Review of the Mathematics of Finance - Lecture 25people.hsc.edu/faculty-staff/robbk/Math111/Lectures...1 Simple Interest 2 Municipal Bonds 3 Compound Interest 4 Inflation 5 Installment
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Review of the Mathematics of FinanceLecture 25
Robb T. Koether
Hampden-Sydney College
Fri, Mar 27, 2015
Robb T. Koether (Hampden-Sydney College) Review of the Mathematics of Finance Fri, Mar 27, 2015 1 / 26
1 Simple Interest
2 Municipal Bonds
3 Compound Interest
4 Inflation
5 Installment Loans
6 Annuities
7 Assignment
Robb T. Koether (Hampden-Sydney College) Review of the Mathematics of Finance Fri, Mar 27, 2015 2 / 26
Outline
1 Simple Interest
2 Municipal Bonds
3 Compound Interest
4 Inflation
5 Installment Loans
6 Annuities
7 Assignment
Robb T. Koether (Hampden-Sydney College) Review of the Mathematics of Finance Fri, Mar 27, 2015 3 / 26
Simple Interest Formula
Simple Interest Formula
F = P(1 + rt)
P =F
1 + rt
whereP is the present value.F is the future value.r is the APR.t is the term.
Robb T. Koether (Hampden-Sydney College) Review of the Mathematics of Finance Fri, Mar 27, 2015 4 / 26
Example
Example (Simple Interest)Invest $1,000 at 5% simple interest for 10 years.Find the future value.
F = 1000(1 + (0.05)(10))
= 1000(1.5)
= $1, 500.00
Robb T. Koether (Hampden-Sydney College) Review of the Mathematics of Finance Fri, Mar 27, 2015 5 / 26
Example
Example (Simple Interest)Invest $1,000 at 5% simple interest for 10 years.Find the future value.
F = 1000(1 + (0.05)(10))
= 1000(1.5)
= $1, 500.00
Robb T. Koether (Hampden-Sydney College) Review of the Mathematics of Finance Fri, Mar 27, 2015 5 / 26
Example
Example (Simple Interest)Invest $1,000 at 5% simple interest for 10 years.Find the future value.
F = 1000(1 + (0.05)(10))
= 1000(1.5)
= $1, 500.00
Robb T. Koether (Hampden-Sydney College) Review of the Mathematics of Finance Fri, Mar 27, 2015 5 / 26
Example
Example (Simple Interest)Invest $1,000 at 5% simple interest for 10 years.Find the future value.
F = 1000(1 + (0.05)(10))
= 1000(1.5)
= $1, 500.00
Robb T. Koether (Hampden-Sydney College) Review of the Mathematics of Finance Fri, Mar 27, 2015 5 / 26
Outline
1 Simple Interest
2 Municipal Bonds
3 Compound Interest
4 Inflation
5 Installment Loans
6 Annuities
7 Assignment
Robb T. Koether (Hampden-Sydney College) Review of the Mathematics of Finance Fri, Mar 27, 2015 6 / 26
Municipal Bonds Formula
Municipal Bonds Formula
face value = (purchase price)(1 + rt)
purchase price =face value
1 + rt
wherer is the APR.t is the term.
Robb T. Koether (Hampden-Sydney College) Review of the Mathematics of Finance Fri, Mar 27, 2015 7 / 26
Example
Example (Municipal Bonds)A municipal bond has a face value of $5,600 at 2% simple interestfor 6 years.Find the purchase price.
purchase price =5600
1 + (0.02)(6)
=56001.12
= $5000.00
Robb T. Koether (Hampden-Sydney College) Review of the Mathematics of Finance Fri, Mar 27, 2015 8 / 26
Example
Example (Municipal Bonds)A municipal bond has a face value of $5,600 at 2% simple interestfor 6 years.Find the purchase price.
purchase price =5600
1 + (0.02)(6)
=56001.12
= $5000.00
Robb T. Koether (Hampden-Sydney College) Review of the Mathematics of Finance Fri, Mar 27, 2015 8 / 26
Example
Example (Municipal Bonds)A municipal bond has a face value of $5,600 at 2% simple interestfor 6 years.Find the purchase price.
purchase price =5600
1 + (0.02)(6)
=56001.12
= $5000.00
Robb T. Koether (Hampden-Sydney College) Review of the Mathematics of Finance Fri, Mar 27, 2015 8 / 26
Example
Example (Municipal Bonds)A municipal bond has a face value of $5,600 at 2% simple interestfor 6 years.Find the purchase price.
purchase price =5600
1 + (0.02)(6)
=56001.12
= $5000.00
Robb T. Koether (Hampden-Sydney College) Review of the Mathematics of Finance Fri, Mar 27, 2015 8 / 26
Outline
1 Simple Interest
2 Municipal Bonds
3 Compound Interest
4 Inflation
5 Installment Loans
6 Annuities
7 Assignment
Robb T. Koether (Hampden-Sydney College) Review of the Mathematics of Finance Fri, Mar 27, 2015 9 / 26
Compound Interest Formula
Compound Interest Formula
F = P(
1 +rk
)kt
P =F(
1 + rk
)kt
whereP is the present value.F is the future value.r is the APR.t is the term.k is the number of compounding periods per year.
