Interest is the price paid for borrowing money or payment received for lending money, usually a percentage of the amount borrowed (i.e. the only reason a sane person would allow someone else to touch his/her money). Without interest, the global economy would come to a standstill. Companies won't be able to sell their stocks, no one will be enticed to invest, no one would be able to borrow money because no one is "interest"-ed to lend, etc. In this article, the basic principles that govern interest will be covered.
Before jumping to simple interest, an even simpler topic is often taken for granted. Interest is price paid for borrowing money, but it can be payment at a constant value, or payment increasing relative to the original loan. For payments paid at constant value, time is simple. But for payments with increasing value, time needs to be analyzed meticulously to avoid errors. It is helpful to think of time as a variable of interest, proven theoretically in latter discussions.
Simple Interest
$ I \ = \ P*i*n $
Easy mnemonics to remember this formula for Filipino readers - a Tagalog term for teeth "ngipin" sounding like 'ipin'. Interest (I) is a function of
• Present value (P)
• Interest rate (i) in days
• Number of days (n)
Interest grows linearly.
Fig. 1 is a visual of the amount of interest returned for 10 pesos and 20 pesos in 10 days invested at different rates of simple interest. It is evident from Fig. 1 that interest and interest rates share a linear directly proportional relationship with each other.
From Fig. 1, slope can simply be taken by dividing present amount by number of days. 10 pesos/10 days = slope of 1. 20 pesos/10 days = slope of 2. Notice higher slopes yield higher returns. Thus, a better slope/ratio between present amount and number of days yields better returns for simple interest. Note that no matter what the values, the output graph is always a straight line, which confirms linearity.
Future worth can be calculated by adding interest to the present value.
$ F \ = \ P \ + \ I $
Algebraic manipulation yields other forms of the above equation such as
Fig. 1 is a visual of the amount of interest returned for 10 pesos and 20 pesos in 10 days invested at different rates of simple interest. It is evident from Fig. 1 that interest and interest rates share a linear directly proportional relationship with each other.
Figure 1. Interest Rate vs. Interest for differing Present Values
From Fig. 1, slope can simply be taken by dividing present amount by number of days. 10 pesos/10 days = slope of 1. 20 pesos/10 days = slope of 2. Notice higher slopes yield higher returns. Thus, a better slope/ratio between present amount and number of days yields better returns for simple interest. Note that no matter what the values, the output graph is always a straight line, which confirms linearity.
Future worth can be calculated by adding interest to the present value.
$ F \ = \ P \ + \ I $
Algebraic manipulation yields other forms of the above equation such as
$ F \ = \ P*(1+i*n) $
$ P \ = \ F*((1+i*n)^{-1}) $
Simple interest can be divided into 2 categories when it comes to time, ordinary simple interest and exact simple interest. Ordinary simple interest is based on a calendar where each month is 30 days long and a year is (30x12) 360 days long regardless whether the year is a leap year or not. Exact simple interest is by semantics, more exact when it comes to time. The number of days in a month is based on the actual number, for example, February would have 28 days and for May, 31 days. Leap years are also considered so if the year is divisible by 4, that year has 366 days in it except for the usual 365.
Sample Problem:
Joseph wants to invest for a certification exam that costs 18,000 pesos. He wants to take the exam in 1 year. Right now, he has 10,000 pesos. What should be the interest rate for an investment that earns ordinary simple interest?
Solution:
$ F \ = \ P*(1+i*n) $
$ (1+i*n) \ = \ \large \frac{F}{P} $
$ i \ = \ \frac{(( \frac{F}{P})-1)}{n} $
$ i \ = \ \large \frac{ \frac{18000}{10000}-1}{360} $
$ i \ = \ 0.002222 $
Answer:
Simple interest can be divided into 2 categories when it comes to time, ordinary simple interest and exact simple interest. Ordinary simple interest is based on a calendar where each month is 30 days long and a year is (30x12) 360 days long regardless whether the year is a leap year or not. Exact simple interest is by semantics, more exact when it comes to time. The number of days in a month is based on the actual number, for example, February would have 28 days and for May, 31 days. Leap years are also considered so if the year is divisible by 4, that year has 366 days in it except for the usual 365.
Sample Problem:
Joseph wants to invest for a certification exam that costs 18,000 pesos. He wants to take the exam in 1 year. Right now, he has 10,000 pesos. What should be the interest rate for an investment that earns ordinary simple interest?
Solution:
$ F \ = \ P*(1+i*n) $
$ (1+i*n) \ = \ \large \frac{F}{P} $
$ i \ = \ \frac{(( \frac{F}{P})-1)}{n} $
$ i \ = \ \large \frac{ \frac{18000}{10000}-1}{360} $
$ i \ = \ 0.002222 $
Answer:
Joseph must find an investment that earns 0.22% ordinary simple interest.
