Hi, This is about Constant Maturity Swaps, exhibit Spreadsheet T5.b.4 1- In the bottom calculations, I understand the $2500 are the fixed payments. Why were they made at t=0.5 but not at t=1? 2- I'm not sure I understand why the negative payment -$2500 in the lower line 3- When we discounted back through the tree, why did we multiply the fixed payment of $2500 by the probabilities ?! These payments are not tied to the probabilities. Real world or risk neutral. They are the cash flows due to the fixed receiving leg of the swap. The way I see it is that we should've kept them separate : the cash flows and the PVs of swap value at the second node. The latter have to be discounted and tied to their probabilities, whereas the cash flows are certain (p=100%) and need only be discounted to time. Please tell me if I understand correctly. Thanks.

Hi Hend, 1. In blue cells are the cash flows. Following Tuckman's example, the swap pays $ 1 million notional * (yield - 5%) but pays semi-annual (like all of Tuckman's examples). So one counterparty is paying floating at $ 1 million notional * yield / 2 each six months and receiving fixes $1 million * 5%/2, such that the netted cash flow = $1 MM * (y - 5%)/2 each six months. At T = 0.5, +/- $2,500 is the netted cash flow at either 5.5% or 4.5%; at T = 1, $5,000 is netted CF at 6% or 5% per the binomial tree. 2. The -2,500 and the -5,000, in this way, are just the one counterparty's negative cash flow (the one we are valuing) as the interest rate tree is symmetrical; since the fixed (strike) is 5%, this counterparty nets + $2,500 @ 5.5% but nets -$2,500 at 4.5% 3. That's correct, but note that we don't multiply the current cash flow by the probabilities. The formula at T = 0.5 is =($I$10*J20+$I$12*J23)/(1+G13/2)+G21. In English, that is two terms: on the left is the discounted, weighted average of the two forward nodes (0 and +/- $5,000) PLUS the current +/- $2,500. This is a PV at T = 0.5 that combines the current cash "unadjusted" flow (+/- $2,500) with the PV of future node values, where the future node values are both discounted and weighted by probability. I hope that explains, thanks, David

Hi David, About this slide : I tried pricing the bonds at every node given the evolved rates, but non of my answeres matched the ones in blue here except when the coupon rate equaled the yield, which I thought meant my other calculations were correct....so how were the prices in the slide calculated? For example: I used the following input for the first node : FV = 100 I/Y = 5/2=2.5 PMT = 3 N = 10 And so on. Thanks

Hi Hend, I don't like my example here, frankly. I think you are fine to apply your approach. In the interest rate trees, Tuckman's contrasts both: a market price for the bond; i.e., discounting at the spot rate like you did an expected discounted value for the bond. What i did here (following Tuckman 177 & 199-), for example, $105.17 = ((102.34+ $3) *40% + (106.43+$3) * 60%)/(1+5%/2). See how i employed backward induction? (although including the coupon is dubious as i look at it now ... ) How can I equivocate? This equivocation is sort of the key point of Tuckman Chapter 9: the risk-neutral idea that the market price only equals the expected discounted value only if the probabilities (here, 40% and 60%) are RISK-NEUTRAL probabilities.Now, in truth, I was not thinking this at the time, is just a justification (a way out) to suggest that mine is okay if you assume the probabilities are not risk-neutral ... but it's weak (I was just following Tuckman) ... your approach is more natural and has the added benefit that pricing the bond like you are is how the FRM would do it (clearly) even as the option would continue to use discounted expected value. Long story short: IMO, yours is better. Thanks, David

Hi, I'm sorry, but I don't have Tuckman's book to refer to, or I probably wouldn't have to ask this: Where did he get the last node prices in order to work back towards what I see here? The second part of my question was going to be that I tried evaluating the options given the bond pricing as they appear here, but that didn't work to match your figures. Now I know two reasons: 1- In my understanding, every node here refers to a point in time when a bond/option will be worth either X or Y in dollars in certain probabilities... and every node gets one payment of coupon, not for every possible value of bond, but for the node (period in time). Because that node will in reality have a single bond regardless of its value...is that what you mean by "including the coupon is dubious"? 2- You seem to have confused the probabilities in your calculations, so that you weighted the up move values with the down move probability and vice versa. Still, I redid the calculations considering these discripancies and didn't get your answers. I feel like I spent too much time on this, given your assertion that this is a low testability subject But I think all adds up in the end. Thanks

Hi Hend, I attached the underlying XLS http://db.tt/WDONRWmz 1. Yes. Each node in blue wants a present value. As above, the PV can either be a market price (per your valid approach) or discounted expected value (as shown). The dubious refers to (eg) in the final node (I had to fix the above, sorry i am in a rush due to content creation, as usual): $105.17 = ((102.34+ $3) *40% + (106.43+$3) * 60%)/(1+5%/2) It isn't obvious, IMO, why the $3 are added, instead of just weighting and discounting the future prices. However, now that i've come back here today, I do see the point: the T0 price of the bond discounts the t+0.5 years prices but adds the $3 coupon that will be received because that it not already including in their prices! 2. Sorry, I don't see that? The up is an up in the interest rate, from 5.0% to 5.5%, with a corresponding DOWN in the price, and with 40%. So i think i weighting the UP (to 5.5%) with 40%. but sorry, i am rushed and my previous formula was sloppy, hopefully the above and the XLS reconcile. Although, you are also correct: the particulars have extremely low testibility. The relevant concepts are: discounting to nodes (backward induction; has been tested); the tree concept; maybe the risk neutral idea. ... as you can see, what won't be tested, in a way to assume a correct answer, is the "philosophical" debate of using market prices or discounting expected value. Your ORIGINAL idea to simply price the bond is absolutely good! Thanks, David