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    Mathematics of Finance

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    Ashekin Mahadi
    Mathematics of Finance 1) Simple interest and future value formulas involve four variables: principal (p), interest rate (i), time (n), and interest (I) or future value (F). Given any three, the fourth can be calculated. 2) Quoted interest rates are usually annual rates unless otherwise specified. Rates can be compounded more frequently than annually, like monthly or quarterly. 3) The compound interest formula calculates future value and involves the additional concept of conversion periods to adjust the interest rate for more frequent compounding.

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    Mathematics of Finance

    Uploaded by

    Ashekin Mahadi
    126 views11 pages
    Mathematics of Finance 1) Simple interest and future value formulas involve four variables: principal (p), interest rate (i), time (n), and interest (I) or future value (F). Given any three, the fourth can be calculated. 2) Quoted interest rates are usually annual rates unless otherwise specified. Rates can be compounded more frequently than annually, like monthly or quarterly. 3) The compound interest formula calculates future value and involves the additional concept of conversion periods to adjust the interest rate for more frequent compounding.

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    Mathematics of Finance 1) Simple interest and future value formulas involve four variables: principal (p), interest rate (i), time (n), and interest (I) or future value (F). Given any three, the fourth can be calculated. 2) Quoted interest rates are usually annual rates unless otherwise specified. Rates can be compounded more frequently than annually, like monthly or quarterly. 3) The compound interest formula calculates future value and involves the additional concept of conversion periods to adjust the interest rate for more frequent compounding.

    Copyright:

    © All Rights Reserved
    Download as PPTX, PDF, TXT or read online from Scribd
    Download as pptx, pdf, or txt
    126 views11 pages

    Mathematics of Finance

    Uploaded by

    Ashekin Mahadi
    Mathematics of Finance 1) Simple interest and future value formulas involve four variables: principal (p), interest rate (i), time (n), and interest (I) or future value (F). Given any three, the fourth can be calculated. 2) Quoted interest rates are usually annual rates unless otherwise specified. Rates can be compounded more frequently than annually, like monthly or quarterly. 3) The compound interest formula calculates future value and involves the additional concept of conversion periods to adjust the interest rate for more frequent compounding.

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    © All Rights Reserved
    Download as PPTX, PDF, TXT or read online from Scribd
    Download as pptx, pdf, or txt
    of 11

    Mathematics of Finance

    Simple Interest and the Future Value


    Interest Rates are generally quoted in percentage form and for use in calculation, must be
    converted to the equivalent decimal value by dividing the percentage by 100.
    1
    For example, i = 8 4 % = 8.25% =8.25/100 =.0825.
    Unless otherwise stated, a quoted rate is a rate per year.
    Formula for calculating Simple Interest.

    I = pin,
    where, I = interest in taka/dollar
    p = principal, the sum of money on which interest is being earned.
    i = Rate of interest per period ( assumed to be one year).
    n = Number of years or fraction of one year.
    Example:
    1
    Find the interest on $600 at 7 2 % for 10 months.
    Solution: I = pin, Here p = $600, i= 7 12 % =15/2% = 7.5/100 = .075
    n =10 months = 10/12
    I =pin = (600)(.075)(10/12) = $37.50

    Exercise: Compute the interest on $480 at 6- 1/4 % for 9 months. Ans: $22.50
    The simple interest formula involves four variables. Given any three of the four variables, the
    fourth can be found by solving the formula.

    Example: Find the interest rate, if $1000 earns $45 interest in 6 months.

    I = pin Here, I=$45 p=$1000 n = 6 months = 6/12 =.5


    45=1000(i)(.5) 45=500i i=45/500=.09(Decimal Rate) To obtain the percent rate we
    multiply by 100, i=.09x100=9%
    The Future Value
    The Future value is the sum of principal plus interest.
    Formula for the future value:
    F = p + pin = p(1 + in)
    Example: Find the future value if $20,000 is invested at 6 percent for 3 months.

    F = p(1+in) Here p = 20,000 i=6% = 6/100 = .06 n =3 months = 3/12 = ¼


    F =20000[1+(.06)(1/4)] =20000(1+.015) =$20,300

    Exercise: Find the future value of $5000 at 10 percent for 9 months. Ans: $5375.00
    Future Value Formula involves four variables. Given any three, fourth can be found.
    Example: Jan received $50 for a diamond at a Jewelry shop and a month later paid $53.50 to
    get the diamond back. Find the percent interest rate.
    Solution: F =p(1+in)
    Here p =$50 F = $53.50, n =1/12year
    53.50 = 50[1+(i)(1/12)]
    Dividing both sides by 50
    53.50/50 =(1+i/12)
    12(53.50/50) = 12+I
    12.84 = 12+I
    i=.84
    So, i=.84x100= 84%
    Exercise: Frank has placed $500 in an employees’ savings account that pays 8 percent
    simple interest. How long will it be,in months, until the investment amounts to $530?
    Present Value
    Present value is analogous to the principal
    F=p(1+in) Divide both sides by (+in)
    F/(1+in) = p
    P=F/(1+in) In practice, accountants and financiers use the present value concept very
    often.