Compound Interest
With simple interest, growth is linear. But with compound interest, increments are exponential. Why exponential though? In this way, loyalty to investment is given credit, like a loyalty award. You are rewarded interest on your previous interest. Now who would turn down that kind of offering to dull simple interest? That's not to say compound interest is more profitable than simple interest, because interest rates between the two can be varied.
Future value of a compound interest investment/debt
$ F \ = \ P*(1+i)^{n} $
This time, present worth increases by a factor of (1+i)^n. A factor raised to n describes exponential growth, and is called the "single payment compound amount factor" or SPCAF. If this factor is transposed to the other side to find present value needed given a future amount, then the factor is called the "single payment present worth factor" or SPPWF $ (1+i)^{-n} $.
Fig. 2 aims to achieve a comprehensive understanding of these growth rates.
According to the graph, interest earned by compound interest becomes greater than that of simple interest and continuous to do so as time goes by. If the interest rate of simple interest is increased, we can expect the slope of the magenta-colored line to increase (that is, become steeper with the highest point at the right-most side). This increase can overtake the compound interest curve. But if the time axis curve is extended further, no matter how much slope is increased, the compound interest curve will always overtake it at some point in the future.
With regards to compound interest, the effective interest rate for 1 year can be taken by:
$ Ieff \ = \ (1+i)^{m}-1 $
Where Ieff is the effective interest rate in years, interest i=r/m where r is the nominal interest and m is the mode of compounding and n equals the number of m periods in 1 year.
Compound Interest
With simple interest, growth is linear. But with compound interest, increments are exponential. Why exponential though? In this way, loyalty to investment is given credit, like a loyalty award. You are rewarded interest on your previous interest. Now who would turn down that kind of offering to dull simple interest? That's not to say compound interest is more profitable than simple interest, because interest rates between the two can be varied.
Future value of a compound interest investment/debt
$ F \ = \ P*(1+i)^{n} $
This time, present worth increases by a factor of (1+i)^n. A factor raised to n describes exponential growth, and is called the "single payment compound amount factor" or SPCAF. If this factor is transposed to the other side to find present value needed given a future amount, then the factor is called the "single payment present worth factor" or SPPWF $ (1+i)^{-n} $.
Fig. 2 aims to achieve a comprehensive understanding of these growth rates.
Fig. 2 Simple Interest vs. Compound Interest Growth Rates
According to the graph, interest earned by compound interest becomes greater than that of simple interest and continuous to do so as time goes by. If the interest rate of simple interest is increased, we can expect the slope of the magenta-colored line to increase (that is, become steeper with the highest point at the right-most side). This increase can overtake the compound interest curve. But if the time axis curve is extended further, no matter how much slope is increased, the compound interest curve will always overtake it at some point in the future.
With regards to compound interest, the effective interest rate for 1 year can be taken by:
$ Ieff \ = \ (1+i)^{m}-1 $
Where Ieff is the effective interest rate in years, interest i=r/m where r is the nominal interest and m is the mode of compounding and n equals the number of m periods in 1 year.
Sample Problem:
Let us say that we want to find the annual effective interest of an investment with a nominal interest of 5% compounded semi-annually. Then:
$ Ieff=(1+ \large \frac{0.05}{2})^2-1 $
m=2 because there are 2 semi-annual periods in a year;
$ \large \frac{0.05}{2} $ because $ i= \large \frac{r}{m} $
Answer: 0.0506 or 5.06% annual effective interest
An interest rate of 5% compounded semi-annually is equivalent to an interest rate of 5.06% compounded annually.
Doesn't the formula of effective interest look familiar? Actually it does. It can be related to the way we find the future amount of a compound interest investment.
Let a=effective annual interest rate
b=interest rate ( $ i \ = \ \frac{r}{m} $ )
na=mode of compounding for annual compounding (which is implicitly 1)
nb=mode of compounding for b
Future worth with interest compounded annually=Future worth with interest compounded nb times in a year
$ F \ = \ P*(1+i)^n $
$ P*(1+a)^{na} \ = \ P*(1+b)^{nb} $
-since na=1 for 1 year and P cancels out
$ (1+a)^1 \ = \ (1+b)^{nb} $
$ (1+a) \ = \ (1+b)^{nb} $
$ a \ = \ (1+b)^{nb} \ - \ 1 $
a=Ieff; b=i; nb=m;
$ Ieff \ = \ (1+i)^{m} \ - \ 1 $
Hence, the formula for effective annual interest.
