Interest

A basic definition of interest is the price paid for obtaining, or price received for providing, money or capital in a credit transaction. For example, today Brian borrows $10,000 from Ivan and agrees to pay him $10,500 in one year. In this example the $10,000 is the principal that Ivan invested over a period of 1 year. After 1 year, the accumulated value is $10,500. $500 is the amount of interest Brian paid Ivan.

${\displaystyle I_{simp}=k\cdot i\cdot t}$
${\displaystyle \ I=10,000(.05\cdot 2)}$
${\displaystyle \ I=\$1,000}$
${\displaystyle \ \$10,000+\$1,000=\$11,000}$

${\displaystyle \ A(t)=k(1+i)^{t}}$
${\displaystyle \ A(t)=10,000(1+.05)^{2}}$
${\displaystyle \ A(t)=11025}$

Amount and Accumulation function

${\displaystyle i_{[s,t]}={\frac {A(t)-A(s)}{A(s)}}}$
${\displaystyle i_{[s,t]}={\frac {ke^{0.5\cdot t\cdot i}-ke^{0.5\cdot s\cdot i}}{ke^{0.5\cdot s\cdot i}}}}$
${\displaystyle i_{[3,4]}={\frac {e^{0.5\cdot 4\cdot 0.05}-e^{0.5\cdot 3\cdot 0.05}}{e^{0.5\cdot 3\cdot 0.05}}}}$
${\displaystyle i_{[3,4]}=0.0253}$

${\displaystyle A(t)=k\cdot a(t)}$
${\displaystyle A(4)=k\cdot a(4)}$
${\displaystyle 80=k\cdot 4^{2}}$
${\displaystyle k=5}$

Present and Future value

As we've seen money has a time value so that $1 today is worth more than $1 a year from now. Given an effective annual interest rate of ${\displaystyle i}$ compounded annually over ${\displaystyle t}$ years:

${\displaystyle PV(1+i)^{t}=FV}$

where ${\displaystyle PV}$ is the present value and ${\displaystyle FV}$ is the future value. ${\displaystyle (1+i)}$ is called the interest factor. Note that

${\displaystyle PV=FV(1+i)^{-t}}$

${\displaystyle v={\frac {1}{1+i}}}$ is called the discount factor and is denoted as ${\displaystyle v}$.

Example

In exchange for your car John has promised to pay you $5,000 in 1 year and $10,000 in 3 years. Using an annual effective interest rate of 5% find the present value of these payments.

${\displaystyle PV=5,000v+10,000v^{3}}$
${\displaystyle v=1.05^{-1}}$
${\displaystyle PV=13400.28}$

Using accumulation function ${\displaystyle a(t)}$ notation

• ${\displaystyle {\frac {a(n)}{a(n-1)}}=}$The n-th year interest factor.
• ${\displaystyle i_{n}={\frac {a(n)-a(n-1)}{a(n-1)}}=}$the effective rate of interest in the n-th year.
• ${\displaystyle v_{n}={\frac {a(n-1)}{a(n)}}=}$ the n-th year discount factor.
• ${\displaystyle d_{n}={\frac {a(n)-a(n-1)}{a(n)}}=}$ the effective rate of discount in the n-th year.

Note that

• ${\displaystyle v_{n}=1-d_{n}}$
• ${\displaystyle d_{n}={\frac {i_{n}}{1+i_{n}}}}$
• ${\displaystyle i_{n}={\frac {d_{n}}{1-d_{n}}}}$

${\displaystyle d={\frac {i}{1+i}}={\frac {0.065}{1.065}}=6.103}$
${\displaystyle v=(1+i)^{-1}=1.065^{-1}=.93896}$ OR
${\displaystyle v=1-d=1-0.06103=.93896}$ 

${\displaystyle PV=v^{n}FV}$
${\displaystyle PV=(1-d)^{n}\cdot 9,000}$ 
${\displaystyle PV=(1-.08)^{1}0\cdot 9,000}$
${\displaystyle PV=3,909.5}$

${\displaystyle PV=v^{n}FV}$ 
${\displaystyle 260=0.95^{3.75}FV}$
${\displaystyle FV=315.15}$ 

${\displaystyle i={\frac {260-250}{250}}}$
${\displaystyle i=.04}$
${\displaystyle d={\frac {260-250}{260}}}$
${\displaystyle d=.03846}$

Nominal vs Effective rates

So far when dealing with compound interest we've only seen the interest compounded annually. That is to say, the interest is only added to the principal once a year. To illustrate this, Ivan invests $1000 into a bank account that offers 10% interest compounded annually. At the end of the first year, the account is $1000 + $100 in interest. At this point the principal earning interest is $1100 so that at the end of the second year the interest earned is $111 and the principal is $1100. However, the interest could be compounded more frequently. Suppose that instead of being compounded annually, the interest is compounded semi-annually. At the end of 6 months, the interest is $50 and added to the $1000 principal. The new principal is $1,050. At the end of the first year, the interest earned over the last 6 months is $52.5. After one year, the bank account contains $1,102.5. Notice that the interest rate is still 10% annually, however the effective annual rate of interest is 10.25%.

${\displaystyle i_{n}={\frac {1,102.5-1000}{1000}}=0.1025}$

10% is the nominal interest rate and 10.25% is the effective rate. A nominal rate of interest compounded m times a year is denoted as ${\displaystyle i^{(m)}}$. The relationship between the effective rate and nominal rate of interest is:

${\displaystyle i_{effective}={(1+{\frac {i^{(m)}}{m}})}^{m}-1}$

Example 1

Loki borrows $200 from Thor and repays him $205 in one month.

1. Find the monthly effective interest rate, which Loki is charged for the loan.
2. Find the annual nominal interest rate compounded monthly.
3. Find the annual effective interest rate.

${\displaystyle i={\frac {205-200}{200}}=0.025}$
${\displaystyle i^{(m)}=12\cdot 0.025=0.3}$
${\displaystyle i=(1.025)^{12}-1=0.3448}$

Example2

Xiao Yu invests $7,500 into a bank account with a nominal interest rate of 8% convertible quarterly. How much does she have in the account after 6 years?

${\displaystyle 7,500(1+{\frac {0.08}{4}})^{6\cdot 4}=12,063.28}$  

Inflation and real rate of interest

Force of interest

So far we've said that the effective rate of interest earned in a period ${\displaystyle n}$ is

${\displaystyle i_{n}={\frac {a(n)-a(n-1)}{a(n-1)}}}$.

This gives the interest rate earned over a period however. We can also find the rate of interest over an interval smaller than the period.

${\displaystyle i={\frac {a(t)-a(t-h)}{h\cdot a(t-h)}}}$.

If we have a continuous accumulation function we can also measure the interest acting at a given moment called the force of interest which is denoted as ${\displaystyle \delta }$.

${\displaystyle \delta _{t}=\lim _{h\to 0}{\frac {a(t)-a(t-h)}{h\cdot a(t-h)}}={\frac {a'(t)}{a(t)}}={\frac {d}{dt}}\ln a(t)}$

${\displaystyle e^{\int _{0}^{t}\delta _{r}dr}=a(t)}$

Equation of value

References

1. ↑ http://www.beanactuary.org/exams/syllabi/Notation_term2FM.pdf