Robb T. Koether (Hampden-Sydney College) Review of the Mathematics of Finance Fri, Mar 27, 2015 10 / 26
Example
Example (Compound Interest)Invest $4,000 at 8% compound interest compounded quarterly for10 years.Find the future value.
F = 4000(
1 +0.08
4
)4×10
= 4000(1.02)40
= $8, 832.16
Robb T. Koether (Hampden-Sydney College) Review of the Mathematics of Finance Fri, Mar 27, 2015 11 / 26
Example
Example (Compound Interest)Invest $4,000 at 8% compound interest compounded quarterly for10 years.Find the future value.
F = 4000(
1 +0.08
4
)4×10
= 4000(1.02)40
= $8, 832.16
Robb T. Koether (Hampden-Sydney College) Review of the Mathematics of Finance Fri, Mar 27, 2015 11 / 26
Example
Example (Compound Interest)Invest $4,000 at 8% compound interest compounded quarterly for10 years.Find the future value.
F = 4000(
1 +0.08
4
)4×10
= 4000(1.02)40
= $8, 832.16
Robb T. Koether (Hampden-Sydney College) Review of the Mathematics of Finance Fri, Mar 27, 2015 11 / 26
Example
Example (Compound Interest)Invest $4,000 at 8% compound interest compounded quarterly for10 years.Find the future value.
F = 4000(
1 +0.08
4
)4×10
= 4000(1.02)40
= $8, 832.16
Robb T. Koether (Hampden-Sydney College) Review of the Mathematics of Finance Fri, Mar 27, 2015 11 / 26
Outline
1 Simple Interest
2 Municipal Bonds
3 Compound Interest
4 Inflation
5 Installment Loans
6 Annuities
7 Assignment
Robb T. Koether (Hampden-Sydney College) Review of the Mathematics of Finance Fri, Mar 27, 2015 12 / 26
Inflation Formula
Inflation Formula
F = P (1 + i)t
P =F
(1 + i)t
whereP is the past price.F is the future price.i is the annual inflation rate.t is the number of years.
Robb T. Koether (Hampden-Sydney College) Review of the Mathematics of Finance Fri, Mar 27, 2015 13 / 26
Example
Example (Inflation)A new car costs $25,000 now and the inflation rate is 3%.How much would that car cost 10 years from now?Find the future price.
F = 25000 (1 + 0.03)10
= 25000(1.03)10
= $33, 597.91
Robb T. Koether (Hampden-Sydney College) Review of the Mathematics of Finance Fri, Mar 27, 2015 14 / 26
Example
Example (Inflation)A new car costs $25,000 now and the inflation rate is 3%.How much would that car cost 10 years from now?Find the future price.
F = 25000 (1 + 0.03)10
= 25000(1.03)10
= $33, 597.91
Robb T. Koether (Hampden-Sydney College) Review of the Mathematics of Finance Fri, Mar 27, 2015 14 / 26
Example
Example (Inflation)A new car costs $25,000 now and the inflation rate is 3%.How much would that car cost 10 years from now?Find the future price.
F = 25000 (1 + 0.03)10
= 25000(1.03)10
= $33, 597.91
Robb T. Koether (Hampden-Sydney College) Review of the Mathematics of Finance Fri, Mar 27, 2015 14 / 26
Example
Example (Inflation)A new car costs $25,000 now and the inflation rate is 3%.How much would that car cost 10 years from now?Find the future price.
F = 25000 (1 + 0.03)10
= 25000(1.03)10
= $33, 597.91
Robb T. Koether (Hampden-Sydney College) Review of the Mathematics of Finance Fri, Mar 27, 2015 14 / 26
Outline
1 Simple Interest
2 Municipal Bonds
3 Compound Interest
4 Inflation
5 Installment Loans
6 Annuities
7 Assignment
Robb T. Koether (Hampden-Sydney College) Review of the Mathematics of Finance Fri, Mar 27, 2015 15 / 26
Installment Loans Formula
Installment Loans Formula
M =Pr(1 + r)n
(1 + r)n − 1=
Pr1− (1 + r)−n
whereP is the principal.M is the periodic payment.r is the periodic interest rate.n is the number of payments.
Robb T. Koether (Hampden-Sydney College) Review of the Mathematics of Finance Fri, Mar 27, 2015 16 / 26
Example
Example (Compound Interest)A person takes out a mortgage for $200,000 at 3% for a term of20 years.Find the monthly payments.
