Interest On January 1,2010, Jack deposited 1000 into Bank X to earn interest at the rate of j p

Interest On January 1,2010, Jack deposited 1000 into Bank X...

Key Concepts

Compound Interest

Compound interest refers to the process where interest is earned not only on the initial principal but also on the accumulated interest from previous periods, resulting in exponential growth of an investment. The standard formula A = P(1 + r/n)^(n*t) is used, where P is the principal, r is the nominal annual interest rate, n is the number of compounding periods per year, and t is the time in years.

Effective Annual Rate

The effective annual interest rate is the actual annual rate that an investment earns after accounting for the effects of compounding over a year. It provides a basis for comparing interest rates with different compounding frequencies by transforming a nominal rate into an annualized rate that reflects the true economic return.

Nominal Interest Rate

A nominal interest rate is the stated annual rate that does not take into account the effects of compounding within the year. When interest is compounded more frequently than annually, the nominal rate must be adjusted using the compounding frequency to determine actual accumulation of interest over time.

Compounding Frequency

Compounding frequency refers to the number of times per year that interest is calculated and added to the investment. Different compounding frequencies—such as semiannual or quarterly—affect the total accumulation of interest, and understanding this concept is essential when comparing different financial scenarios or converting between nominal and effective interest rates.

Interest Rate Ratio Analysis

Interest rate ratio analysis involves comparing different interest rates, especially when applied under varying compounding conditions, by establishing relationships between them. In problems that require finding the ratio of two rates, an understanding of how compounding frequency and time intervals affect the accumulation of interest is crucial for forming and solving the necessary equations.

Transcript

00:01 In this problem, we're told that on january 1st of 2010, jack deposited $1 ,000 into bank x to earn interest at the rate of j per year compounded semi -annually.

00:12 On january 1st of 2015, he transferred that amount to bank y to earn interest at a rate of k per year compounded quarterly.

00:20 On january 1st of 2018, the balance at bank y was $1 ,990 .76.

00:28 But if jack could have earned interest at the rate of k per year, compounded, counted quarterly for the entire eight -year period from 2010 to 2018, his balance would have been $2 ,203 .76.

00:41 We're then asked to find the ratio k divided by j of the interest rate at the two banks.

00:47 This problem is pretty long and has a lot of parts, so let's try and break it down and do one part at a time.

00:54 Step one is to find out how much did jack have when he transferred from bank x to bank y, bank x to bank y.

01:16 So we know the formula is amount equals p times one plus r over m to the power of m times t where p is the starting amount which in this case is $1 ,000.

01:39 R is the interest rate.

01:45 We're not given what r is as a number but we're just told that the rate is some number that we're going to call j.

01:50 So this is our j.

01:53 M is the number of times per year.

01:57 The interest is compounded.

02:03 And in bank x, we're told that the interest is compounded semi -annually.

02:08 So that's twice a year.

02:09 So m is equal to 2.

02:11 And t is the number of years.

02:15 And in this case, he kept his money in bank x from the year 2010 to the year 2015.

02:21 So that's a total of five years.

02:23 So t is equal to five.

02:25 So we can put this all together.

02:29 A is equal to 1 ,000 times 1 plus j divided by 2 to the power of 2 times 5, or we can simplify this exponent 1 ,000 times 1 plus j over 2 to the power of 10.

02:44 So this amount is the amount that he has after 5 years when he decides to transfer from bank x to bank y.

02:54 So now, step two is to figure out a formula for how much did he have on january 1, 2018, so after the money was in bank y, after the money has been in bank y.

03:20 So again, we use the same formula.

03:22 A equals p times 1 plus r over m to the power of mt.

03:28 So once again, p is the amount we started with.

03:36 But now this is no longer $1 ,000 because this is the amount that he has when he deposits it into bank y.

03:43 So this is the amount that he has that he transfers out of bank x.

03:47 And we just calculated on the previous page that the amount that he has after he takes his money out of bank x is given by this formula.

03:57 So this is going to be the p on the next page, because this is the amount that he starts with when he transfers his money from bank x to bank y.

04:08 So that's 1 ,000 times 1 plus j over 2 to the power of 10.

04:15 That's his starting amount in 2015.

04:20 R is the rate, the interest rate, which in this case, we're not given a specific number.

04:24 We're just told that it's some number called k.

04:27 T is the number of years.

04:29 And so we know that he spent three years in bank y going from 2010 or excuse me, 2005 to 2008.

04:39 And m is the number of times we compound annually times we compound our interest per year.

04:48 And we're told that in bank y, the interest is compounded quarterly.

04:53 So that's four times per year.

04:55 So m is equal to four.

04:58 So at the end of 2008, or the end of his tenure at bank y, so on january 1st, 2018, the amount of money he has is 1 ,000 times 1 plus j over 2 to the power of 10.

05:14 That's our p times 1 plus k divided by 4 to the power of 3 times 4.

05:27 So that's the information from our new bank is coming into here.

05:35 So this formula represents the amount that he has.

05:39 At the actual amount that he has on january 1st, 2018.

05:43 That simplifier exponent 1 plus j over 2 to the power of 10 times 1 plus k over 4 to the power of 12.

05:52 So this is the amount that he actually has at 2018.

05:58 And the problem tells us, in fact, that we know that at the start of 2018, he has a total of $1 ,990.

06:08 And, oops, $1 ,990, and 76 cents.

06:14 So we need to remember this equation.

06:16 We're going to come back to this equation, but before we can really use this equation, we have to use the rest of the information of the problem.

06:23 So remember this equation, kind of keep it on hold, and we're going to come back to it in just a minute...

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