62 compound interest

Compound InterestWhen an investment or account offers interest on top of previously accumulated interest, it is called compound interest.

Compounded return on $1,000 with annual interest rate r = 20% (Wikipedia)

Compound Interest

Example A. We deposit $1,000 in an account that gives 1%interest compounded monthly. How much money is there after 1 month? 2 months? 3 months? and after 4 months?

Compound Interest

Example A. We deposit $1,000 in an account that gives 1%interest compounded monthly. How much money is there after 1 month? 2 months? 3 months? and after 4 months?After the 1st month: 1000 + 1000*0.01 = 1010 $

Compound Interest

Example A. We deposit $1,000 in an account that gives 1%interest compounded monthly. How much money is there after 1 month? 2 months? 3 months? and after 4 months?After the 1st month: 1000 + 1000*0.01 = 1010 $After the 2nd month: 1010 + 1010*0.01

Compound Interest

Example A. We deposit $1,000 in an account that gives 1%interest compounded monthly. How much money is there after 1 month? 2 months? 3 months? and after 4 months?After the 1st month: 1000 + 1000*0.01 = 1010 $After the 2nd month: 1010 + 1010*0.01 = 1020.10 $

Compound Interest

Example A. We deposit $1,000 in an account that gives 1%interest compounded monthly. How much money is there after 1 month? 2 months? 3 months? and after 4 months?After the 1st month: 1000 + 1000*0.01 = 1010 $After the 2nd month: 1010 + 1010*0.01 = 1020.10 $After the 3rd month: 1020.10 + 1020.10*0.01

Compound Interest

Example A. We deposit $1,000 in an account that gives 1%interest compounded monthly. How much money is there after 1 month? 2 months? 3 months? and after 4 months?After the 1st month: 1000 + 1000*0.01 = 1010 $After the 2nd month: 1010 + 1010*0.01 = 1020.10 $After the 3rd month: 1020.10 + 1020.10*0.01 = 1030.30 $

Compound Interest

Example A. We deposit $1,000 in an account that gives 1%interest compounded monthly. How much money is there after 1 month? 2 months? 3 months? and after 4 months?After the 1st month: 1000 + 1000*0.01 = 1010 $After the 2nd month: 1010 + 1010*0.01 = 1020.10 $After the 3rd month: 1020.10 + 1020.10*0.01 = 1030.30 $After the 4th month: 1030.30 + 1030.30*0.01 = 1040.60 $

Compound Interest


Compound Interest Formula (PINA)

Compound Interest


Compound Interest Formula (PINA) Let P = principal (the money deposited)

Compound Interest


Compound Interest Formula (PINA) Let P = principal (the money deposited) i = periodic rate (interest rate for a period)

Compound Interest


Compound Interest Formula (PINA) Let P = principal (the money deposited) i = periodic rate (interest rate for a period) N = number of periods

Compound Interest


Compound Interest Formula (PINA) Let P = principal (the money deposited) i = periodic rate (interest rate for a period) N = number of periods A = total accumulated value after N periods

Compound Interest


Compound Interest Formula (PINA) Let P = principal (the money deposited) i = periodic rate (interest rate for a period) N = number of periods A = total accumulated value after N periods Compound Interest Formula: P(1 + i )N = A

Compound Interest


Compound Interest Formula (PINA) Let P = principal (the money deposited) i = periodic rate (interest rate for a period) N = number of periods A = total accumulated value after N periods Compound Interest Formula: P(1 + i )N = A With this formula, we may compute the return after N periods directly.

Compound Interest

Example B. We deposit $1,000 in an account that gives 1%interest compounded monthly. How much money is there after 3 months? and after 4 months?

