Compound Interest Formula

BREAKING: Target Test Prep releases Brand New 2026 On Demand GMAT prep course

Redeem

This topic has expert replies

Post new topic Post Reply

Previous Topic Next Topic

knight247: Legendary Member; Posts: 504; Joined: Tue Apr 19, 2011 1:40 pm; Thanked: 114 times; Followed by:11 members

Compound Interest Formula

by knight247 » Sat Aug 06, 2011 8:42 am

I would appreciate a clarification on the simple interest formula. I've come across two different formulas in different books

I've got
P+I=P(1+R)^n

Where P=Principal, I=Interest amount, R=Rate
and N=Total number of payment intervals=(Total duration of the loan in months)/Total duration of each compounding period in months

But I've also come across this formula

P+I=P(1+(R)/n)^n

Would like to confirm, which of the above are correct. Thanks

Quote

Source: Beat The GMAT — Problem Solving | Sat Aug 06, 2011 8:42 am

Touseef: Junior | Next Rank: 30 Posts; Posts: 19; Joined: Sat Jul 30, 2011 10:39 pm; Location: India; Thanked: 3 times

by Touseef » Sat Aug 06, 2011 9:00 am

Hi,

Both these formulas for Compound Intrest are correct but they are used in different situations.

Scenario1 P+I=P(1+R)^n
when intrest is compunded annually.

Scenario2 P+I=P(1+(R)/n)^n
When intrest is compunded quarterly or semi-annually.

For ex:if P=5000 and intrest is 10% compunded semi-annually(every 6 months)
Then interest will be 5% each every 6 months.
hence,I=$250 at the end of first six months and Amount will be $5250.
The Intrest for the next six months will be calculated for P=$5250 and Intrest rate=5%

Quote

edge: Senior | Next Rank: 100 Posts; Posts: 49; Joined: Sun Jun 26, 2011 5:45 pm; Thanked: 2 times

by edge » Sat Aug 06, 2011 9:15 am

Touseef wrote:Hi,

Both these formulas for Compound Intrest are correct but they are used in different situations.

Scenario1 P+I=P(1+R)^n
when intrest is compunded annually.

Scenario2 P+I=P(1+(R)/n)^n
When intrest is compunded quarterly or semi-annually.

For ex:if P=5000 and intrest is 10% compunded semi-annually(every 6 months)
Then interest will be 5% each every 6 months.
hence,I=$250 at the end of first six months and Amount will be $5250.
The Intrest for the next six months will be calculated for P=$5250 and Intrest rate=5%

A note of caution: n is the number of compounding periods per year. For semi-annual compounding (6 months), n = 12/6 = 2. Similarly, n = 4 and 12 for quarterly and monthly compounding respectively.

Hence, Future Value (at the end of one year) = $5000 * (1 + (10/2))^2 = $5512.50. Interest = FV - P = $512.50.

Last edited by edge on Sat Aug 06, 2011 12:12 pm, edited 2 times in total.

Quote

Touseef: Junior | Next Rank: 30 Posts; Posts: 19; Joined: Sat Jul 30, 2011 10:39 pm; Location: India; Thanked: 3 times

by Touseef » Sat Aug 06, 2011 9:26 am

You are Right Edge.

But if u look closely you will observe that my logic is the same.

At the end of 6 months,I=$250 and the Amount will be $5250
For the next 6 months,$5250 will become the principal and 5% will be the intrest rate.
Hence,I=$262.5
Amount at the end of the year=5250+262.5=$5512.5
TOTAL ACCUMULATED INTREST=5512.5-5000=$512.5

Quote

goalevan: Master | Next Rank: 500 Posts; Posts: 111; Joined: Tue Dec 30, 2008 1:25 pm; Location: USA; Thanked: 28 times; GMAT Score:770

by goalevan » Sat Aug 06, 2011 4:47 pm

It's easiest to use the variables t and m instead of n:

P + I = P(1 + r)^t with one compound per period and t periods.
P + I = P(1 + r/m)^(mt) with m compounds per period and t periods.

As an aside, when the limit of n to infinity is taken, the quantity (1 + 1/n)^n is equal to the mathematical constant e. We can show that when m compounds per period approaches a very high number, the interest formula approaches "continuous compounding", P + I = P*e^(rt):

P[1 + r/m]^(mt)
P[1 + 1/(m/r)]^[(mt)*(r/r)]
P[1 + 1/(m/r)]^[(m/r)(rt)]
Pe^(rt) (with the limit of m to infinity)