{"id":58,"date":"2023-12-31T02:50:47","date_gmt":"2023-12-31T02:50:47","guid":{"rendered":"https:\/\/openbook.ums.edu.my\/financialmathematicsineconomics\/chapter\/compound-interest\/"},"modified":"2024-10-08T07:39:22","modified_gmt":"2024-10-08T07:39:22","slug":"compound-interest","status":"publish","type":"chapter","link":"https:\/\/openbook.ums.edu.my\/financialmathematicsineconomics\/chapter\/compound-interest\/","title":{"raw":"3.1 Compound Interest","rendered":"3.1 Compound Interest"},"content":{"raw":"

An interest paid only on the principal is called simple interest. When the interest of each period is added to the principal in computing the interest for the next period, it is called compound interest.<\/p>\r\n

Interest amount computed at the end of a period is added to a single principal sum. That is, at the end of each interest period the interest earned for that period is added to the principal so that the interest also earns interest over the next interest period. In the same way, interest due on a debt at the end of a period is subject to interest in the next period.<\/p>\r\n

Suppose that $P$ is deposited at a rate of interest $r$ per year. The amount on deposit at the end of the first year is found by the simple interest formula with t = 1;<\/p>\r\n

\\begin{equation}<\/p>\r\n

$S = P(1 + r .1)$\u00a0 \u00a0 \u00a0 \u00a0 \\label{eq:1}<\/p>\r\n

\\end{equation}<\/p>\r\n

If the deposit earns compound interest, the interest earned during the second year is paid on the total amount on the deposit at the end of the first year. Using the formula $S = P(1 + rt )$ again, with $P$ replaced by $P(1 + r )$ and t = 1, gives the total amount on deposit at the end of the second year;<\/p>\r\n

\\begin{equation}<\/p>\r\n

$S = [P(1 + r )] (1 + r .1) $ \\label{eq:2}<\/p>\r\n

\\end{equation}<\/p>\r\n

In the same way, the total amount on deposit at the end of the third year is;<\/p>\r\n

[latexpage]<\/p>\r\n

\\begin{equation}<\/p>\r\n

$S = P(1+r)^{3}$\u00a0 \\label{eq:3}<\/p>\r\n

\\end{equation}<\/p>\r\n

Generalizing, in t years the total amount on deposit is;<\/p>\r\n

\\begin{equation}<\/p>\r\n

$S = P(1+r)^{t}$\u00a0 \u00a0 \\label{eq:4}<\/p>\r\n

\\end{equation}<\/p>\r\n

called the compound amount.<\/p>\r\n\r\n

3.1.1 The Formula \u00a0<\/strong><\/h1>\r\n

The compound amount can be calculated using the following formula:<\/p>\r\n

Compound amount:<\/p>\r\n

\\begin{equation}<\/p>\r\n

$S = P(1 + i)^{n}$ \\label{eq:5}<\/p>\r\n

\\end{equation}<\/p>\r\n

where i = r\/m and n = mt ;<\/p>\r\n

S = future (maturity) value, or the compound amount of P, or the accumulated value of P<\/p>\r\n

P = original principal, or the present value of P, or the discounted value of S<\/p>\r\n

i = interest rate per period<\/p>\r\n

r = annual interest rate<\/p>\r\n

m = number of compounding periods per year<\/p>\r\n

n = number of compounding periods<\/p>\r\n

t = number of years<\/p>\r\n\r\n

Example 3.1<\/header>\r\n
\r\n\r\nConsider a sum of RM8,200 is deposited into a time deposit account\u00a0 today that pays 5% p.a. How much will it be in the next 5 years if compounded quarterly?\r\n\r\n[h5p id=\"10\"]\r\n\r\n<\/div>\r\n<\/div>\r\n
Example 3.2<\/header>\r\n
\r\n\r\nAli invests RM5000 at 6.2% p.a. with interest compounded\u00a0 monthly. What would his investment be worth after five years? What\u00a0 amount of interest has been earned during the five years?\r\n\r\n[h5p id=\"11\"]\r\n\r\n<\/div>\r\n<\/div>\r\n
Example 3.3<\/header>\r\n
\r\n\r\nHow much money will be required on 31 December 2003 to repay a loan of RM2000 made on 31 December 2000 if i=12% compounded quarterly?\r\n\r\n[h5p id=\"12\"]\r\n\r\n<\/div>\r\n<\/div>\r\n
Example 3.4<\/header>\r\n
\r\n\r\nWhat amount of money invested today will grow to RM1000 at the end of 5 years if i=18% compounded quarterly?\r\n\r\n[h5p id=\"13\"]\r\n\r\n<\/div>\r\n<\/div>\r\n

<\/p>","rendered":"

An interest paid only on the principal is called simple interest. When the interest of each period is added to the principal in computing the interest for the next period, it is called compound interest.<\/p>\n

Interest amount computed at the end of a period is added to a single principal sum. That is, at the end of each interest period the interest earned for that period is added to the principal so that the interest also earns interest over the next interest period. In the same way, interest due on a debt at the end of a period is subject to interest in the next period.<\/p>\n

Suppose that \"P\" is deposited at a rate of interest \"r\" per year. The amount on deposit at the end of the first year is found by the simple interest formula with t = 1;<\/p>\n

<\/a><\/p>\n

(1) <\/span>   <\/span>\"\begin{equation*}<\/p>\n

If the deposit earns compound interest, the interest earned during the second year is paid on the total amount on the deposit at the end of the first year. Using the formula \"S = P(1 + rt )\" again, with \"P\" replaced by \"P(1 + r )\" and t = 1, gives the total amount on deposit at the end of the second year;<\/p>\n

<\/a><\/p>\n

(2) <\/span>   <\/span>\"\begin{equation*}<\/p>\n

In the same way, the total amount on deposit at the end of the third year is;<\/p>\n

\n

<\/a><\/p>\n

(3) <\/span>   <\/span>\"\begin{equation*}<\/p>\n

Generalizing, in t years the total amount on deposit is;<\/p>\n

<\/a><\/p>\n

(4) <\/span>   <\/span>\"\begin{equation*}<\/p>\n

called the compound amount.<\/p>\n

3.1.1 The Formula \u00a0<\/strong><\/h1>\n

The compound amount can be calculated using the following formula:<\/p>\n

Compound amount:<\/p>\n

<\/a><\/p>\n

(5) <\/span>   <\/span>\"\begin{equation*}<\/p>\n

where i = r\/m and n = mt ;<\/p>\n

S = future (maturity) value, or the compound amount of P, or the accumulated value of P<\/p>\n

P = original principal, or the present value of P, or the discounted value of S<\/p>\n

i = interest rate per period<\/p>\n

r = annual interest rate<\/p>\n

m = number of compounding periods per year<\/p>\n

n = number of compounding periods<\/p>\n

t = number of years<\/p>\n

\n
Example 3.1<\/header>\n
\n

Consider a sum of RM8,200 is deposited into a time deposit account\u00a0 today that pays 5% p.a. How much will it be in the next 5 years if compounded quarterly?<\/p>\n

\n