Compound Interest

Example B. We deposit $1,000 in an account that gives 1%interest compounded monthly. How much money is there after 3 months? and after 4 months?Use the formula P(1 + i )N = A

Compound Interest

Example B. We deposit $1,000 in an account that gives 1%interest compounded monthly. How much money is there after 3 months? and after 4 months?Use the formula P(1 + i )N = A P = 1000, i = 0.01, after 3 months, N = 3,

Compound Interest

Example B. We deposit $1,000 in an account that gives 1%interest compounded monthly. How much money is there after 3 months? and after 4 months?Use the formula P(1 + i )N = A P = 1000, i = 0.01, after 3 months, N = 3,A = 1000(1 + 0.01)3

Compound Interest

Example B. We deposit $1,000 in an account that gives 1%interest compounded monthly. How much money is there after 3 months? and after 4 months?Use the formula P(1 + i )N = A P = 1000, i = 0.01, after 3 months, N = 3,A = 1000(1 + 0.01)3 = 1000(1.01)3 = 1030.30 $

Compound Interest

Example B. We deposit $1,000 in an account that gives 1%interest compounded monthly. How much money is there after 3 months? and after 4 months?Use the formula P(1 + i )N = A P = 1000, i = 0.01, after 3 months, N = 3,A = 1000(1 + 0.01)3 = 1000(1.01)3 = 1030.30 $ After 4 months, N = 4,

Compound Interest

Example B. We deposit $1,000 in an account that gives 1%interest compounded monthly. How much money is there after 3 months? and after 4 months?Use the formula P(1 + i )N = A P = 1000, i = 0.01, after 3 months, N = 3,A = 1000(1 + 0.01)3 = 1000(1.01)3 = 1030.30 $ After 4 months, N = 4, A = 1000(1 + 0.01)4

Compound Interest

Example B. We deposit $1,000 in an account that gives 1%interest compounded monthly. How much money is there after 3 months? and after 4 months?Use the formula P(1 + i )N = A P = 1000, i = 0.01, after 3 months, N = 3,A = 1000(1 + 0.01)3 = 1000(1.01)3 = 1030.30 $ After 4 months, N = 4, A = 1000(1 + 0.01)4 = 1000(1.01)4 =1040.60 $

Compound Interest


Compound Interest


In real life, instead of giving the periodic rate i of an account–which usually is a small decimal number that is difficult to track, the total yearly interest rate and the frequency of compounding are prescribed.

Compound Interest


In real life, instead of giving the periodic rate i of an account–which usually is a small decimal number that is difficult to track, the total yearly interest rate and the frequency of compounding are prescribed. For example, if the given annual compound interest rate isr = 8% and compounded 4 times a year,

Compound Interest


In real life, instead of giving the periodic rate i of an account–which usually is a small decimal number that is difficult to track, the total yearly interest rate and the frequency of compounding are prescribed. For example, if the given annual compound interest rate isr = 8% and compounded 4 times a year, then the periodic ratei = = 0.02. 4

0.08

Compound Interest


In real life, instead of giving the periodic rate i of an account–which usually is a small decimal number that is difficult to track, the total yearly interest rate and the frequency of compounding are prescribed. For example, if the given annual compound interest rate isr = 8% and compounded 4 times a year, then the periodic ratei = = 0.02. If it's compounded 12 times a year, then i = .

40.08

120.08

Compound Interest

Example C. We deposited $1000 in an account with annual compound interest rate r = 8%, compounded 4 times a year. How much will be there after 20 years?

Compound Interest

Example C. We deposited $1000 in an account with annual compound interest rate r = 8%, compounded 4 times a year. How much will be there after 20 years?We have

40.08P = 1000, i = = 0.02,

Compound Interest


40.08P = 1000, i = = 0.02,

N = (20 years)(4 times per years) = 80

Compound Interest


40.08P = 1000, i = = 0.02,

N = (20 years)(4 times per years) = 80Hence A = 1000(1 + 0.02 )80

Compound Interest


40.08P = 1000, i = = 0.02,

N = (20 years)(4 times per years) = 80Hence A = 1000(1 + 0.02 )80 = 1000(1.02)80 4875.44 $

Compound Interest


40.08P = 1000, i = = 0.02,


Compound Interest


40.08P = 1000, i = = 0.02,


Compound Interest

Let r = annual rate (r = 8% in example C)f = frequency of compounding (f = 4 times / yr. in example C)


40.08P = 1000, i = = 0.02,


Compound Interest

Let r = annual rate (r = 8% in example C)f = frequency of compounding (f = 4 times / yr. in example C)

Then the periodic rate i =rf (i = ), 0.08

4


40.08P = 1000, i = = 0.02,


Compound Interest

Let r = annual rate (r = 8% in example C)f = frequency of compounding (f = 4 times / yr. in example C)t = number of years (t = 20 years in example C)

Then the periodic rate i =rf (i = ), 0.08

4


40.08P = 1000, i = = 0.02,


Then the periodic rate i =

Let r = annual rate (r = 8% in example C)f = frequency of compounding (f = 4 times / yr. in example C)t = number of years (t = 20 years in example C)

rf (i = ), and0.08

4the total number of period N = ft (N = 4*20 in example C.).

