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Chapter 9: Mathematics of Finance

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Title: Chapter 9: Mathematics of Finance


1
Chapter 9 Mathematics of Finance
2
9-1 Time Value of Money Problems
  • Whenever we take out loans, make investments, or
    deal with money over time, common questions
    arise.
  • How much will this be worth in 10 years?
  • How will inflation eat into my retirement savings
    over the next 30 years?
  • How much do I need to save each month to have
    1,000,000 at age 60?
  • How much do I need to pay each month on a 30 year
    mortgage of 400,000 if the interest rate is
    5.25
  • These are called Time Value of Money (TVM)
    problems.

3
Common Notations
  • P Present Value the current value of an
    investment or loan or sum of money.
  • F Future Value The value of an investment,
    loan or sum of money in the future.
  • r Annual Percentage Rate
  • m Number of periods per year. Example If you
    make monthly loan payments, them m 12

4
Common Notationscontd
  • i Interest Rate per Period Example The annual
    interest rate is 6 and there are 12 payments per
    year. Then i .06/12 .005, or 0.5 per month.
  • i r/m
  • t Time, measured in years
  • n Total number of periods. Example You have a
    10 year loan that is paid monthly. Then you have
    n 1012120 total periods. n mt
  • R The amount of a Rent. This is the regular
    payment made on a loan or into an investment.

5
Example
  • Suppose you deposit 1000 into an account that
    compounds interest quarterly. The annual rate of
    interest is 2.3 and you are going to keep it in
    the account for 4.5 years. At the end of this
    time, the account will be worth 1108.72
  • P 1000 F 1108.72 r .023
  • m 4 i .023/4 .00575 t 4.5
  • n 18

6
9-2 Percent Increase/Decrease
  • If a quantity increases by some percent, we can
    create a multiplier that helps us convert a
    beginning value to an ending value.
  • To create the appropriate multiplier
  • Percent Increase 1 i
  • Percent Decrease 1 - i

7
Example
  • This year, the SCCC student population is 11,350.
  • The administration estimates that will increase
    by 2 next year. How many students can we expect
    next year?
  • The multiplier 10.02 1.02
  • New student population 11,350(1.02) 11577

8
Example
  • The current balance of my retirement account is
    244,350.
  • If the value of the account drops by 5.2 over
    the next year, what will be the new value?
  • Multiplier 1 - .052 .948
  • New Value 244,350(.948) 231,643

9
9-3 Compound Interest
  • When we invest money, interest may be applied to
    the account on a regular basis. For example, if
    we invest in an account that pays interest
    monthly, we say the interest compounds monthly.
    Anything in the account at the time of
    compounding gets interest added to it. In the
    monthly case, we have 12 compoundings per year,
    with each compounding representing 1/12 of the
    total annual interest rate.

10
Example
  • We invest 1000 in an account that compounds
    monthly. The annual interest rate is 3.6. If we
    keep it in the account for 5 years, adding or
    removing nothing, how much will be in the account
    at the end of 5 years?
  • First, we need to note that the interest per
    period is i .036/12 .003.
  • Lets begin by building a table

11
Example
Period Previous Balance New Balance
0 0 1000
1 1000 1000(1.003)
2 1000(1.003) 1000(1.003)(1.003)
3 1000(1.003)(1.003) 1000(1.003) (1.003) (1.003)

12
Example
Period New Balance
0 1000 1000(1.003)0
1 1000(1.003) 1000(1.003)1
2 1000(1.003)(1.003) 1000(1.003)2
3 1000(1.003) (1.003) (1.003) 1000(1.003)3

13
Example
Period New Balance
3 1000(1.003) (1.003) (1.003) 1000(1.003)3
4 1000(1.003)4
5 1000(1.003)5
After 5 years, or 60 periods, we have After 5 years, or 60 periods, we have After 5 years, or 60 periods, we have
60 1000(1.003)60
After n total periods, we have After n total periods, we have After n total periods, we have
n 1000(1.003)n
14
Compound Interest Formula
  • If P dollars earn an annual interest rate of i
    per period for n periods, with no additional
    principal added or removed, then the future
    value (F) is given by
  • F P(1i)n

15
Example
  • A bank account is opened with 4,000 in the
    account. It earns 6 annual interest. If it earns
    interest quarterly (four times per year), then
    what is in the account after 10 years?

