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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
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 Formula (PINA)
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 Formula (PINA) Let P = principal (the money deposited)
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 Formula (PINA) Let P = principal (the money deposited) i = periodic rate (interest rate for a period)
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 Formula (PINA) Let P = principal (the money deposited) i = periodic rate (interest rate for a period) N = number of periods
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 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
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 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
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 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
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
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 $
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
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 $
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
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 $
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
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 $
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
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,
N = (20 years)(4 times per years) = 80
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,
N = (20 years)(4 times per years) = 80Hence A = 1000(1 + 0.02 )80
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,
N = (20 years)(4 times per years) = 80Hence A = 1000(1 + 0.02 )80 = 1000(1.02)80 4875.44 $
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,
N = (20 years)(4 times per years) = 80Hence A = 1000(1 + 0.02 )80 = 1000(1.02)80 4875.44 $
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,
N = (20 years)(4 times per years) = 80Hence A = 1000(1 + 0.02 )80 = 1000(1.02)80 4875.44 $
Compound Interest
Let r = annual rate (r = 8% in example C)f = frequency of compounding (f = 4 times / yr. in example C)
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,
N = (20 years)(4 times per years) = 80Hence A = 1000(1 + 0.02 )80 = 1000(1.02)80 4875.44 $
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
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,
N = (20 years)(4 times per years) = 80Hence A = 1000(1 + 0.02 )80 = 1000(1.02)80 4875.44 $
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
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,
N = (20 years)(4 times per years) = 80Hence A = 1000(1 + 0.02 )80 = 1000(1.02)80 4875.44 $
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
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 , N by ft
From the periodic compound interest formula based on i and N,
Compound Interest
rf
A = P (1 + i )N
we get the compound interest Prffta formula based on r, f, and t
r = annual rate f = frequency of compoundingt = number of yearsreplacing i by , N by ft
P(1 + )ft = Arf
From the periodic compound interest formula based on i and N,
Compound Interest
rf
A = P (1 + i )N
r = annual rate f = frequency of compoundingt = number of yearsreplacing i by , N by ft
From the periodic compound interest formula based on i and 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
we get the compound interest Prffta formula based on r, f, and t
A = P (1 + i )N
r = annual rate f = frequency of compoundingt = number of yearsreplacing i by , N by ft
From the periodic compound interest formula based on i and 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
we get the compound interest Prffta formula based on r, f, and t
A = P (1 + i )N
r = annual rate f = frequency of compoundingt = number of yearsreplacing i by , N by ft
From the periodic compound interest formula based on i and 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
we get the compound interest Prffta formula based on r, f, and t
A = P (1 + i )N
r = annual rate f = frequency of compoundingt = number of yearsreplacing i by , N by ft
From the periodic compound interest formula based on i and 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
we get the compound interest Prffta formula based on r, f, and t
A = P (1 + i )N
r = annual rate f = frequency of compoundingt = number of yearsreplacing i by , N by ft
From the periodic compound interest formula based on i and 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
we get the compound interest Prffta formula based on r, f, and t
A = P (1 + i )N
r = annual rate f = frequency of compoundingt = number of yearsreplacing i by , N by ft
From the periodic compound interest formula based on i and 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
we get the compound interest Prffta formula based on r, f, and t
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
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
P (1 + )12 * 40 = 250,0000.0912
rf
ft
Compound Interest
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
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
r = annual rate = 9% = 0.09 f = frequency of compounding = 12t = number of years = 40A = total return = 250,000
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
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
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