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1 Project 2- Stock Option Pricing Mathematical Tools -Today we will learn Compound Interest

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2 Compounding Suppose that money left on deposit earns interest. Interest is normally paid at regular intervals, while the money is on deposit. This is called compounding.

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3 Compound Interest Discrete CompoundingDiscrete Compounding -Interest compounded n times per year Continuous CompoundingContinuous Compounding -Interest compounded continuously

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4 Compound Interest Discrete Compounding P- dollars invested r -an annual rate n- number of times the interest compounded per year t- number of years F- dollars after t years.

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5 Yield for Discrete Compounding The annual rate that would produce the same amount as in discrete compounding for one year. Such a rate is called an effective annual yield, annual percentage yield, or just the yield. Compounded once a year for one year Compunded n times for one year

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6 Yield for Discrete Compounding Interest at an annual rate r, compounded n times per year has yield y.

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7 Discrete Compounding Example 1 (i)What is the value of $74,000 after 3-1/2 years at 5.25%,compounded monthly? (ii) What is the effective annual yield?

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8 Example1 (i) Using Discrete Compounding formula Given P=$74,000 r=0.0525 n=12 t=3.5 Goal- To find F

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9 Example 1 (ii) Using yield formula Given r=0.0525 n=12 Goal- To find y

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10 Discrete Compounding Example 2 (i)What is the value of $150,000 after 5 years at 6.2%, compounded quarterly? (ii) What is the effective annual yield?

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11 Example 2 (i) Using Discrete Compounding formula Given P=$150,000 r=0.062 n=4 t=5 Goal- To find F

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12 Example 2 (ii) Using yield formula Given r=0.062 n=4 Goal- To find y

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13 Annual rate for Discrete Compounding

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14 Annual rate for Discrete Compounding Interest compounded n times per year at a yield y, has an annual rate r.

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15 Discrete Compounding Example 3 (i)What rate, r, compounded monthly, will yield 5.25%?

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16 Example 3 (i) Using Annual rate formula Given y=0.0525 n=12 Goal- To find r

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17 Compound Interest Continuous Compounding The value of P dollars after t years, when compounded continuously at an annual rate r, is F = P e r t

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18 Yield for Continuous Compounding Interest at an annual rate r, compounded continuously has yield y.

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19 Continuous Compounding Example 1 (i)Find the value, rounded to whole dollars, of $750,000 after 3 years and 4 months, if it is invested at a rate of 6.1% compounded continuously. (ii) What is the yield, rounded to 3 places, on this investment?

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20 Example1 (i)Using Continuous Compounding formula Given P=$750,000 r=0.061 t=(40/12) Goal- To find F F = P e r t F = 750,000 e 0.061 (40/12) =$ 919,111

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21 Example 1 (ii) Using yield formula Given r=0.061 Goal- To find y

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22 Logarithms Why do we need logarithms for compound interest ? To find r (since r is an exponent) Recall: yield formula for continuous compounding

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23 Review of Logarithms For any base b, the logarithm function log b (x) The equations u = b v and v = log b u are equivalent Eg: 100=10 2 and 2=log 10 100 are equivalent Two types -Common Logarithms (base is 10) -Natural Logartihms (base is e)- Notation: ln

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24 Review of Logarithms 1.The logarithm log b (x) function is the INVERSE of exp b (x) 2. log b (x) is defined for any positive real number x

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25 Review of Logarithms log b (u v) = log b u + log b v log b (u/v) = log b u log b v log b u v = v log b u. b u b v = b u+v and (b u ) v = b u v, The basic properties of exponents, yield properties for the logarithm functions.

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26 Review of Logarithms ln u = ln v if and only if u=v Most commonly used to obtain solution of equations We can transform an equation into an equivalent form by taking ln of both sides

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27 Review of Logarithms Example1 Find the annual rate, r, that produces an effective annual yield of 6.00%, when compounded continuously.

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28 Example 1 (ii) Using yield formula Given y=6.00% Goal- To find r Taking ln on both sides

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29 Review of Logarithms Example 2 Find the annual rate, r, that produces an effective annual yield of 5.15%, when compounded continuously. Round your answer to 3 places.

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30 Example 2 (ii) Using continuous compounding formula Given y=5.15% Goal- To find r Taking ln on both sides

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31 Review of Logarithms Example 3 How long will it take $10,000 to grow to $15,162.65 if interest is paid at an annual rate of 2.5% compounded continuously?

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32 Example 3 (ii) Using yield formula Given F=$15,162.65 P=$10,000 r=0.025 Goal- To find t

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33 Example 3

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34 Value of Money Discrete compounding Present value (P) and Future value(F) of money We need to rearrange the formula to find P Recall The present value of money for discrete compounding

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35 Value of Money Continuous compounding Present value (P) and Future value(F) of money We need to rearrange the formula to find P Recall The present value of money for continuous compounding

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36 Ratio (R) Under continuous compounding-The ratio of the future value to the present value This allows us to convert the interest rate for a given period to a ratio of future to present value for the same period

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37 Recall- Class Project We suppose that it is Friday, January 11, 2002. Our goal is to find the present value, per share, of a European call on Walt Disney Company stock. The call is to expire 20 weeks later strike price of $23. stocks price record of weekly closes for the past 8 years(work basis). risk free rate 4% (this means that on Jan 11,2002 the annual interest rate for a 20 week Treasury Bill was 4% compounded continuously)

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38 Project Focus I Walt Disney- r =4%, compounded continuously The risk-free weekly ratio for the Walt Disney The weekly risk-free rate for the Walt Disney

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39 Project Focus II Suppose we know the future value ( fv ) for our 20 week option at the end of 20 weeks risk-free rate annual interest 4% Can find the Present value ( pv )

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