    Example: Find the present value of $530 receivable 9 months from now if the interest rate is 8
    percent.

    P=F/(1+in) = 530/[1+.08(9/12)]=530/1.06 =$500


    Exercise: How much will Fran have to invest now in the employees’ 8 percent savings account
    in order to have $600 a year from now?
    Compund Interest and the Future value
    Compound interest includes interest on the interest.

    Future Value,
    F = p(1+i)n It is the future value of $p for n periods at an interest rate of i percent.

    Example: Find the future value of $1000 at 7 percent compounded yearly for 10 years.

    Solution:

    F = p(1+i)n here,
    p=$1000 i=7% = 7/100 = .07 n=10years.

    F =1000(1+.07)10 =1000(1.07)10 = $1967.15


    EXERCISE: If $500 is invested at 6 percent compounded annually,
    what will be the future value 30 years later? Ans:$2871.74
    The Conversion Period

    Quoted interest rates are rates per year if not accompanied by a qualifying statement such as 1.5
    percent per month.
    In the absence of a qualifier, the quoted annual rate is called the nominal rate.
    Nominal Rate = Rate per year.
    Although the nominal rate is the rate per year, it is common practice to compound interest more
    frequently than once a year.
    Many banks compound interest on savings accounts on a daily basis (365 times a year).
    In other transactions, interest is compounded monthly, quarterly, or semiannually.
    No. of conversions per year
    Conversion No. of conversions per year
    Daily 365
    Monthly 12
    Quarterly 4
    Semiannually 2
    Example
    Find the future value of $500 at 8 percent compounded quarterly for 10 years.

    Solution: F = P(1+i)n

    Here, no. of conversions per year = 4


    So n = (No. of years)(No. of conversions per year) = (10)(4) = 40
    The quoted 8% is a nominal rate or rate per year.

    i = Nominal rate per year/No. of conversions per year = 8%/4 = 2% =2/100 =.02 per period.
    F=500(1+.02)40 =500(1.02)40 =$1104.02
    Exercise: If $500 is invested at 6 percent compounded semiannually , what will be the amount
    5 years?
    Example: Compute the future value of $5,000at 9 percent compounded monthly for 10 years.

    Solution: F = P(1+i)n
    Here P=$5000 n=(10)(12) = 120 i = .09/12 = .0075

    F = 5000(1+.0075)120 =$12256.79

    Exercise: A bank pays 7.25 percent compounded daily on 90-day notice accounts. If $500is
    deposited in such an account, what will be the amount in 90 days? Ans. :$509.02
    Finding the time and the interest rates
    The compound interest formula involves four variables. If we know one, the other three can be found.

    Example: At 8 percent compounded annually, how many years will it take for $2000 to grow to $3000?

    Solution: F=p(1+i)n n?

    3000=2000(1+.08)n 2000(1.08)n = 3000


    (1.08)n =3000/2000 =1.5
    Taking natural logarithms on both sides, ln(1.08) n = ln(1.5)
    n ln(1.08) = ln(1.5)
    n = ln1.5/ln1.08 = 5.268 years.
    .268 years .268*12 =3.216 months. .216 months = .216*30 =6.48 days =6 days.
    It will take 5 years 3 months and 6 days.
    Exercise: Find how many years it will take at 9 percent compounded annually for $1000 to grow to $2000.
    Ans: About 8.043 years.
    Example: At what interest rate compounded annually will a sum of money double in 10 years?
    Solution: Here doubling means $100 grows to $200, $25 grows to $50, $1 grows to $2 and so
    on and the time required is the same in any doubling.
    We shall use p= $1 and F = $2
    Hence F =p(1+i)n 2 =1 (1+i)10 1(1+i)10 =2 (1+i)10 =2
    Taking natural logarithms on both sides, 10ln(1+i) =ln2 ln(1+i) = ln2/10=.069314718

    (1+i) =Antiln(.069314718) =1.071773462 So, i=1.071773462-1 = .071773462

    i= .071773462*100 =7.177%
    Exercise: Find the rate of interest that, compounded annually, will result in tripling a sum of
    money in 10 years?
    Ans. 11.612 percent.

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