Annuity
Annuity is defined as payments made at equal intervals of time. A good example would be allocating one's monthly salary to meet a financial goal.
Fig. 3 Future Worth vs. Time at 3%, 5%, and 7% interest.
Fig. 3 shows annuity investments at 3%, 5% and 7% interest. Even if payments are made periodically, the interest growth rate shares striking similarities to that of compound interest (an exponential curve). The stem plot emphasizes payments being made at equal intervals of time.
Ordinary Annuity
$ F \ = \ \large \frac{A}{i}*((1+i)^{n}-1) $
With ordinary annuity payments are made at the end of each period.
To get present worth:
$ F=P*(1+i)^n $
$ P*(1+i)^n \ = \ \large \frac{A}{i}*((1+i)^{n}-1) $
$ P \ = \ \large \frac{A}{i*(1+i)^n}*((1+i)^{n}-1) $
Deferred Annuity
For deferred annuity, payments are not made at the beginning of the period, but after m periods have elapsed. The formulas are the same with ordinary annuity except for present worth, since start of payment was delayed:
$ F \ = \ P*(1+i)^{n+m} $
$ P \ = \ \large \frac{A}{i*(1+i)^{n+m}}*((1+i)^{n}-1) $
Annuity Due
When payments are made at the beginning of each period, it is called annuity due. Just as with deferred annuity, formulas are similar to ordinary annuity except for present worth, since payments started at the beginning.
$ F \ = \ P*(1+i)^{n-1} $
$ P \ = \ \large \frac{A}{i*(1+i)^{n-1}}*((1+i)^{n}-1) $
$ F=P*(1+i)^n $
$ P*(1+i)^n \ = \ \large \frac{A}{i}*((1+i)^{n}-1) $
$ P \ = \ \large \frac{A}{i*(1+i)^n}*((1+i)^{n}-1) $
Deferred Annuity
For deferred annuity, payments are not made at the beginning of the period, but after m periods have elapsed. The formulas are the same with ordinary annuity except for present worth, since start of payment was delayed:
$ F \ = \ P*(1+i)^{n+m} $
$ P \ = \ \large \frac{A}{i*(1+i)^{n+m}}*((1+i)^{n}-1) $
Annuity Due
When payments are made at the beginning of each period, it is called annuity due. Just as with deferred annuity, formulas are similar to ordinary annuity except for present worth, since payments started at the beginning.
$ F \ = \ P*(1+i)^{n-1} $
$ P \ = \ \large \frac{A}{i*(1+i)^{n-1}}*((1+i)^{n}-1) $
*Essentially, one only has to be familiar with the formulas used in ordinary annuity and then tweak a variable to get the others.
Perpetuity
What if saving has grown into an unstoppable addiction? The formula then shrinks to just 2 variables. Looking back at the formula for ordinary annuity:
$ P \ = \ \large \frac{A}{i*(1+i)^n}*((1+i)^{n}-1) $
rearranging...
$ P \ = \ \large \frac{A}{i*(1+i)^n}*(1- \large \frac{1}{(1+i)^n})*(1+i)^{n} $
$ P \ = \ \large \frac{A}{i}*(1- \large \frac{1}{(1+i)^n}) $
since n approaches infinity, $ (1+i)^n $ becomes infinite and $ \large \frac{1}{(1+i)^n} $ approaches 0 (or becomes 0)
thus,
$ P \ = \large \frac{A}{i}*(1-0) $
$ P \ = \large \frac{A}{i} $
Capitalized Cost
Philosophers would argue capitalized cost as mere cost. But those initiated in finance would beg to differ, as capitalized cost has a much deeper meaning. It is the present worth of property assumed to last forever. From this definition, present worth must be equal to the amount of expenses one initially has to pay for property plus expenses to maintain or replace said property.
Case 1: No replacement, only maintenance.
Then:
Capitalized Cost=First Cost+Perpetual Maintenance Fee(i.e. Perpetuity)
$ CC \ = \ FC \ + \ \large \frac{A}{i} $
A=maintenance fee
Case 2: No maintenance, only replacement.
Capitalized Cost=First Cost+Perpetual Replacement Fee(Perpetuity)@effective interest rate
$ CC \ = \ FC \ + \ \large \frac{P}{Ieff} $
$ CC \ = \ FC \ + \ \large \frac{P}{(1+i)^{n}-1} $
P=replacement fee every n periods
Case 3: Both maintenance and replacement required.
$ \ Capitalized \ Cost \ = \ Sum \ up \ Case \ 1 \ and \ Case \ 2 $
$ CC \ = \ FC \ + \ \large \frac{A}{i} \ + \ \large \frac{P}{(1+i)^{n}-1} $
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