Calculating Forward Price Given Monthly Risk-Free Rates

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bigdu

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Hello,
I am having trouble pricing a six-month forward contract given monthly risk free rates. In this particular example you are given the following monthly risk free rates:
3-month: .5%
6-month: .5%
12 months: 1%
Assume that the price of the stock today is $100 and we will receive $1 dividends in 90 and 180 days. The steps to find the answer (according to the solution given) are:
Find PV of dividends:
$1/(1.005)^90/365
$1/(1.005)^180/365
Once the PV of dividends are calculated and added together, you subract them from the stock price today and multiply by the (1+RFR)^T as follows:
($100-PV Dividends) x (1.005)^180/365
I do not understand the discount rates they are using. Why, for example, would you use (1.005)^90/365 to find the PV of the first dividend? Wouldn’t you use (1 + annual RFR)^90/365?
Every example in the curriculum as far as I can tell gives the annual risk free rate so my rationale was to use the 12 month rate as given (1%) and find the PV of the dividend as 1/(1.01)^90/365.
Similarly, why would you multiply (100-PV of Dividends) by (1 + 6 month rate)^180/365?
Any input on this would be appreciated. Thanks.
 
If they give you specific rates applicable for the periods needed, you should de-annualize them to get your discount factor and multiply, or divide by (1+r)^ t/365.
The PV of dividends/coupons must be accounted for and discounted. Alternatively, you could grow the 100 to the FV at one year, grow the dividends to the FV at one year and remove that from the FV of the 100. Either way, the answer will be the same.
Are you asking why do you not use the longest term (12 month) rate to calculate the 6 month forward on the equity? In theory, the 12 month rate isn’t applicable on a 6 month forward. You do not need to extrapolate the 6 month forward rate using the 12 and 6 month rates. Simply take it as given.
Hope that helps.
 
Thanks for the help. My main concern is the discount rates. As I mentioned (as far as I can tell) in the curriculum they only provide annual risk free rates for calculating forward contracts. So let’s say the annual RFR is 5%. If I want to find the present value of a $1 dividend in 180 days given that annual risk free rate, I would do 1/(1.05)^180/365. That makes sense…given an annual rate, I have to deannualize it to the point at which I’m receiving the dividend.
I guess what’s not clicking with me is if they give the 6 month risk free rate, why do I need to deannualize it to calculate a cash flow being received in 6 months?
Thanks again.
 
The author of the question probably didn’t specify that those rates were spot rates; sloppy on the author’s part.
The one-year spot rate applies to payments received in one year. Not payments received in less than one year, nor payments received in more than one year; only to payments received in exactly one year.
If you have a payment coming in 6 months, you have to use the 6-month spot rate to discount it.
Such is the nature of spot rates.
 
The rates were given in an “exhibit” labeled labeled risk-free interest rates with the maturity and the corresponding rate. So you’re right…they did not specify that they were spot rates. Makes sense now though, thanks.
 
i think i’m in trouble. i remember that calculation and did it correctly (ie. found the answer the way shown above). BUT, now i’m having a very brain foggy moment, and not understanding why in certain cases (ie. calculating the PV/discount factors on term structures, you don’t take the exponent on the rate to discount, you simply multiply by 30/360, or 540/360, etc. But then there’s other instances where you obv take the annual rate to the exponent (90/365), instead of simply multiplying the annual rate by 90/365 to discount.
very scary that this was completely intuitive for all these months, and now i’m questioning basic PV laws…if anyone can help, do chime in.
 
yeah this makes sense given the only times i recall using simple interest is when calculating LIBOR 360 day term structure rates. thanks guys.
 
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