), \(p_1 = 600\)
$$ \text{Price Relative} = \frac{600}{500} \times 100 = 1.2 \times 100 $$
Answer: 120
Example 3: The price of a KRA tax form rose from Ksh 200 to Ksh 220. Find the price relative.
Given: \(p_0 = 200\), \(p_1 = 220\)
$$ \text{Price Relative} = \frac{220}{200} \times 100 = 1.1 \times 100 $$
Answer: 110
Calculate the price index for a bank loan interest rate that increased from 9% to 11%. (3 marks)
A retail chain’s sales volume increased from 10,000 units to 12,000 units. Compute the quantity index. (3 marks)
An investment portfolio has base prices [Ksh 100, 200] and quantities [50, 30]. Current prices are [Ksh 110, 210]. Calculate the weighted price index. (5 marks)
Find the price relative for a commodity whose price changed from Ksh 80 to Ksh 96. (3 marks)
A SACCO’s service fee increased from Ksh 150 to Ksh 165. Calculate the price relative. (3 marks)
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Create a free accountThis chapter focused on the use of index numbers in business mathematics and statistics. It began by introducing the formulae necessary for computing various types of index numbers, establishing the foundational calculations. The chapter then detailed the computation methods for four key index numbers: Laspeyre's, which uses base period quantities; Paasche's, which relies on current period quantities; Fisher's ideal, which combines Laspeyre's and Paasche's indices for a balanced measure; and Marshal's, which is an average of the Laspeyre's and Paasche's indices. Each method was explained with its respective formula and calculation approach. Finally, the chapter highlighted the practical application of index numbers in preparing reports and aiding business decision making, emphasizing their importance in analyzing economic and financial trends. The understanding of these index numbers equips students with tools to measure changes in prices, quantities, and values over time effectively.
The price of a banking service in 2019 was Ksh 120 and in 2020 it increased to Ksh 132. Calculate the simple price index number for 2020 using 2019 as the base year. (5 marks)
A bank’s loan portfolio had values of Ksh 50 million in 2018 and Ksh 60 million in 2019. The number of loans disbursed was 100 in 2018 and 120 in 2019. Calculate Laspeyre’s quantity index for 2019 using 2018 as the base year. (6 marks)
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