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in this video we're going to talk about how you can calculate the forward rate based on yields from zero coupon bonds so when we say the forward rate what we're talking about is the interest rate that we could guarantee today right now for a loaner investment that's going to occur at some point in the future could be two years from now three years from now but we're going to lock in this rate today so in an example we could say for let's say that you the forward rate for year two for the second year would be the rate that you could get today assuming that you wanted a one-year investment that was going to start one year from right now so let's say we're right here we're at period zero and we're wanting to lock in an interest rate in this interest rate is going to be for a one-year investment that's going to go from year one to year two the end of year one so that's one year from now to the end of year two okay so right here we're not going to have any investment but we're locking in the rate right now and we'll have six point one percent that'll be our rate for this period here right so at the end of one year from now that's right here this starts and then it ends right here at the end of year two and we're locking in that rate today even though this investment or loan this doesn't start until the year from now so that's a forward rate and we just can denote it with an F and a subscript two to mean forward rate for year two so we can go ahead and compute the forward rate obviously we just have a given here in the example but if we didn't know it if we didn't know the the forward rate for a given period we could actually use yields from zero coupon bonds to go ahead and calculate foot and here's how this works we would say well look we know the rate for a one year zero coupon bond let's say that's 3.5% and then the rate for a two year zero coupon bond is 4.25 percent then we can actually back out the forward rate so let me come put some numbers to it we'll walk through it basically just make sure you understand conceptually though so when we say that we know the rate for a one year zero coupon and the rate for two years zero coupon what we're saying is if you bought a zero coupon bond for one year let me actually it so here's your zero that'sthat's right now you're one year too so what we're saying is that if we know the yield for a one year zero coupon that's three point five percent but if you also instead bought a zero coupon for two years for the whole two year period you get charged for point two five percent so now we're wanting to know the specific what if you bought what what if you bought something in you wanted the rate just for this just this portion right here right we know the whole thing is four point two five percent we know this part is three point five percent from the yields right from these these numbers here but we want to know this part and that's the forward rate right this investment wouldn't start until right here and it would end here we want to know that rate and we can back it out so let's go about doing that so basically we can think about having two different strategies right so one strategy would be to buy a zero coupon bond right here for 3.5% a one-year zero coupon and then to also simultaneously lock in a forward rate for year two right now alternatively we could just buy a two year zero coupon right and that would be this strategy right here that would be your return if you bought a zero year or excuse me a two year zero coupon bond your return would be just you know one point zero four two five squared because we're just taking this this rate of return for those of those two periods now we know that these two strategies that the buying a one-year coupon and then locking into the forward rate for year two and then as opposed to just buying a zero coupon for two years that those strategies should be equal in terms of what the return is because if they weren't then there'd be some kind of arbitrage opportunity here right so really we're just looking at either way there's risk free right these are should be kind of equivalently we're thinking about it's the same return because either way we're locking in the interest rates today so we can just go ahead and we can calculate this out and just solve and see what we get so we'll get let's see if we have one point zero three five times the forward rate for year two it's going to be equal to that's going to be and I'm going to just round here so if my numbers are off a little bit from yours it's going to be one point zero eight six a I'm giving it up a little more space and so then we're going to have the forward rate for year two we just divide each side by this one point zero three five and that's going to give us one point zero five again I'm just I'm just rounding here and then we can also just think of that as that's going to be a five percent return so what does that mean that means that if we come back here this number this return that our that forward rate for year two is five percent so if we bought a zero coupon bond for one year one year coupon bond and then entered into a forward rate for year two at five percent that would give us roughly the same return as just buying a two year zero coupon bond and if it's off just by a decimal or so it's just due to rounding and so that that's for just looking at something with just two periods and you might say well how can we generalize this to multiple periods what's really nice we've got this formula so the forward rate for any year when I say what I have this little n the subscript and that's just the period that we're talking about so like the forward rate for year two the forward rate for year three we can actually calculate that using this formula and we have white YTM here yield to maturity that's just the yield on a zero coupon bond so yield of zero coupon and