Inflation and the value of money

Finance and economics, grade 11 (3 units) Lesson 5 of 14Learning

Inflation and the value of money

Inflation

Inflation is a general rise in prices. In Israel it is measured by the consumer price index: the price of a fixed basket of goods and services – food, housing, transport, electricity – every month.

Inflation of 3% a year means that the basket costs $\displaystyle 1.03$ times more every year. A basket that costs 1,000 ₪ today will cost in five years:

1000\cdot1.03^{5}\approx1159.27

The value of money

When prices grow $\displaystyle q^{n}$ times, the same sum of money buys only $\displaystyle \frac{1}{q^{n}}$ of what it buys today. So to know what a sum $\displaystyle A$ will be worth in $\displaystyle n$ years in today’s prices, divide:

\displaystyle \frac{A}{q^{n}}

Grandpa kept 1,000 ₪ in a drawer for five years, and inflation was 3% a year:

\frac{1000}{1.03^{5}}\approx862.61

At the end they buy only what 862.61 ₪ buys today: the money lost almost 14% of its value.

A bank’s interest against inflation

A deposit keeps the value of the money only if the interest is higher than inflation: the money grows by the interest’s factor, and prices – by inflation’s factor.

A deposit at 2% interest a year, when inflation is 3% a year:

the money grows $\displaystyle 1.02$ times in a year;
prices grow $\displaystyle 1.03$ times;
what the money buys grows $\displaystyle \frac{1.02}{1.03}\approx0.99$ times – that is, it shrinks by about 1%.

In numbers the deposit holds more shekels, but they buy less.

Pay against prices

The same sum shows whether a pay rise is a real rise.
Adi’s pay rose by 2%, and prices rose by 4% that year:

\frac{1.02}{1.04}\approx0.98

With her new pay Adi can buy about 2% less than last year.

How much to deposit today

When you know how much money you want at the end, $\displaystyle A$ , you find how much to deposit today, $\displaystyle A_0$ , by dividing by the same factor: $\displaystyle {A_0}={\frac{A}{q^{n}}}$ .

Parents want their daughter to have 20,000 ₪ in five years, in a deposit at a compound interest of 5% a year:

A_0=\frac{20000}{1.05^{5}}\approx15670.52

Exercise 1

A cinema ticket costs 50 ₪ today.
Inflation is 4% a year.
What will the ticket cost in three years, if its price rises like all prices? Round to agorot.

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

Exercise 2

Noga kept 10,000 ₪ at home for four years.
Inflation was 2.5% a year.
What is her money worth at the end, in the prices of the day she started? Round to agorot.

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

Exercise 3

A five-year deposit pays a compound interest of 3% a year.
In the same years inflation is 3.5% a year.

a

How many times does the money in the deposit grow in the five years? Round to three digits after the point.

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

b

How many times do prices grow in the five years? Round to three digits after the point.

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

c

At the end, does the money from the deposit buy more or less than at the start? Explain.

Exercise 4

Jonathan’s pay is 9,000 ₪ a month, and it rises by 3% every year.
His rent is 3,500 ₪ a month, and it rises by 6% every year.
Today the rent is about 38.9% of the pay.

a

What will Jonathan’s pay be in five years? Round to agorot.

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

b

What will the rent be in five years? Round to agorot.

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

c

What percent of the pay will the rent be then? Round to two digits after the point.

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

Exercise 5

Michal wants to have 30,000 ₪ in six years, for a trip after the army.
The bank offers a deposit at a compound interest of 4% a year.
How much must she deposit today? Round to agorot.

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

Formulas: finance and economics, grade 11 (3 units)

Growth and decay

Simple interest – $\displaystyle A_0$ the principal, $\displaystyle p$ the interest rate a year in percent, $\displaystyle n$ the number of years, $\displaystyle A$ the sum at the end
$\displaystyle A=A_0+n\cdot A_0\cdot\frac{p}{100}$
The growth factor – a rise of $\displaystyle p$ percent
$\displaystyle q=1+\frac{p}{100}$
The decay factor – a fall of $\displaystyle p$ percent
$\displaystyle q=1-\frac{p}{100}$
Growth and decay – $\displaystyle A_0$ the value at the start, $\displaystyle q$ the factor for a period, $\displaystyle n$ the number of periods, $\displaystyle A$ the value at the end
$\displaystyle A$ $\displaystyle =$ $\displaystyle A_0\cdot q^{n}$
The factor of two changes, one after the other
$\displaystyle q_1\cdot q_2$
Today’s value of a sum $\displaystyle A$ in $\displaystyle n$ years, when prices grow $\displaystyle q$ times every year
$\displaystyle \frac{A}{q^{n}}$
The rule of 72 – about how many years until a sum doubles, growing by $\displaystyle p$ percent a year
$\displaystyle n\approx\frac{72}{p}$

The mean and the spread

The mean of $\displaystyle n$ data
$\displaystyle \bar{x}=\frac{x_1+x_2+\dots+x_n}{n}$
The range
$\displaystyle x_{\max}-x_{\min}$
The variance – the mean of the squares of the deviations from the mean
$\displaystyle \sigma^{2}=\frac{(x_1-\bar{x})^{2}+(x_2-\bar{x})^{2}+\dots+(x_n-\bar{x})^{2}}{n}$
The standard deviation
$\displaystyle \sigma=\sqrt{\sigma^{2}}$

Quartiles and deciles

The place of the median in an ordered list of $\displaystyle n$ data
$\displaystyle \frac{n+1}{2}$
The interquartile range – $\displaystyle Q_1$ the lower quartile, $\displaystyle Q_3$ the upper quartile
$\displaystyle Q_3-Q_1$
The gap between the top and the bottom decile – how many times the mean income in decile 10 is that in decile 1
$\displaystyle \frac{\bar{x}_{10}}{\bar{x}_{1}}$