## Thursday, April 5, 2012

### "Expected" is Not the Same as "Most Likely" -- the Case of Bond Valuation

A common valuation approach is the discounted cash flow method.  Future cash flows are discounted to present value using a risk adjusted rate. The mechanics behind this valuation approach are important in establishing an appropriate valuation mindset.  In the case of standard bond valuation, I believe the widely accepted discounted cash flow valuation mechanics are incorrect, and fail to incorporate an important valuation consideration.

The standard bond valuation equation is:

V = SUM{C/ [(1+d)^(t)]} + Px / [(1+d)^(x)], where

V = bond value
C = bond coupon
t = time period, beginning at t=1 and ending at t=x
d = risk adjusted discount rate
P = bond principal
x = remaining life of bond, expressed in number of periods

Here and here are two websites that provide bond valuation equations like the one shown above.

The problem with such a valuation lies in the cash flows used.  The appropriate cash flows to use are expected cash flows, a standard valuation concept.  Coupon and principal payments represent the maximum cash flow possible, not the expected cash flow.   Therefore, the cash flows used to value bonds should be lower than the coupon and principal amount if there is any risk of default.  Even if the bond will probably pay off in full, expected cash flows should incorporate probabilities of possible scenarios.  Thus, I draw an important distinction between "expected" and "most likely."  Coupon and principal payments can only represent expected cash flow if the investor (naively) believes default probability is zero.

For highly rated, low risk bonds, I concede the difference between the standard approach and the probability adjusted approach is likely negligible.  Nevertheless, the importance of illuminating the theoretical shortfall in the standard valuation approach remains.  Thinking in probabilistic terms is challenging. Substituting a shortcut approach for a theoretically sound approach encourages intellectual laziness.  I think students and practitioners would benefit from reinforcing the need to assess alternative scenarios and the probability of occurrence for each.

An approach incorporating scenario probabilities is useful for more than determining the expected cash flows of an investment.  It also is highly relevant in evaluating the appropriate risk adjusted discount rate.*  After all, in the standard bond equation noted above, the expected cash flows are the promised, or maximum, cash flows.  Implicitly, using them as expected cash flows implies no chance of default.  Conversely, the discount rate used in the equation is a risk adjusted discount rate, with credit risk a component of the risk premium.  This creates a schizophrenic proposition--that bond's cash flows have no risk, yet are discounted using a risk adjusted rate.

Even if it does not always result in a meaningful valuation difference, better to use probability adjusted cash flows with a risk adjusted discount rate.  When done properly, the cash flow scenarios will be consistent with the expected cash flows (numerator) and the risk adjusted discount rate (denominator).used in the valuation equation.*

* The risk adjusted discount rate depends on more than simply the variability of the cash flows.  The correlation of the cash flows  with those of other assets is another factor in determining the appropriate risk adjusted discount rate.