16
Example
  • Suppose you invest 500 today at an annual rate
    of 1.5, compounded daily. How long before the
    balance doubles?

17
9-4 Rule of 72
  • Given some investment that grows at an annual
    interest rate, r, (not expressed as a decimal),
    then the amount of time in years it takes for the
    investment to double is approximately

18
Example
  • How long will it take for an investment to double
    if it earns 5 annual interest?
  • Note that estimating the doubling time does not
    require that we know how much is originally
    invested!

19
Example
  • If you want your investment to double in 30
    months, what annual interest rate do you need to
    secure?

20
9-5 Yield
  • Because each compounding acts on the original
    balance and any interest that has been previously
    earned, the net interest earned will not be the
    same as the annual interest rate at the end of
    the investment. The true interest rate earned
    is called the Yield.

21
Example
  • Invest 100 for 1 years, compounded monthly at an
    annual rate of 12.
  • F 100(1.01)12 112.68
  • This represents a yield of 12.68, which is
    higher than the original 12 stated above. The
    yield is often called the Annual Percentage Yield
    (APY). Always ask what this is when you take out
    a loantime, compounding and bank fees can
    substantially increase your rate of interest and
    therefore your total payments due!
  • APR 12
  • APY 12.68

22
Yield Formula
  • The formula for yield in the t 1 year case is

23
Where did the formula come from?
24
Why is it important to know the APY?
  • The APY, or yield, is helpful since it simplifies
    calculations. If we know the APY, then it does
    not matter how many times we compound per year
    because the APR will give us actual percentage
    increase at the end of a year.
  • APR Annual Percentage Rate or nominal rate or
    rate
  • APY Annual Percentage Yield or effective rate
    or yield

25
Example
  • If 375 is invested with an APY of 5.22 for 8
    years and 3 months, what is the Future value of
    the investment?

26
9-6 Annuities
  • When additional payments or deposits (rents) are
    made at regular intervals into an investment,
    then we call these annuities
  • Ordinary Annuity Payment is due at the END of
    each period.
  • Annuity Due Payment is due at the START of each
    period.
  • This will complicate our Future Value
    calculations.

27
Example
  • We invest in an account that compounds annually.
    The annual interest rate is 3. If we add 800 at
    the end of each year (an ordinary annuity), how
    much will be in the account at the end of 5 years
    total?
  • Lets look at a picture of what is going on here.

28
Example
Year 1
Year 2
Year 3
Year 4
Year 5
Start

Investment Period
800
800
800
800
800
Each of these 800 investments earns interest for
a different period of time. Hence, the value of
each of these deposits is different at the end of
the 5-year period.
29
Example
The End
End Year 1
End Year 2
End Year 3
End Year 4
End Year 5
Start

Investment Period
800
800
800
800
800
This one is worth 800(1.03)4 at the End 900.41
This one is worth 800(1.03)3 at the End 874.18
This one is worth 800 at the End 800
This one is worth 800(1.03)1 at the End 824
This one is worth 800(1.03)2 at the End 848.72
30
Example
  • We can add all of these up
  • 800(1.03)4 800(1.03)3 800(1.03)2
    800(1.03)1 800
  • 900.41 874.18 848.72 824 800
  • 4247.31

31
A General Formula
  • Now imagine if the monthly payments were
    deposited and monthly interest credited. We would
    then have 512 60 different deposits to find
    the values for so we can add them up.
  • To avoid this inefficiency, we instead use the
    following formula, which is equivalent to going
    through that process.

32
A General Formula
  • If R dollars are paid at the end of each period,
    with an interest rate of I per period, then the
    Future Value of the Annuity is

33
Where did the formula come from?
  • We can generalize the example before and think
    about adding R R(1i) R(1i)2 R(1i)3
    R(1i)n-1. Let us call this sum S.
  • Hence, (1i)S - S R(1i)n - R
  • S (1i - 1) R(1i)n - R
  • Hence, the sum S is
  • (R(1i)n - R)/i

34
Check
  • We invest in an account that compounds annually.
    The annual interest rate is 3. If we add 800 at
    the end of each year (an ordinary annuity), how
    much will be in the account at the end of 5 years
    total?

