Comparing banks' offers

Finance and economics, grade 11 (3 units) Lesson 6 of 14Learning

Comparing banks' offers

The factor for the same period

To compare two interest rates, work out the factor of each for the same period – usually a year.

Bank A: 4% a year – the factor for a year $\displaystyle 1.04$ .
Bank B: 2% every half-year – a year has two halves, and the factor for a year is $\displaystyle {1.02^{2}}={1.0404}$ .
Bank C: 0.33% a month – the factor for a year $\displaystyle 1.0033^{12}\approx1.0403$ .

Bank B gives the most: 4.04% a year. Interest that joins the deposit more often starts earning interest sooner.

An offer with a gift

When an offer has a fixed sum too, work out the sum at the end in each offer – and the answer can change with the number of years.

10,000 ₪ in a deposit:
Bank A – a compound interest of 4% a year.
Bank D – a compound interest of 3.5% a year, and a gift of 300 ₪ at the end.

	After 3 years	After 10 years
Bank A	$\displaystyle 10000\cdot1.04^{3}\approx11248.64$	$\displaystyle 10000\cdot1.04^{10}\approx14802.44$
Bank D	$\displaystyle 10000\cdot1.035^{3}+300\approx11387.18$	$\displaystyle 10000\cdot1.035^{10}+300\approx14405.99$

For three years Bank D is better; for ten years – Bank A. The gift is fixed, and the gap in interest grows from year to year.

A loan: interest against a fee

For a loan, compare how much is paid back altogether; the cheaper offer is the one where less is paid back.

Amit takes a loan of 20,000 ₪, and will pay all of it back in two years:
Offer A – a compound interest of 6% a year.
Offer B – no interest, and a fee of 2,000 ₪.

Offer A: $\displaystyle {20000\cdot1.06^{2}}={22472}$ ₪.
Offer B: $\displaystyle {20000+2000}={22000}$ ₪.

Offer B is cheaper by 472 ₪.

Exercise 1

Bank X offers a compound interest of 6% a year.
Bank Y offers a compound interest of 1.5% every quarter (three months).

a

Work out the factor for a year in Bank Y. Round to four 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

Which bank gives more? By how many percent a year?

Exercise 2

Lior wants to deposit 15,000 ₪ for four years.
Offer A – a compound interest of 5% a year.
Offer B – a simple interest of 5.5% a year.

a

How much money will the deposit hold at the end, in offer A? 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

How much money will the deposit hold at the end, in offer B?

/ – 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

Which offer should Lior choose?

Exercise 3

Shiran takes a loan of 12,000 ₪, and will pay all of it back in three years.
Offer A – a compound interest of 7% a year.
Offer B – a compound interest of 0.5% a month.

a

How much will Shiran pay back in offer A? 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

How much will Shiran pay back in offer B? 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

Which offer is cheaper, and by how much?

/ – 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 4

A bank’s advert says: “A 12% return in three years” – the money will grow by 12%.
Another bank offers a compound interest of 3.9% a year.
Which offer gives more in three years? Work it out and explain.

/ – 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
$\displaystyle \frac{\bar{x}_{10}}{\bar{x}_{1}}$