32h - calculating z-spread

r1 given as 3%
r2 given as 4%
r3 given as 5%

a 5%, 3 year annual bond is being sold for (a given price of) $99. I need to solve for z, so:

99 = 5/((1.03+z)^1) + 5/((1.04+z)^2) + 5/((1.05+z)^3)

Not sure if the obstacle here is my algebra or my familiarity with the financial calculators. How would I quickly solve for z?

In general you can solve it using Cardano’s formula, but it won’t be quick.

In practice, you’ll approximate it using numerical techniques, such as by using Excel’s Solver.

Note, by the way, that your equation is wrong; the last payment is 105, not 5.

Oh, thanks man. That was sloppy of me.

Yes, I would have used “Goal Seek” in Excel to find the answer in practice, but (to the best of my knowledge), using an Excel clone isn’t an option for the L2, even the digital version, right?

So for the purposes of the L2 exam, where they only allow you 3 mins/question, what should I do? Just assume that the CFAI wouldn’t make us do a Z calculation on the real exam?

They’ll give you three possible answers, say, 40bp, 80bp, and 120 bp.

The proper approach is to try the middle number (answer choice b). Add that to each spot rate, discount the cash flows, and see if you get the correct price. If so, 80bp is the answer. If your calculated price is too low, then 80bp is too high: 40 bp is the correct answer. If your calculated price is too high, then 80bp is too low: 120bp is the correct answer.

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There is no simple way to do this on a BAII, but I did have a thought on this:

If I assume that the z-spread effect for years 1 and 2 will be very small and the year 3 effect will predominate, then I deduct the unloaded PV for years 1 and 2 from the price of 99 and solve for the interest rate that will get me to 105 at the end of year 3.

99- 5/1.03 - 5/(1.04^2) = 89.52285

N 3 PV 89.52285 FV =-105 CPT I 5.4593604 which implies a spread of approximately 0.4593604%

Now the actual z-spread is 0.4362%, so I think this is a reasonable method to get you close!