M =(200000)(0.03
12 )(1 + 0.03
12
)12×20(1 + 0.03
12
)12×20 − 1
=(200000)(0.0025) (1.0025)240
(1.0025)240 − 1
=910.3770.82075
= $1, 109.20
Robb T. Koether (Hampden-Sydney College) Review of the Mathematics of Finance Fri, Mar 27, 2015 17 / 26
Example
Example (Compound Interest)A person takes out a mortgage for $200,000 at 3% for a term of20 years.Find the monthly payments.
M =(200000)(0.03
12 )(1 + 0.03
12
)12×20(1 + 0.03
12
)12×20 − 1
=(200000)(0.0025) (1.0025)240
(1.0025)240 − 1
=910.3770.82075
= $1, 109.20
Robb T. Koether (Hampden-Sydney College) Review of the Mathematics of Finance Fri, Mar 27, 2015 17 / 26
Example
Example (Compound Interest)A person takes out a mortgage for $200,000 at 3% for a term of20 years.Find the monthly payments.
M =(200000)(0.03
12 )(1 + 0.03
12
)12×20(1 + 0.03
12
)12×20 − 1
=(200000)(0.0025) (1.0025)240
(1.0025)240 − 1
=910.3770.82075
= $1, 109.20
Robb T. Koether (Hampden-Sydney College) Review of the Mathematics of Finance Fri, Mar 27, 2015 17 / 26
Example
Example (Compound Interest)A person takes out a mortgage for $200,000 at 3% for a term of20 years.Find the monthly payments.
M =(200000)(0.03
12 )(1 + 0.03
12
)12×20(1 + 0.03
12
)12×20 − 1
=(200000)(0.0025) (1.0025)240
(1.0025)240 − 1
=910.3770.82075
= $1, 109.20
Robb T. Koether (Hampden-Sydney College) Review of the Mathematics of Finance Fri, Mar 27, 2015 17 / 26
Example
Example (Compound Interest)A person takes out a mortgage for $200,000 at 3% for a term of20 years.Find the monthly payments.
M =(200000)(0.03
12 )(1 + 0.03
12
)12×20(1 + 0.03
12
)12×20 − 1
=(200000)(0.0025) (1.0025)240
(1.0025)240 − 1
=910.3770.82075
= $1, 109.20
Robb T. Koether (Hampden-Sydney College) Review of the Mathematics of Finance Fri, Mar 27, 2015 17 / 26
Outline
1 Simple Interest
2 Municipal Bonds
3 Compound Interest
4 Inflation
5 Installment Loans
6 Annuities
7 Assignment
Robb T. Koether (Hampden-Sydney College) Review of the Mathematics of Finance Fri, Mar 27, 2015 18 / 26
Annuities Formulas
Annuities FormulasBuilding up:
F =P((1 + r)n − 1)
r
whereP is the period deposit.F is the future value.r is the periodic interest rate.n is the number of deposits.
Robb T. Koether (Hampden-Sydney College) Review of the Mathematics of Finance Fri, Mar 27, 2015 19 / 26
Annuities Formulas
Annuities FormulasDrawing down:
M =Pr(1 + r)n
(1 + r)n − 1=
Pr1− (1 + r)−n
whereP is the principal (F from previous formula).M is the periodic withdrawal.r is the periodic interest rate.n is the number of withdrawals.
Robb T. Koether (Hampden-Sydney College) Review of the Mathematics of Finance Fri, Mar 27, 2015 20 / 26
Example
Example (Annuities)A person wishes to retire in 40 years and live for 20 years off theannuity. He can earn an APR of 9%.If he and his employer together invest $800 a month for 40 years,how much can he withdraw each month from the annuity for thenext 20 years?
F =800((1.0075)480 − 1)
0.0075
=28087.920.0075
= $3, 745, 056.22
Robb T. Koether (Hampden-Sydney College) Review of the Mathematics of Finance Fri, Mar 27, 2015 21 / 26
Example
Example (Annuities)A person wishes to retire in 40 years and live for 20 years off theannuity. He can earn an APR of 9%.If he and his employer together invest $800 a month for 40 years,how much can he withdraw each month from the annuity for thenext 20 years?
F =800((1.0075)480 − 1)
0.0075
=28087.920.0075
= $3, 745, 056.22
Robb T. Koether (Hampden-Sydney College) Review of the Mathematics of Finance Fri, Mar 27, 2015 21 / 26
Example
Example (Annuities)A person wishes to retire in 40 years and live for 20 years off theannuity. He can earn an APR of 9%.If he and his employer together invest $800 a month for 40 years,how much can he withdraw each month from the annuity for thenext 20 years?
