# change of swap value over time

one key point schweser highlighted for los 35sa is “the values of swaps change over time even if market rates and prices do not change.” to understand this, i did following experiment, but i don’t seem to be able to reach that conclusion. ===================================== a two-year swap at a rate = S one year interest rate = r1, one year forword rate = r2; forward price in one year = F1; in two year = F2 the basics of swap tells me that: at time 0 (F1 - S) / (1 + r1) + (F2 - S) / (1+r1)*(1 + r2) = 0 (1) so, valuation of swap, S, can be derived from equation (1). suppose nothing changed, except i am 6 months into the contract. now, i multiply square root of (1+r1) to both side of equation (1), i have (F1 - S) / (1 + r1)^(0.5) + (F2 - S) / (1+r1)^(0.5)*(1 + r2) = 0 (2) meaning, my valuation of this swap after 6 months is still S. (or the value of this swap is zero at time=0; it is still zero at time = 6 months) ======================================== what did i do worng? or my interpretation of schweser’s note is wrong?

Suppose that I have a really simple two period swap. The 6 month forward LIBOR rate is 1% and the 1 yr forward LIBOR rate is 25%. This swap rate is 10% (I dunno) and the swap is fairly priced at 10% today. 6 months from now if the interest rates evolve according to current expectations, the swap to trade 10% for 25% is pretty valuable.

JoeyDVivre Wrote: ------------------------------------------------------- > Suppose that I have a really simple two period > swap. The 6 month forward LIBOR rate is 1% and > the 1 yr forward LIBOR rate is 25%. This swap > rate is 10% (I dunno) and the swap is fairly > priced at 10% today. 6 months from now if the > interest rates evolve according to current > expectations, the swap to trade 10% for 25% is > pretty valuable. i see your point. it’s absolutely valid. would it still be valid if it’s 5.9 months instead of 6 months from now, i.e. before the point of reset? or the change is really a step function of time instead of continuous function of time?

Sure, because in 5.9 months I am discounting the 10 -> 1 and 10 -> 25 differently than I did on the swap origination day.

JoeyDVivre Wrote: ------------------------------------------------------- > Sure, because in 5.9 months I am discounting the > 10 -> 1 and 10 -> 25 differently than I did on the > swap origination day. if (suppose it’s true, as you said) (1 - 10) / (1.1) ^ (0.5) + (25 - 10) / (1.1) ^ (0.5) * (1.25) ^ (0.5) = 0 => (1.1)^(0.45) * {(1 - 10) / (1.1) ^ (0.5) + (25 - 10) / (1.1) ^ (0.5) * (1.25) ^ (0.5)} = 0 => (1 - 10) / (1.1) ^ (0.05) + (25 - 10) / (1.1) ^ (0.05) * (1.25) ^ (0.5) = 0 the equality still holds

rand0m Wrote: ------------------------------------------------------- > JoeyDVivre Wrote: > -------------------------------------------------- > ----- > > Sure, because in 5.9 months I am discounting > the > > 10 -> 1 and 10 -> 25 differently than I did on > the > > swap origination day. > > if (suppose it’s true, as you said) > (1 - 10) / (1.1) ^ (0.5) + (25 - 10) / (1.1) ^ > (0.5) * (1.25) ^ (0.5) = 0 > Better check that. I need the PV of the floating leg = PV of the fixed leg. You’ve got two things that are > 0 adding up to 0. > => > > (1.1)^(0.45) * {(1 - 10) / (1.1) ^ (0.5) + (25 - > 10) / (1.1) ^ (0.5) * (1.25) ^ (0.5)} = 0 > > => > > (1 - 10) / (1.1) ^ (0.05) + (25 - 10) / (1.1) ^ > (0.05) * (1.25) ^ (0.5) = 0 > > the equality still holds

> > if (suppose it’s true, as you said) > > (1 - 10) / (1.1) ^ (0.5) + (25 - 10) / (1.1) ^ > > (0.5) * (1.25) ^ (0.5) = 0 > > > > Better check that. I need the PV of the floating > leg = PV of the fixed leg. You’ve got two things > that are > 0 adding up to 0. the equation indeed says pv(float) - pv(fix) = 0, exactly what you asked for. the 1st part is negative cause i pay 10 receive 1; the 2nd part is positive cause i pay 10 receive 25.

Just remember, that at the initiation the SWAP value = 0, but any time after initiation the SWAP value will not = 0, well i guess it could possibly, but I’ll say 99% of the time it will not.

bigwilly Wrote: ------------------------------------------------------- > Just remember, that at the initiation the SWAP > value = 0, but any time after initiation the SWAP > value will not = 0, well i guess it could > possibly, but I’ll say 99% of the time it will > not. it did remember that by heart. but, somehow, my math screws me up.

Sometimes I try to take things for face value and not get caught up in all the math and how the formula is derived as it will tend to make thing smore difficult and or confusing for this test.

rand0m, I really admire your analyical ability. Any quant firm will be lucky to have you. I am serious!! Me, on the other hand, I am like bigwilly, face-value baby!!!

I just try to take things for face-value or else I’ll be here all day analyzing every single formula, derivative or concept. If I’m intersted enough in it, I’ll go back to it after June 7th, not before bigwilly, I thought you are on you way out (workplace) 10 mintues ago.

eh, traffic ws Wrote: ------------------------------------------------------- > rand0m, I really admire your analyical ability. > Any quant firm will be lucky to have you. I am > serious!! > > Me, on the other hand, I am like bigwilly, > face-value baby!!! given what happened last august, quant does not taste as good as before. i heard money are going out of quant as if there is bad flu …

bigwilly Wrote: ------------------------------------------------------- > eh, traffic r u trading japan … i am thinking of going home now …

rand0m Wrote: ------------------------------------------------------- > i heard money are going > out of quant as if there is bad flu … Don’t worry, just like a bad flu, people will catch on it soon or later down the road again. SH*T flows around on the street.

Japan, no. But I do, feel like I’m turning Japanese, I’m turning Japanse, I really think so.