Jun 22

Lognormal Distribution. Part 2: Bond Price

by David Harper, CFA, FRM, CIPM

FRM | Risk | Quant | |

teacherPart2

I previously introduced the lognormal distribution. We saw how a random variable comes to be lognormal: its natural log is normal.

Let's briefly look at the simplest application, bond pricing. We might assume the bond's yield is normally distributed. (By yield I mean the 'yield to maturity' or YTM. The YTM is the common meaning; the YTM is the internal rate of return [IRR]. It's the rate implied if the future coupon payments and principal redemption are discounted to the current price). If we assume the yield is normally distributed, then the bond's price is log-normally distributed.

Is any formula more elegant than the formula that gives the price of a zero-coupon bond? Given the yield (r), the term (number of years = T) and the par/face value (F), the price of the bond (V) is given by V=(F)exp(-rT). If we rearrange to solve for the rate, we see that the rate is a function of the natural log:

zerobond

 

If the yield (r) is normally distributed, the price (V) has a lognormal distribution.

You can better understand by reviewing the EditGrid spreadsheet below. You can open a read/write copy here.

Two distributions are plotted in the charts. The first is the standard lognormal. Notice that I hand-calculated the probability density function for a lognormal distribution, displayed in column H (I don't think it is directly available as an Excel function); the cumulative distribution function for the lognormal, however, has a built-in function =LOGNORMDIST().

The second distribution is the plot of the bond price, for a zero-coupon bond based on the assumptions; e.g., term = 20 years.

EditGrid Spreadsheet by bt/admin.

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