Valuing an Option-Free Bond with Pathwise Valuation
A three-year, 8% annual coupon bond was valued with this interest rate tree, with backward induction to produce a value of 106.8256.
But maybe you'd rather move forward than backward.
The process of __pathwise valuation__ is to count all possible interest rate paths, calculate the bond's present value for each, and average these present values. Sounds simple enough, and it's a little simpler to explain and understand than backward induction, perhaps.
To start with, a two-period interest rate tree has just two paths: up and down. And each path can give a different value, so they all need to be counted. How many individual paths do you think would be in a four-period tree, with t=3?
No. More than that. It's a geometric pattern.
No. Consider that each period is like adding a second branch onto every existing branch.
That's right!
It's a geometric series, and it grows fast. Since the number of nodes grows linearly, the pattern follows Pascal's triangle, if you're familiar with that. If not, then just know that there are many ways to get to nodes in the middle of the branches. For example, take five periods. In Time 4, there are just five nodes. There's a single path to the very top and the very bottom, but there are four ways to get to the second nodes to the end, and six ways of getting to the middle node. That's 16 paths in total, following the same geometric series of doubling the number with each period.
But back to the example now. It's just three periods, so four paths in total. Not so bad. Looking back at the interest rate tree, the paths can be listed in table form.
| Time | 0 | 1 | 2 | (What It Is) |
|---|---|---|---|---|
| Path 1 | 5% | 6% | 7% | (Up-Up) |
| Path 2 | 5% | 6% | 6% | (Up-Down) |
| Path 3 | 5% | 5% | 6% | (Down-Up) |
| Path 4 | 5% | 5% | 5% | (Down-Down) |
Now there aren't spot rates. These are "discount this much for this year" rates. In other words, just like in backward induction, the cash flow of 108 in this example won't be discounted by a single rate in Path 1; it will be discounted by all three. With that in mind, what is your Path 1 present value?
No. That's the value if you pretended that these were spot rates. It should be 105.49.
You got it!
Here is that present value.
$$\displaystyle PV = \frac{8}{1.05} + \frac{8}{(1.05)(1.06)} + \frac{108}{(1.05)(1.06)(1.07)} = 105.4938 $$
It looks a little different, but again, think about the fact that backward induction was using this same calculation in a sense. It was "discount a period, add the coupon, discount again (and average)." This is the same math, but you'll average at the end. Here are the rest of the PVs.
$$\displaystyle PV = \frac{8}{1.05} + \frac{8}{(1.05)(1.06)} + \frac{108}{(1.05)(1.06)(1.06)} = 106.3493 $$
$$\displaystyle PV = \frac{8}{1.05} + \frac{8}{(1.05)(1.05)} + \frac{108}{(1.05)(1.05)(1.06)} = 107.2896 $$
$$\displaystyle PV = \frac{8}{1.05} + \frac{8}{(1.05)(1.05)} + \frac{108}{(1.05)(1.05)(1.05)} = 108.1697 $$
The last step again is just to average these, and you have the same price as with backward induction.
$$\displaystyle \frac{105.4938 + 106.3493 + 107.2896 + 108.1697}{4} = 106.8256 $$
This works with zero-coupon bonds or coupon bonds (obviously, since you just saw one). If it seems a little straightforward, and especially if you have a spreadsheet handy, it might be the way to go.
To summarize:
[[summary]]
5
6
8
102.90
105.49
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