(B) What Is R(S), the Forward Rate Function?

Math 53 – Math of Finance

Friday, February 2, 2007 (revised Feb. 6)

Homework 3 --- Due Wednesday, February 7, 2007

1. An explicit P(t) function. Suppose for all t ≥ 0.

(a) What is D(t) ?

(b) What is r(s), the forward rate function?

(This assumption is theoretically possible, but not very plausible. In reality there is no reason for P(t) to be given by a simple formula.)

2. For this problem, use this present value curve (not the same as problem 1):

t 0 0.5 1 1.5 2 2.5 3

P(t) 1.000 0.990 0.980 0.969 0.957 0.946 0.934

You can take these values as exact, but you’ll have to guess at the values between.

(a) What is D(2) ?

(b) What is ?

3. For this problem, use the present value curve implied by columns H and I of the spreadsheet “strip-prices-29jan07.xls” (on the course website). (These are the columns represented by the blue dots in the graph.) (Warning: The “P(t)” values in column I are multiplied by 100. So if the table appears to say that P(t)=98.553, then P(t) is really 0.98553.)

(a) Make reasonable estimates of P(10), D(10), and . (“10” means t = 10 years.)

(b) OK, now it gets harder. Make a reasonable estimate of r(10) and explain your method. [ Hint: 6.567% is a good start, but it isn’t a reasonable estimate. ]

4. Flat Dollars. We defined a flat dollar as follows:

“A flat dollar delivered at time t is the same as (1/P(t)) US dollars delivered at time t. Equivalently: a US dollar delivered at time t is the same as P(t) flat dollars. ”

The following contract is written in terms of US dollars:

“I pay you $2.80 today. Six months from now, if the share price S of MSO is

above K = $30.00, then at that time you will pay me the excess, S – K.”

Translate this into a contract expressed in terms of flat dollars. Use the P(t) function from problem 2 (in particular, P(1/2 year)=0.99).

5. Pick a publicly traded corporation, and download daily closing share prices for that company’s common stock for at least all of 2001-2006 (and a longer period if you like).

Work with others if you like, but choose one company for each person.

Adjust the prices for dividends, or trust (for example) Yahoo to do it for you. (That is, use the “adjusted closing prices” from Yahoo’s site.)

Compute R(t), the logarithmically-defined daily return, for each day.

a. Symbol for your company: ______

b. Time period for prices you downloaded: ______to ______

c. Largest absolute value of R(t) (logarithmically-defined daily return) you found

during this period: ______

d. Mean (average) of the R(t) values: ______

e. Standard deviation of the R(t) values: ______

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