Mathematics of Finance
description
Transcript of Mathematics of Finance
![Page 1: Mathematics of Finance](https://reader036.fdocuments.us/reader036/viewer/2022081502/56815a68550346895dc7b433/html5/thumbnails/1.jpg)
Mathematics of Mathematics of FinanceFinance
Mathematics of Mathematics of FinanceFinance
![Page 2: Mathematics of Finance](https://reader036.fdocuments.us/reader036/viewer/2022081502/56815a68550346895dc7b433/html5/thumbnails/2.jpg)
We can use our knowledge of exponential functions and logarithms to see how interest works. When customers put money into a savings account, the money is a loan to the bank and the bank pays interest to the customer for the use of their money. If money is borrowed the customer will pay interest to the bank for the use of the banks money.
![Page 3: Mathematics of Finance](https://reader036.fdocuments.us/reader036/viewer/2022081502/56815a68550346895dc7b433/html5/thumbnails/3.jpg)
Suppose that a principal (beginning) amount P dollars in invested in an account earning 3% annual interest. How much money would be there at the end of n years if no money is added or taken away? (Let t = time and A(t) = amount in account after time t)
![Page 4: Mathematics of Finance](https://reader036.fdocuments.us/reader036/viewer/2022081502/56815a68550346895dc7b433/html5/thumbnails/4.jpg)
Suppose that a principal (beginning) amount P dollars in invested in an account earning 3% annual interest. How much money would be there at the end of n years if no money is added or taken away?
)03.01(03.0)1( PPPA
)03.01(03.0)03.01()2( PPA
))03.01(out (Factor )03.01)(03.01( PP2)03.01( P
![Page 5: Mathematics of Finance](https://reader036.fdocuments.us/reader036/viewer/2022081502/56815a68550346895dc7b433/html5/thumbnails/5.jpg)
By extending this pattern we find that, where P is the principal and r is the constant interest rate expressed as a decimal. This is the compound interest formula where interest is compounded annually.
( ) (1 )tA t P r
![Page 6: Mathematics of Finance](https://reader036.fdocuments.us/reader036/viewer/2022081502/56815a68550346895dc7b433/html5/thumbnails/6.jpg)
Example. Joe invests $500 in a savings account earning 2% annual interest compounded annually. How much will be in his account after 5 years?
trPtA )1()( 5)02.01(500)5( A
04.552$)5( A
![Page 7: Mathematics of Finance](https://reader036.fdocuments.us/reader036/viewer/2022081502/56815a68550346895dc7b433/html5/thumbnails/7.jpg)
What happens when interest is compounded more than one time a year?
![Page 8: Mathematics of Finance](https://reader036.fdocuments.us/reader036/viewer/2022081502/56815a68550346895dc7b433/html5/thumbnails/8.jpg)
Let P =principal, r=annual interest rate, k=number of times the account is compounded per year, and t=time in years. Thus, r/k=interest rate per compounding period, and kt=the number of compounding periods. The amount A in the account after t years is:
![Page 9: Mathematics of Finance](https://reader036.fdocuments.us/reader036/viewer/2022081502/56815a68550346895dc7b433/html5/thumbnails/9.jpg)
Let P =principal, r=annual interest rate, k=number of times the account is compounded per year, and t=time in years. Thus, r/k=interest rate per compounding period, and kt=the number of compounding periods. The amount A in the account after t years is:
A(t) P 1r
k
kt
![Page 10: Mathematics of Finance](https://reader036.fdocuments.us/reader036/viewer/2022081502/56815a68550346895dc7b433/html5/thumbnails/10.jpg)
Example. Suppose P=$1500, r=7%, t=5, k=4
54
4
07.011500)5(
A
17.2122$)5( A
![Page 11: Mathematics of Finance](https://reader036.fdocuments.us/reader036/viewer/2022081502/56815a68550346895dc7b433/html5/thumbnails/11.jpg)
Example. Finding time. If John invests $2300 in a savings account with 9% interest rate compounded quarterly, how long will it take until John’s account has a balance of $4150?
![Page 12: Mathematics of Finance](https://reader036.fdocuments.us/reader036/viewer/2022081502/56815a68550346895dc7b433/html5/thumbnails/12.jpg)
Example. Finding time. If John invests $2300 in a savings account with 9% interest rate compounded quarterly, how long will it take until John’s account has a balance of $4150? t4
4
09.0123004150
t4)0225.1(8043.1 0225.1ln48043.1ln tt4525.26 tyears 63.6
![Page 13: Mathematics of Finance](https://reader036.fdocuments.us/reader036/viewer/2022081502/56815a68550346895dc7b433/html5/thumbnails/13.jpg)
Compounding Continuously
A(t) Pert
![Page 14: Mathematics of Finance](https://reader036.fdocuments.us/reader036/viewer/2022081502/56815a68550346895dc7b433/html5/thumbnails/14.jpg)
Example. Suppose P=$3350, r=6.2%, t=8 yrs
8062.03350)8( eA
17.5501$)8( A
![Page 15: Mathematics of Finance](https://reader036.fdocuments.us/reader036/viewer/2022081502/56815a68550346895dc7b433/html5/thumbnails/15.jpg)
It can be very difficult to compare investment with all the different compounding options. A common basis for comparing investments is the annual percentage yield (APY) – the percentage that, compounded annually, would yield the same return as the given interest rate with the given compounding period.
