How do you solve for risk free rate/or any superscript variable?

[DavidM]

How do you solve for risk free rate/or other variables in the superscript?

i.e. F=Se^(rf)T
F=Se^(rf-c)T



What is rf?

What is c=convineince yield?

How do you algebraically solve for anything that is in power form?

I think I saw this question in the practice exam 2009 or some variation of it?

I think this question could be applied to many formulas?

Could you PLEASE show the algabraic breakdown for all the steps!

I would greatly appreciate it,

Thanks!

David.

 

[David Harper CFA FRM]

David,

Yes, agreed, this is a vital operation...thank you for surfacing to remind folks
What is rf? riskless rate
What is c=convineince yield? it is the intangible equivalent of a dividend/income: non-financial benefit of ownership which reduces cost of carry; since it is difficult to directly observe, whereas futures prices are not, often is it the solved for "plug variable"

F = S*e^((r-c)*T) = S*EXP[(r-c)T); both are exponential function, I note this only b/c we tend to use EXP() here and I like EXP() a little better if the formula gets hairy

divide both sides by S
F/S = EXP[(r-c)T]

so here is the key: LN() and EXP() are inverse functions
e.g., EXP(LN(x)) = x and LN(EXP(x)) = x

we have F/S = EXP[(r-c)T] and we need to "liberate" (r) from the exponent
so take LN() of both sides
LN(F/S) = (r-c)*T
...now you can get to the riskless rate

Okay, so now let me connect this to another similar formula, and ask you a question:
under continuos compounding, the price of a zero-coupon bond is given by:
D = F*EXP(-y*T), D = price, F = Face, y = yield
but yield is riskless rate + spread, so y = r + s, so:

D = F*EXP(-(r+s)*T)

now, Stulz gives the formula for the spread as a function of debt price (D), face value (F), riskless rate (r), and term to maturity (T). Can you show the derivation of (s)?

David

 

[DavidM]

D=F*EXP(-(r+s)*T)

divide both sides by F:
D/F=EXP(-(r+s)*T)
take the log of both sides:
LN(D/F)=(-(r+s)*T)
divide both sides by T:
LN(D/F)/T= -(r+s)

NOT TOO SURE ABOUT THAT NEGATIVE BEOFORE (r +s)

Subtract r from both sides r:

{LN(D/F)/T}-r=s

is that correct?

 

[DavidM]

Oh and thanks for the initial answer that really has helped me!

 

[David Harper CFA FRM]

Sure thing (the basics are the most important, and they never stop teaching me)

Almost... I know *exactly* that feeling at the negative ... the doubt creeps in ... but maybe trust the math a little more to do the right thing for you:

LN(D/F)/T= -(r+s)

LN(D/F)/T = -r -s
LN(D/F)/T + r = -s
-1*(LN(D/F)/T + r) = s
s = -LN(D/F)/T - r; i.e., this is Stulz 18.6 formula for credit sread

...and keep in mind, you can almost always give yourself a sanity check at the end; e.g.,
if problem is: 4% rf, 5 yr term, debt price of $70, then
spread = -LN(70/100)/5 - 4% = 3.13% ... okay how can i check that?

70*EXP((4%+3.13%)*5) =? 100

David