35
Example
  • What is the future value if you invest 95 per
    month for 7 years at an annual rate of 3.75?
  • R 95
  • i .0375/12
  • n 127 84

Note that I try to keep as many decimal places as
possible until the end
36
FV for Annuities Due
  • When payments or deposits are made at the
    beginning of a period (rather than at the end as
    in the previous examples), an adjustment is
    needed.
  • We can view each payment as if it were made at
    the end of the preceding period. This would
    require one more payment (n1 total) than usual
    and would require that we subtract the last
    payment so we dont overpay.

37
FV for Annuities Due
38
Example
  • If 300 payments are made at the beginning of the
    month for 18 years (a college savings fund), what
    is the Future Value if the annual interest rate
    is 5.5

39
9-7 Future Value (FV) on Excel
  • The FV command will do these computations for us
    automatically.
  • Command Format
  • FV(rate, nper, pmt, pv, type)

This is i, the rate per period
This is n, the total of periods
This is R, the amount of rent
This is P, the Present Value
Blank for ordinary annuity, 1 for annuity due
40
Excel Examples
  • If 300 payments are made at the beginning of the
    month for 18 years (a college savings fund), what
    is the Future Value if the annual interest rate
    is 5.5
  • What is the future value if you invest 95 per
    month (paid at the end of the month) for 7 years
    at an annual rate of 3.75?

41
9-8 PV of Annuities
  • Suppose you have an ordinary 20-year annuity that
    you pay 500 into at the end of each quarter. The
    annual interest rate is 7.
  • What is the lump-sum of money which should be
    deposited at the start of the annuity that would
    produce the exact same amount of money at the end
    of the period, without any additional payments?
  • This is know as the Present Value of the Annuity

42
Present Value of an Ordinary Annuity
43
Example
  • Suppose you set up an ordinary annuity account
    which is to last 10 years and earn 4 annual
    interest rate. If your rent payment is 150 per
    month, how much do you need to deposit as a lump
    sum up front to achieve the same end result
    without any regular payments?

44
9-9 Present Value (PV) on Excel
  • The PV command will do these computations for us
    automatically.
  • Command Format
  • PV(rate, nper, pmt, fv, type)

This is i, the rate per period
This is n, the total of periods
This is R, the amount of rent
This is the FV you want after the last
pmt Optional
Blank for ordinary annuity, 1 for annuity due
45
  • Can you figure out how this comes from the
    formula for the Future Value of an Ordinary
    Annuity?

46
Loan Payment Formula
Start here with the original formula
Divide both sides by to get R alone
Here is the formula for the loan payment.
47
Loan Payment Formula
  • Using basic algebra, we can rewrite this as

48
Example
  • Suppose you want to buy a home and take out a
    30-year mortgage for 240,000. The annual
    interest rate is 5.75.
  • What is the monthly payment?
  • What total amount of money do you pay over the
    life of the loan (assuming all regular payments
    are made)?
  • How much of your total payments is interest?

49
Example (a)
  • We use the formula to get 1400.57 per month .

50
Example (b) and (c)
  • The total amount of money we pay is
  • 360x1400.57 504,205.20
  • Hence, the amount of interest paid is
  • 504,205.20 - 240,000 264,205.20

51
69
  • You borrowed 150,000, which you agree to pay
    back with monthly payments at the end of each
    month for the next 10 years. At 6.25 interest,
    how much is each payment?

52
9-10 Loan Payments (PMT) on Excel
  • The PMT command will do these computations for us
    automatically.
  • Command Format
  • PMT(rate, nper, pv, fv, type)

This is i, the rate per period
This is n, the total of periods
This is the present value of the loan
This is the FV you want after the last
pmt Default0
Blank for ordinary annuity, 1 for annuity due
53
9-12 Adjusting for Inflation
  • Inflation can seriously devalue a loan or asset
    over time.
  • For example, if the average inflation rate is
    3.5, how much will 50 be worth in 5 years (in
    terms of todays dollars)? In other words, in
    five years how much can I buy with a 50 bill
    compared to what I can buy today?

54
Example
Important Notice that we substituted 50 for F
since that is what we know we will have in the
future. We solve for P since we want to know what
the Future 50 is worth in Present dollars.
  • F P(1i)n
  • 50 P(1.035)5
  • 50 P(1.187686306)
  • 50/(1.187686306) P
  • 42.10 P

55
Terms
  • Nominal Dollars are those that have not been
    adjusted for inflation.
  • Real Dollars Present Dollars are those that
    have been adjusted for inflation and therefore
    reflect the spending power of some future amount
    of money in terms of todays dollars.

56
  • Well disregard amortization tables.
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