![Page 16: Mathematics of Finance](https://reader036.fdocuments.us/reader036/viewer/2022081502/56815a68550346895dc7b433/html5/thumbnails/16.jpg)
Example. What is the APY for a $8000 investment at 5.75% compounded daily?
Let x=APY. Therefore, the value of the investment using the APY after 1 year is A=8000(1+x). So,
![Page 17: Mathematics of Finance](https://reader036.fdocuments.us/reader036/viewer/2022081502/56815a68550346895dc7b433/html5/thumbnails/17.jpg)
365
365
0575.018000)1(8000
x
365
365
0575.011
x
1365
0575.01
365
x
%92.50592.0 x
![Page 18: Mathematics of Finance](https://reader036.fdocuments.us/reader036/viewer/2022081502/56815a68550346895dc7b433/html5/thumbnails/18.jpg)
Example. Which investment is more attractive, 6% compounded monthly, or 6.1% compounded semiannually?
Let
ratetheforAPYther
ratetheforAPYther
%1.6
%6
2
1
![Page 19: Mathematics of Finance](https://reader036.fdocuments.us/reader036/viewer/2022081502/56815a68550346895dc7b433/html5/thumbnails/19.jpg)
0619.0 0617.0 21 rr
2
2
12
1 2
061.011
12
06.011
rr
12
061.011
12
06.01
2
2
12
1
rr
![Page 20: Mathematics of Finance](https://reader036.fdocuments.us/reader036/viewer/2022081502/56815a68550346895dc7b433/html5/thumbnails/20.jpg)
The 6.1% compounded semiannually is more attractive because the APY=6.19% compared with APY=6.17% for the 6% compounded monthly.
![Page 21: Mathematics of Finance](https://reader036.fdocuments.us/reader036/viewer/2022081502/56815a68550346895dc7b433/html5/thumbnails/21.jpg)
Guided Practice
![Page 22: Mathematics of Finance](https://reader036.fdocuments.us/reader036/viewer/2022081502/56815a68550346895dc7b433/html5/thumbnails/22.jpg)
1. Jean deposits $3000 into a savings account earning 3% annual interest compounded semiannually. How much will be in the account in 5 years?
![Page 23: Mathematics of Finance](https://reader036.fdocuments.us/reader036/viewer/2022081502/56815a68550346895dc7b433/html5/thumbnails/23.jpg)
2. Becky Jo deposits $10,000 into an account earning 2% annual interest compounded continuously. How much will be in the account in 7 years?
![Page 24: Mathematics of Finance](https://reader036.fdocuments.us/reader036/viewer/2022081502/56815a68550346895dc7b433/html5/thumbnails/24.jpg)
3. Which investment is more attractive, 4% compounded daily, or 4.1% compounded semiannually?
![Page 25: Mathematics of Finance](https://reader036.fdocuments.us/reader036/viewer/2022081502/56815a68550346895dc7b433/html5/thumbnails/25.jpg)
Annuities – Future Value
![Page 26: Mathematics of Finance](https://reader036.fdocuments.us/reader036/viewer/2022081502/56815a68550346895dc7b433/html5/thumbnails/26.jpg)
So far we have only discussed when the investor has made a single lump-sum deposit. But what if the investor makes regular deposits monthly, quarterly, yearly – the same amount each time. This is an annuity
![Page 27: Mathematics of Finance](https://reader036.fdocuments.us/reader036/viewer/2022081502/56815a68550346895dc7b433/html5/thumbnails/27.jpg)
Annuity: a sequence of equal periodic payments
![Page 28: Mathematics of Finance](https://reader036.fdocuments.us/reader036/viewer/2022081502/56815a68550346895dc7b433/html5/thumbnails/28.jpg)
We will be studying ordinary annuities – deposits are made at the end of each period at the same time the interest is posted in the account.
![Page 29: Mathematics of Finance](https://reader036.fdocuments.us/reader036/viewer/2022081502/56815a68550346895dc7b433/html5/thumbnails/29.jpg)
Suppose Jill makes quarterly $200 payments at the end of each quarter into a retirement account that pays 6% interest compounded quarterly. How much will be in Jill’s account after 1 year?
![Page 30: Mathematics of Finance](https://reader036.fdocuments.us/reader036/viewer/2022081502/56815a68550346895dc7b433/html5/thumbnails/30.jpg)
Since interest is compounded quarterly, Jill will not earn the full 6% each quarter.
She will earn 6%/4=1.5% each quarter. Following is the growth pattern of Jill’s account:
![Page 31: Mathematics of Finance](https://reader036.fdocuments.us/reader036/viewer/2022081502/56815a68550346895dc7b433/html5/thumbnails/31.jpg)
End of quarter 2: $200 + 200(1+0.015)=$403
End of quarter 1: $200
End of quarter 3: $200 + 200(1.015) + 200(1.015)2=$609.05
2
End of the year: 200+200(1.015)+200(1.015)2+200(1.015)3= $818.19
![Page 32: Mathematics of Finance](https://reader036.fdocuments.us/reader036/viewer/2022081502/56815a68550346895dc7b433/html5/thumbnails/32.jpg)
This is called future value. It includes all periodic payments and the interest earned. It is called future value because it is projecting the value of the annuity into the future.