when we're talking about we've got this little subscript n so that's the period n for like above right for for year two YTM two would be the rate on the yield on the two years zero coupon bond so I'm going to this a little abstract so let's jump into an example and show how we would go about using this right here this formula so let's say make sure you have to spending a plenty of space to see this one we'll scroll down in a moment here so let's say that we have a set of yields here here are zero coupon yields we know for a one year zero coupon a two year zero coupon three-year and so forth all the way to a five-year zero coupon bond and we have these different yields here's our yield to maturity for each of these zero coupon bonds right so these are our yields and now we can use those yields to calculate the forward rates forward rate for year one is easy that's just the yield of maturity right now on the one year zero coupon so that's just going to be 4.1 percent right that's just mechanical but now we want to say well what about this forward rate for year two and my let me to move change colors here we'll mix it up a bit so we want to know the forward rate for year two now what we're going to do is we're just going to use this formula right so we have one plus the yield for period n what is period n well we're talking about let's see here we've got a forward rate for year two so n is 2 right n is 2 because we're just looking at forward rate for year two here's that n so then we're going to have the 1 over the yield for a period n ok so that's 3.8% so one plus three point eight percent is going to be one point zero three eight and we square that because of this we raise it to the nth power right and again we're talking about period a period two here n equals two so it's just one point zero three eight raised to the second power that's this part right here and then in the denominator we're going to have the yield from the previous year so see this n minus one so we said well n is two because we're looking at the forward rate for period two so we just go and say okay well actually the yield for the previous the n minus one that yield is four point one percent so we look if we say okay look here we go we'll just add that one plus the yield is going to be one point zero four one now we raise it to the N minus one power but n minus one it was n is two so that we're raising it basically to the first power let's put that there that one we don't even need that there though obviously right because you just raise anything to the power it's just itself so now we can just go out in them of course we subtract one just like we do up here so now we can go ahead we could calculate this and our forward rate for your two is going to be one point zero three five a three point five percent I'm just saying that this here just means equivalent so just think of it as one point zero three five is just saying we have a three point five percent forward rate for year two now similarly when we go to do forward rate for year three we're just going to say okay well let's take the yield for year three all right so that for point three percent and we just add that in there we add one to it and then we raise it to the nth power and the N over here is going to be three right so we just take one point zero four three that's just one plus the yield raise it to the nth power which is three and then we divide it by the yield from the previous year the yield from in this case n minus one so it's three minus one is two so the yield from year two which was three point eight percent right so now in the denominator we have one point zero three eight and we raise that to the second power because we're raising it to the N minus one power and then of course we subtract one again and so now we can go ahead and that will give us our yield and we're going to have one point zero five three and then we can just think of that as five point three percent okay now I've actually calculated all the way through the fifth period here so I give you all the forward rates and again here just n is going to be four and it's going to be five I'm not going to belabor the point and go through all these I'm just going to give you the rates real quick in case you want to calculate these out for yourself here we're going to have five point one percent is going to be our forward rate and then six point zero percent now I would like to you you might have noticed something already now and this is this is kind of an interesting fact that we have here is when the yield and we're talking about this your coupon yield here for period and so so let's say right here for Period three let's say n is three that we're talking about when that yield when that yield is greater than the previous year's yield okay so let's take a look for example period three is that greater that's this yield greater than this one yes when that's the case then the forward rate for that period for that period and so forward rate for n is this would be right here 5.3 percent that's going to be greater than the yield of the zero coupon bond for period n so in other words let's think about this since 4.3% is greater than three point eight then the previous years zero coupon yield then that means that the forward rate for that period and for in this case Period three the forward rate for year three is going to be higher that's going to be higher than the yield is so five point three percent so because four point three percent is higher than the three point eight from the previous year then the forward rate for that period three is actually going to be higher than the zero coupon yield and that will always be true now the reverse is also true so if the yield is actually less than the previous year's yield in terms of zero coupon yields then the forward rate for period n is actually going to be less than the yield of the zero coupon bond for period end