 # Pricing Fwds/futures

A couple quick questions that hopefully y’all can help me with:

1.) When pricing these derivatives, i understand adjusting the price for monetary benefits (eg interest or dividends) but why are they discounted by the full Risk-free annual rate? That is, if you receive interest in 30 days for example, why is that amount discounted by (1+Rf) instead of (1+Rf / # of compounding periods) ? We used the latter previously when discounting something other than annual periods, no? Seems odd that we use an annual rate to discount a non-annual cash flow.

2.) The text equates shorting to borrowing. How are the two essentially the same? In cash and carry arbitrage, you first borrow the money to buy the security. In reverse C&C, you short first and then invest the money. The term and process of actually “borrowing” is used in the former yet the book relates shorting to borrowing. Which would mean that for reverse C&C you basically start the process by shorting/borrowing, no??

Thanks in advance for any help!!!

You are using ^T for the discounting definitely. E.g. if coupon is in 6 months and this is a 2 year contract (u use (1+rf)^(18/12), e.g.)

When you short - you are giving up something you do not have yet, getting cash in hand and at the end of the contract, buying and replacing the asset e.g. So it is akin to borrowing, since you are borrowing the asset.

but why not (1+Rf / 2)^T ?? In the fixed income study session, we would discount the semiannual coupons by cpn / (1+r/n)^T*n. However those were stated nominal rates which are not effective rates. Is the Risk-free rate applied here to futures/forwards assumed to be an effective rate, in which case i totally agree with and understand your example ???

you are looking at it from the point of view of the investor. the dividend / coupon has been received. He is depositing that amount in the bank and he gets the full risk free rate for it.

for a bond - that calculation is from the issuer’s perspective. He is assumed to be reinvesting it at the YTM to arrive at a “future” / “present” value.

I think these are different applications.

You first point is interesting, haven’t quite thought of it like that…altough i’m not still entirely understanding. Are you referring to the viewpoint of SHORT investor to a future/forward ? In that case i guess i could see that bc they’re the ones receiving the coupon/dividend.

I think the whole point is that we’re dealing with annual Rf rates in which case compounding frequency doesn’t matter (ie the n portion of 1+r/n) and we’re only concerned about the period of the year to discount or future (ie T). Agree?

Then the only thing i’m wondering is why if say the Forward expires in 3 mos, why don’t we use a 3 month treasury Rf?

Guess i’m having second thoughts on my previous post. For coupons if you have semiannual coupons, you do indeed have a compounding frequency other than annual, so again, why not (1 / 1+r/n)?

you are looking at it from the perspective of the person who is LONG THE FUTURES CONTRACT.

He has a position on the underlying - and the underlying is paying the coupons e.g.

if he owned the bond - he would be receiving the coupons, and depositing that money in his bank, earning the risk free rate. Since the futures position is a substitute for owning the bond - you need to discount that at the risk free rate. and for your other point.

if you had a 5% coupon -> semi annual -> 2,5% per period.

1.05^0.5 -1 = 2.47%

not very different.

The risk-free rate is an annual effective rate. It’s not the six-month rate reported annually; it’s the annual rate. Therefore, when you discount you simply change the time period in the exponent.

You’re right about bonds, but that is sort of specific to bond lingo. Interest rates on bonds are reported as the effective periodic rate * the number of periods. When you use a generic “risk-free rate” in derivatives problems, it’s assumed you have an annual rate (whether it’s continuously compounded or not is another issue) unless stated otherwise, because derivatives by themselves do not spit out interest payments at certain intervals.

Thanks CPk and Aaron for the help!! Let me make sure I’m with y’all.

How do you know Rf rates are implicitly effective rates, does that go for all Rf rates or just thsoe involving derivatives?

I’m in total agreement if that’s the case bc like you mentioned aaron, at that point all you do is change the exponent to reflect the period when using efffective rates. This is unique bc it seems like all we’ve dealt with up till now is stated nominal rates (coupons, discount rates/WACC…).

So do we only use annual rates when dealing with derivatives regardless of the time till expiration? Why not a 3 mos treasury Rf if the contract expires in 90 days (instead of an annual rate)?

If you have nothing to go from (e.g., it’s not obvious from the problem statement that the interest rate isn’t annual), you should assume that the interest rate they give you is annual. That’s pretty typical for non-bond problems that have interest rates. A huge amount of my derivatives study has been put-call parity stuff, and in those problems the interest rates are typically continuous. I don’t know if CFA questions wouldn’t be explicit about whether an interest rate is continuous; I would hope that they’re clear. Just for the sake of saying it, you can switch between continuous and discrete interest rates really easily anyway.

How do you know the risk-free rate is an annual effective rate, as opposed to a nominal rate that we’ve used for so many other problems/applications?

Because when they say, “the risk-free rate is 5%,” there is no payment schedule underlying it. A bond that pays annual coupons would be reported as an effective rate; other frequencies are nominal because they are reported in annualized terms. Referring to a general interest rate, unless specified otherwise, implies the rate is annual by convention.

Thanks, aaron!