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

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.

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

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.

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.
Compunded n times for one year
Compounded once a year
for one year

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

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

8. Example1 (i) Using Discrete Compounding formula Given P=$74,000
Goal- To find F

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

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?

11. Example 2 (i) Using Discrete Compounding formula Given P=$150,000
Goal- To find F

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

13. Annual rate for Discrete Compounding

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

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

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

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

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

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?

20. F = Pert Example1 F = 750,000e0.061(40/12) =$ 919,111
Using Continuous
Compounding formula
Given
P=$750,000
r=0.061
t=(40/12)
Goal- To find F
F = Pert
F = 750,000e0.061(40/12) =$ 919,111

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

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

23. Review of Logarithms For any base b, the logarithm function logb (x)
The equations u = bv and v = logbu are equivalent
Eg: 100=102 and 2=log10100 are equivalent
Two types
-Common Logarithms (base is 10)
-Natural Logartihms (base is e)- Notation: ln

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

25. Review of Logarithms bubv = bu+v and (bu)v = buv,
The basic properties of exponents,
yield properties for the logarithm functions.
bubv = bu+v and (bu)v = buv,
logb(uv) = logbu + logbv
logb(u/v) = logbu  logbv
logbuv = vlogbu.

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

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

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

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.

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

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

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

33. Example 3

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

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

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

37. Recall- Class Project
We suppose that it is Friday, January 11, 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.
stock’s 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)

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

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)