![Page 33: Mathematics of Finance](https://reader036.fdocuments.us/reader036/viewer/2022081502/56815a68550346895dc7b433/html5/thumbnails/33.jpg)
Future Value (FV) of an Annuity:
krkr
RFV
kt
11
where R=payments, k=number of times compounded per year, r=annual interest rate, and t=years of investment.
![Page 34: Mathematics of Finance](https://reader036.fdocuments.us/reader036/viewer/2022081502/56815a68550346895dc7b433/html5/thumbnails/34.jpg)
Example. Matthew contributes $50 per month into the Hoffbrau Fund that earns 15.5% annual interest. What is the value of Matthew’s investment after 20 years?
![Page 35: Mathematics of Finance](https://reader036.fdocuments.us/reader036/viewer/2022081502/56815a68550346895dc7b433/html5/thumbnails/35.jpg)
Example. Matthew contributes $50 per month into the Hoffbrau Fund that earns 15.5% annual interest. What is the value of Matthew’s investment after 20 years?
12
155.0
112
155.01
50
2012
FV
73.367,80$FV
![Page 36: Mathematics of Finance](https://reader036.fdocuments.us/reader036/viewer/2022081502/56815a68550346895dc7b433/html5/thumbnails/36.jpg)
Loans and Mortgages – Present Value
![Page 37: Mathematics of Finance](https://reader036.fdocuments.us/reader036/viewer/2022081502/56815a68550346895dc7b433/html5/thumbnails/37.jpg)
Present Value
the net amount of money put into an annuity
This is how a bank determines the amount of the periodic payments of a loan/mortgage.
![Page 38: Mathematics of Finance](https://reader036.fdocuments.us/reader036/viewer/2022081502/56815a68550346895dc7b433/html5/thumbnails/38.jpg)
Present value (PV) of an annuity:
PV R1 1 r
k
kt
r
k
**Note that the annual interest rate charged on consumer loans is the annual percentage rate (APR).
![Page 39: Mathematics of Finance](https://reader036.fdocuments.us/reader036/viewer/2022081502/56815a68550346895dc7b433/html5/thumbnails/39.jpg)
Example. Calculating a Car Loan Payment:
What is Kim’s monthly payment for a 4-year $9000 car loan with an APR of 7.95% from Century
Bank?
![Page 40: Mathematics of Finance](https://reader036.fdocuments.us/reader036/viewer/2022081502/56815a68550346895dc7b433/html5/thumbnails/40.jpg)
12
0795.012
0795.011
9000
)4(12
R
![Page 41: Mathematics of Finance](https://reader036.fdocuments.us/reader036/viewer/2022081502/56815a68550346895dc7b433/html5/thumbnails/41.jpg)
R
48
12
0795.011
12
0795.09000
R51.219$
![Page 42: Mathematics of Finance](https://reader036.fdocuments.us/reader036/viewer/2022081502/56815a68550346895dc7b433/html5/thumbnails/42.jpg)
Kim’s monthly payment will pay $219.51 for 48 months.
![Page 43: Mathematics of Finance](https://reader036.fdocuments.us/reader036/viewer/2022081502/56815a68550346895dc7b433/html5/thumbnails/43.jpg)
Review:
![Page 44: Mathematics of Finance](https://reader036.fdocuments.us/reader036/viewer/2022081502/56815a68550346895dc7b433/html5/thumbnails/44.jpg)
When would you use the compound interest or continuous interest formulas?
![Page 45: Mathematics of Finance](https://reader036.fdocuments.us/reader036/viewer/2022081502/56815a68550346895dc7b433/html5/thumbnails/45.jpg)
What are the similarities and differences Future Value and Present Value?
![Page 46: Mathematics of Finance](https://reader036.fdocuments.us/reader036/viewer/2022081502/56815a68550346895dc7b433/html5/thumbnails/46.jpg)
Can you determine what you would use
to answer the following?
1. Sally purchases a $1000 certificate of deposit (CD) earning 5.6% annual interest compounded quarterly. How much will it be worth in 5 years?
![Page 47: Mathematics of Finance](https://reader036.fdocuments.us/reader036/viewer/2022081502/56815a68550346895dc7b433/html5/thumbnails/47.jpg)
2. Luke contributes $200 a month into a retirement account that earns 10% annual interest. How long will it take the account to grow to $1,000,000?
![Page 48: Mathematics of Finance](https://reader036.fdocuments.us/reader036/viewer/2022081502/56815a68550346895dc7b433/html5/thumbnails/48.jpg)
3. Gina is planning on purchasing a home. She will need to apply for a mortgage and can only afford to make $1000 monthly payments. The current 30-yr mortgage rate is 5.8% (APR). How much can she afford to spend on a home?