F =800((1.0075)480 − 1)
0.0075
=28087.92
0.0075
= $3, 745, 056.22
Robb T. Koether (Hampden-Sydney College) Review of the Mathematics of Finance Fri, Mar 27, 2015 21 / 26
Example
Example (Annuities)A person wishes to retire in 40 years and live for 20 years off theannuity. He can earn an APR of 9%.If he and his employer together invest $800 a month for 40 years,how much can he withdraw each month from the annuity for thenext 20 years?
F =800((1.0075)480 − 1)
0.0075
=28087.92
0.0075= $3, 745, 056.22
Robb T. Koether (Hampden-Sydney College) Review of the Mathematics of Finance Fri, Mar 27, 2015 21 / 26
Example
Example (Annuities)A person wishes to retire in 40 years and live for 20 years off theannuity. He can earn an APR of 9%.If he and his employer together invest $800 a month for 40 years,how much can he withdraw from the annuity for the next 20 years?
M =(3745056.22)(.0075)
1− (1.0075)−240
=28087.920.83359
= $33, 695.24
Robb T. Koether (Hampden-Sydney College) Review of the Mathematics of Finance Fri, Mar 27, 2015 22 / 26
Example
Example (Annuities)A person wishes to retire in 40 years and live for 20 years off theannuity. He can earn an APR of 9%.If he and his employer together invest $800 a month for 40 years,how much can he withdraw from the annuity for the next 20 years?
M =(3745056.22)(.0075)
1− (1.0075)−240
=28087.920.83359
= $33, 695.24
Robb T. Koether (Hampden-Sydney College) Review of the Mathematics of Finance Fri, Mar 27, 2015 22 / 26
Example
Example (Annuities)A person wishes to retire in 40 years and live for 20 years off theannuity. He can earn an APR of 9%.If he and his employer together invest $800 a month for 40 years,how much can he withdraw from the annuity for the next 20 years?
M =(3745056.22)(.0075)
1− (1.0075)−240
=28087.920.83359
= $33, 695.24
Robb T. Koether (Hampden-Sydney College) Review of the Mathematics of Finance Fri, Mar 27, 2015 22 / 26
Example
Example (Annuities)A person wishes to retire in 40 years and live for 20 years off theannuity. He can earn an APR of 9%.If he and his employer together invest $800 a month for 40 years,how much can he withdraw from the annuity for the next 20 years?
M =(3745056.22)(.0075)
1− (1.0075)−240
=28087.920.83359
= $33, 695.24
Robb T. Koether (Hampden-Sydney College) Review of the Mathematics of Finance Fri, Mar 27, 2015 22 / 26
Example
Example (Annuities and Inflation)If the inflation rate over those 60 years is 2%, how much is$33,695.24 worth when he makes his last withdrawal?
future value =present value
(1.02)60
=33695.243.28103
= $10, 269.71
Robb T. Koether (Hampden-Sydney College) Review of the Mathematics of Finance Fri, Mar 27, 2015 23 / 26
Example
Example (Annuities and Inflation)If the inflation rate over those 60 years is 2%, how much is$33,695.24 worth when he makes his last withdrawal?
future value =present value
(1.02)60
=33695.243.28103
= $10, 269.71
Robb T. Koether (Hampden-Sydney College) Review of the Mathematics of Finance Fri, Mar 27, 2015 23 / 26
Example
Example (Annuities and Inflation)If the inflation rate over those 60 years is 2%, how much is$33,695.24 worth when he makes his last withdrawal?
future value =present value
(1.02)60
=33695.243.28103
= $10, 269.71
Robb T. Koether (Hampden-Sydney College) Review of the Mathematics of Finance Fri, Mar 27, 2015 23 / 26
Example
Example (Annuities and Inflation)If the inflation rate over those 60 years is 2%, how much is$33,695.24 worth when he makes his last withdrawal?
future value =present value
(1.02)60
=33695.243.28103
= $10, 269.71
Robb T. Koether (Hampden-Sydney College) Review of the Mathematics of Finance Fri, Mar 27, 2015 23 / 26
Example
Example (Annuities and Inflation)If the inflation rate is 2.5% and the APR is 10% and the personwants to withdraw, on his last withdrawal, the equivalent of $7,000,how much should he invest each month?
Robb T. Koether (Hampden-Sydney College) Review of the Mathematics of Finance Fri, Mar 27, 2015 24 / 26
Outline
1 Simple Interest
2 Municipal Bonds
3 Compound Interest
4 Inflation
5 Installment Loans
6 Annuities
7 Assignment
Robb T. Koether (Hampden-Sydney College) Review of the Mathematics of Finance Fri, Mar 27, 2015 25 / 26
Assignment
AssignmentPractice.
Robb T. Koether (Hampden-Sydney College) Review of the Mathematics of Finance Fri, Mar 27, 2015 26 / 26
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