Math geeks and CFAs wanna help me out?

Question

How?

pelihu · Answer

Well, I'm not a CFA and definitely not a math geek, but I'll give it a shot.

I'm not sure what the assumptions are for your questions. Generally, American style bonds pay out coupons twice per year, but the question says annual coupons so does that mean one payment per year? I'm going with once per year because its simpler.

Anyhow, for Q7, the equivalent rates for the zero coupon bonds are 4% for the 1 year, 6% for the 2 year and 7% for the 3 year. I'm a little unclear on the next step, but I think you should take either an average or a weighted average of the prices of the zero coupon bonds to get the market value of the coupon bond. I'm going to go with the average of $889.28 because I don't really care that much, but I think you might actually want to take a weighted average of the NPVs to get a present value equal to the 5% coupon; and again, payment frequency could be a factor here.

So, if the price (market value) of the 5% coupon bond is $889.28 (because you'd get the same from the zero coupon bonds), the YTM would be 9.41%. The key to this question is finding the current market price of the 5% coupon bond based on the current prices of the zero coupon bonds. Once you have the price, just plug in the values to find YTM.

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For Q8, the YTM of the 7% coupon bond is 7.5% (it is selling at a discount so the YTM must be greater than the coupon). The YTM on the zero coupon bonds are 5.26% for the 1 year, 5.41% for the 2 year and 6.84% on the three year. Again, you need to match the duration of the 7% coupon bond (3 years) with the zero coupon bonds. You'll probably want to take a weighted average, but I'm just going to take the average which is 5.84%. So if you match the durations and the bonds have the same risk (we can only assume they do because no other info is given), then differences in YTM are an arbitrage opportunity.

So the 7% coupon bond has a YTM of 7.5% and the zero coupon bonds have an average YTM of 5.84%. You should sell zero coupon bonds and take the proceeds and buy the 7% coupon bonds until you get the NPV of $2000 you are looking for.

Note: I just wanted to point out again that I just took the simple average in both problems, but you probably need to take a weighted average to match durations and values.

IHateTheGMAT · Answer

Ryme,

On the first one I believe you need to calculate the no arbitrage price of the 3 year bond using spot rates (unless that's not something your class covers - if its not, let me know). I'm working on the solution now...

edit: pelihu is on the right track for no. 7. The yield for each zero coupon should be the spot rate for that year

rhyme · Answer

We haven't really covered spot rates yet, but that certainly makes sense - if you have a solution, please share. This is helpful boys, thanks!

IHateTheGMAT · Answer

The yield for number 7 is 6.91%

I solved for the yields on the zeros to get the spot rate for each time period. So the 1 year spot was 4%, 2year was 6%, 3yr was 7%. You then discount the cash flows of the coupon bond at the approprate spot rate. For example, the first coupon is 50 in one year, discount that at 4%.The last payment was $1,050 and you discount it at 7% for 3 years. Using this process you get a no arbitrage value of around 950 (I got 949.72). Then use this as your PV, 1000 as FV, 50 as PMT and solve for yield on your calculator. I got 6.91%. Not 100% positive but I think this is right.

I'm not a CFA charterholder but a did sleep at a Holiday Inn Express last night

rhyme · Answer

haha... nice.

I just did the problem again (before reading your answer) and got the same thing, although your explanation is a lot clearer than what I would have come up with. Thanks!

The way I got there was similar --
If 50 is the coupon in year 1, then 50/1000 * 961.54 should be the arbitrage price for that stream... year 2, 50/1000 * 890 ... same thing for year 3... add those up you get 949.69..

IHateTheGMAT · Answer

You're welcome!

You can solve for the no arbitrage price in number 8 the exact same way. I got $1,006.87, so it is mispriced. I think you make an arb profit by buying the zero coupons and packaging them into a bunch of the coupon bond and selling them at $1,006 a piece (a package of zeros with equivalent cash flows will cost less than the coupon bond). If you buy the right amount of the zero coupon bonds you should be able to create the right number of the coupon bonds to get a $200 profit.

IHateTheGMAT · Answer

You got it! That works too

rhyme · Answer

Now to tackle #8

This is so my own fault for sleeping through the last 3 lectures. haha.

rhyme · Answer

meh. GTAIV is more fun.

3underscore · Answer

#8

Right, you have the discount factors, by way of having the prices of the zero's (they are the discount factors, multiplied by a notional, 1000)

1yr - 0.95
2yr - 0.9
3yr - 0.82

So, to get the price, you multiply your cashflows by the discount factors

70 x 0.95 = 66.5
70 x 0.9 = 63
1070 x 0.82 (it matures, so you get the full amount back) = 877.4

So the fair value is (sum DCF) 1006.9. As per the question it is underpriced - we want to buy the coupon bond and sell various bits of the zero's to arbitrage. So, we want to get neutral to get $2000. This is a bit trickier. But we  know  we are right at this point. How? because the difference between the fair value and the market price is $20 per bond. Which is pretty damn convenient, given the arbitrage they want us to do.

We need to buy 100 times the variable bond - the tricky part is funding the thing for nothing. We need to get the split of value from each of the coupon cash flows and the final payment. We want to sell short 7 of the 1yr and 2yr bonds, and 107 of the 3yr.

This will get us value today of $6650, $6300, and $87740 (total $100,690).

To meet the payments required for shorting of those bonds, we need to buy 100 of the coupon bond at market, $986.90, an outlay of $98,690.

The cashflows we generate are a wash - each cashflow from the bond we receive, we pay to the person we did the applicable short trade with. Your Broker can take care of that. We have $2000 and are going to spend it now.

How often would you do this? As often as you can (Stern reference for you all there). I loved doing debt instruments this semester. I completely aced that class - it made sense somehow (even binomial tree swaption pricing).

rhyme · Answer

Exactly what I got for #8. Well done.

3underscore · Answer

the fact that my semester is finished and I got remotely excited by a bond math question is not cause for celebration!

rhyme · Answer

Yea, you are one sick puppy.

Math geeks and CFAs wanna help me out?

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