Annual Percent Yield - YouTube

in this video we're going to compute the annual percent yields also called the

APY here is our formula for the APY where r is the annual interest rate and

n is the number of compoundings per year so the idea of this measurement is that

if we're using compound interest the number of compoundings effect the total

percent yields you get in one year so that's the annual and that's the yield part so

let's compare some of these we're going to look at annual compounding this is

semiannual quarterly and monthly so we're given

that n equals one and we're told here that the interest rate is eight point

five percent we're going to write that as a decimal so that will be 0.085

moving the decimal two places to the left and we're going to round our end

results to three decimal places so here when n equals one we have 1 plus 0.08

five the number of compoundings is 1 the number of compoundings per year is one

and then we're going to subtract one when we're done we'll change us two as a

percent after computing this by multiplying by 100 so in general even

though we don't need to here we're going to start by computing this ratio adding

one raising it to the exponent then subtracting one so this is the order of

operations again if you have a scientific calculator or a graphing

calculator you can enter this entire expression all at once just make sure

you use the corresponding parentheses so we have 0.08 5 divided by 1 and yes

that's just itself then we're going to add one

and we have this result for inside the parentheses raise it to the first power

again that's itself but we just want to do the same process for all of these

subtract one put this part and multiply by 100 to change it to a percent and we

have 8.5% which should look very familiar now that may have seemed

pointless and useless but the idea is since this is an annual percent yield if

we're compounding annually the interest rate is equivalent to this annual

percentage rate is equivalent to the annual percent yield let's see what

happens when we compound twice a year so

we'll have 1 plus 0.08 five we're going to divide the interest rate by two

because we're going to do half the interest rate twice a year so the

exponent is two that's our two compoundings and let's see what we get

so first we'll divide the interest rate by two so that's half of 8.5%

now we'll add one now we're going to raise that result to the second power so

raised to the second and then we'll subtract one for this part and now we

want to rewrite this as a percent so we multiply by 100 and we were asked to

round to three decimal places so that would be eight point six eight here

there's a zero but to the right there's a six so we're going to round that up to

a one so eight point six eight one percent so notice just by compounding

twice a year versus one time a year with the same interest rate we went up by one

tenth and a little bit more of a percent per year let's do four that

change colors so they don't smush together so one plus 0.08 five we're

going to divide the interest rate over for compounding so one fourth of this

interest rate per compounding there's our four compoundings subtract one so

let's start with the interest rate divide it by four add one that's this

result in the parentheses so now we raise it to the fourth power and

then we're going to subtract one and then this is our decimal percentage for

the APY and we multiply by 100 so for three decimal places for is in the third

seven is in the fourth so we'll round up to eight point seven seven five percent

and now let's see what happens with monthly compoundings so one plus 0.08

five we'll divide that yearly amount into twelve equal monthly pieces will

compound 12 times which goes in the exponent and we'll start with point 0 8

5 divided by 12 then we're going to add the 1 raise this result to the 12th

power and then we subtract 1 to change this result to a percent we multiply by

100 and to three decimal places we get eight point eight three nine percent

eight point eight three nine percent so the moral of the story is notice as the

compounding increase we fix the interest rate it's

the same interest rate for all of them but the more you compound the greater

the annual percent yield and if we looked at this if either of these or if

all of these were for 30 years this compounding would accumulate even more