Compound Interest

r = annual rate f = frequency of compoundingt = number of years

Compound Interest

A = P (1 + i )N

r = annual rate f = frequency of compoundingt = number of years

From the periodic compound interest formula based on i and N,

Compound Interest

A = P (1 + i )N

r = annual rate f = frequency of compoundingt = number of yearsreplacing i by r

f


Compound Interest

A = P (1 + i )N

r = annual rate f = frequency of compoundingt = number of yearsreplacing i by , N by ft


Compound Interest

rf

A = P (1 + i )N

we get the compound interest Prffta formula based on r, f, and t


P(1 + )ft = Arf


Compound Interest

rf

A = P (1 + i )N



Example D. We open an account with r = 9%, compounded monthly. and after 40 years the total return would be $250,000, what is the initial principal?

Compound Interest

rf

P(1 + )ft = Arf


A = P (1 + i )N



Example D. We open an account with r = 9%, compounded monthly. and after 40 years the total return would be $250,000, what is the initial principal?We have thatr = annual rate =f = frequency of compounding =t = number of years =A = total return =

Compound Interest

rf

P(1 + )ft = Arf


A = P (1 + i )N



Example D. We open an account with r = 9%, compounded monthly. and after 40 years the total return would be $250,000, what is the initial principal?We have thatr = annual rate = 9% = 0.09 f = frequency of compounding =t = number of years =A = total return =

Compound Interest

rf

P(1 + )ft = Arf


A = P (1 + i )N



Example D. We open an account with r = 9%, compounded monthly. and after 40 years the total return would be $250,000, what is the initial principal?We have thatr = annual rate = 9% = 0.09 f = frequency of compounding = 12t = number of years =A = total return =

Compound Interest

rf

P(1 + )ft = Arf


A = P (1 + i )N



Example D. We open an account with r = 9%, compounded monthly. and after 40 years the total return would be $250,000, what is the initial principal?We have thatr = annual rate = 9% = 0.09 f = frequency of compounding = 12t = number of years = 40A = total return =

Compound Interest

rf

P(1 + )ft = Arf


A = P (1 + i )N



Example D. We open an account with r = 9%, compounded monthly. and after 40 years the total return would be $250,000, what is the initial principal?We have thatr = annual rate = 9% = 0.09 f = frequency of compounding = 12t = number of years = 40A = total return = 250,000

Compound Interest

rf

P(1 + )ft = Arf


P (1 + )ft = Arf

Substitute these values into formula

r = annual rate = 9% = 0.09 f = frequency of compounding = 12t = number of years = 40A = total return = 250,000

we have that

Compound Interest



we have that

P (1 + )12 * 40 = 250,0000.0912

rf

ft

Compound Interest

P (1 + )ft = Arf



we have that

P (1 + )12 * 40 = 250,0000.0912

rf

ft

P (1 + ) 480 = 250,0000.0912

Compound Interest

P (1 + )ft = Arf

Compound Interest

rSubstitute these values into formula


we have that

P (1 + )12 * 40 = 250,0000.0912

rf

ft

P (1 + ) 480 = 250,0000.0912 or

(1 + ) 480P = 250,000

0.0912

P (1 + )ft = Arf



we have that

P (1 + )12 * 40 = 250,0000.0912

rf

ft

P (1 + ) 480 = 250,0000.0912 or

(1 + ) 480P = 250,000

0.0912

P = $6,923.31

by calculator

Hence the initial deposit in $6,923.31.

Compound Interest

P (1 + )ft = Arf

The four graphs shown here are the different returns with different compound methods ranging from compounding once a yearly to “compounding continuously” – which is our next topic.

Compounded return on $1,000 with annual interest rate r = 20% (Wikipedia)

Compound Interest

62 compound interest

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