THE VALUATION OF CONVERTIBLE BONDS
Otto. H. Poensgen ^^'
MAR 17 1966
1. T. UBRA.ulS
The Valuation of Convertible Bonds
Convertible Bonds are bonds that are convertible into another security
at the option of the holder subject to conditions specified in the indenture,
For our paper we will restrict the terra 'convertible' to mean exchangeable
for 'the common stock of the issuing corporation.' The restriction is not
a stringent one: the author in examining publicly traded bonds issued be-
tween 1948 and 1963 by companies that are traded on an organized stock ex-
change (or over the counter) found no bonds which were excluded by that
definition. The vast majority of nation-wide traded convertible bonds is
not only unsecured, but even subordinated to prior or even after-acquired
debt. Deducing from cum hoc to ergo propter hoc this has led many writers
to state or hypothesize that one of the reasons, if not the principal one,
to attach to the bond the convertibility feature was the necessity to have
a sweetener make an otherwise unpalatable instrument acceptable to the tn-
The conversion price indicates how many dollars of face value must be
given up at conversion for each common share. Occasionally, we find a con-
version ratio instead, stating into how many shares one debenture of $1,000
The converse statement also holds: subordinated bonds arc usually
convertible, see K. L. Browman, "The Use of Convertible Subordinated
Debentures by Industrial Firms" 1949-59, Quart . Rev, of Econ . Business
(Spring 1963) Vol. 3 no. 1, p. 65-75. The same point is made by R. Johnson
in "Subordinated Debentures: The Debt That Serves as Equity," Journal of
Finance (March 1955) Vol. 10, no. 1, p. 16.
See Pilcher, oÂ£. cit. p. 57 ff. for sources.
(or $500 etc.) is convertible. In the case of a $1,000 bond, a conversion
ratio of twenty thus would be equivalent to a $50 conversion price. We
will use almost exclusively the term 'conversion price,' since it is both
customary and independent of the denomination of the bond. The conversion
price is usually well above the stock price at the time of issue. ^
The indenture will also specify between which dates the conversion
price applies. There are many bonds for which one conversion price applies
from the date of issue to maturity; occasionally conversion is not permitted
before a certain time, say two or three years, have elapsed since flotation.
More often there is a clause stating that the conversion privilege expires
after ten, twelve, fifteen or so years, well in advance of maturity. An
additional complication, that is even more frequent (and applied to almost
40 per cent of a sample of 165 postwar bonds) , is a conversion price chat
changes at specified dates, usually in five-year intervals, invariably
increasing. Finally, in a few cases, the indenture requires the owner of
the bond to surrender not only his bond at conversion but also a specified
amount of cash per share received.
The conversion ratio is not to be confused with the word 'ratio'
in another frequently found statement such as "The corporation is offering
to the holders of its outstanding Common Stock of record .. .rights to
subscribe for the above debenture in the ratio of $100 in principal amount
of debentures for each 28 shares of Common Stock then held of record.
"Such a statement has nothing to do with conversion but indicates that
the issue is a preemptive one, i.e., is offered to stockholders before it
is offered to the public at large.
with A. T. and T.'s eight postwar convertible issues furnishing
a notable exception.
The conversion privilege is somewhat like a nondetachable warrant
with the conversion price taking the place of the exercise price. If
conversion price and exercise price are the same, a convertible bond-
holder could after conversion achieve the position the owner of a bond-
plus -non-detachable warrant is in after exercising his right by buying
a bond of similar quality, maturity, and coupon rate costing the amount
the bond-plus -warrant holder spends when exercising his right. Cash,
bond, stock positions then would be identical. Vice versa, the bond-
plus-warrant holder could reach the position of the convertible bond
owner after conversion by simply selling the bond after exercising the
A difference, however, is that a conversion price of $ y does not
mean that something worth $ y is given at conversion. If, e.g., the
conversion price is $25 while the straight debt value of a $100 face
value bond is $90 then at conversion a consideration of $25 times â€”
is paid for the share received. This is perfectly normal, the coupon
rate for convertible bonds usually is below the going rate for bonds
of that quality, the full price of $100 is paid because of the conversion
privilege. The difference between straight debt value and par value
For sample of 165 bonds we found as average figures for a $100 face
value bond shortly after its flotation a straight debt value of $93, a
market price of $106, making for an average value placed on the conversion
privilege of $13. Average subscription price was closer to $101, but since
about half of them were preemptive, this figure understates the value of
the bond for the investor.
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(discount) is gradually amortized as maturity approaches. A change in
the quality of the bond will also change the straight debt value. The
same factors that push share price above conversion price may also in-
crease the straight debt value.
Evaluation of the Convertible Bond in Toto
We shall start by looking at the convertible bond in toto - straight
debt part and conversion option. We select arbitrarily a point t^ in time
when we will either sell the bond at its straight debt value or convert it
into common stock and sell the shares. If, at time t, the stock price x
exceeds the straight debt value y, then we convert and receive stock worth
x; otherwise, we keep the straight debt value y. For a given y the expected
value of the stock is the sum of all values of x for which x > y multiplied
by their probability of occurrence h(x | y) and summed:
/ X h (x I y) dx
To this expression we have to add the straight debt value y times the
probability that the stock price x falls short of y, in which case we
do not convert
y / h (x I y) dx.
Bringing the terms together, we state that the expected value of the
convertible bond given a straight debt value y Is
y / h (x I y) dx + / X h (x I y) dx
Since bond yields vary just as prices do, we must do our analysis for
all possible values of y, multiply by the probability of the occurrence
g(y) and integrate from o to Â». The expected value of the bond thus
w y OB
E(P) "â– / [y / h (x I y) dx + / X h (x I y) dx] g(y) dy.
o o y
Writing h(x,y) for h(x j y) g(y) we get
ea y oo
E(p) = / [y / h(x,y) dx + / X h(x,y) dx] dy
o o y
00 00 y
E(p) â– = / [ / xh (x.y) dx + / (y-x)h(x,y) dx] dy
o o o
V ^ 1 s ^ /
expected stock value of floor
In words, owning a convertible bond is like owning a stock with the
guarantee that a loss due to a fall below a floor y will be made up.
Rearranging the terms in E(p) in the original form differently and using
^ h(x I y) d X = 1
we may write
E(p) = / yg(y) dy + / / (x-y)h(x,y) dxdy
o o y
^ ^ ^ ^ '
expected straight expected value of
debt value the conversion option
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This Is simply the other side of the same coin. The second term is the
gain from conversion - value of stock received minus value of straight debt
given up - weighted by the probability density and summed over the range
where conversion occurs, x > y, and the range of possible straight debt
values , o < y < OB.
We have succeeded in splitting the convertible bond into a straight
bond and conversion option. The present value of the straight bond part,
inclusive of all interest payments on the convertible bond, Is equal to
the price of a bond of comparable quality without the convertibility
feature. This value has been computed for us by Moody's Investors' Service.
This organization regularly rates large, widely held U. S. convertible
bonds as to their investment quality. They specifically assign equivalent
straight debt yields and straight debt values to the bonds. These last two
measures have, of course, a one-to-one correspondence once the coupon race
and maturity are given.
Baumol, Malkiel, and Quandt in a paper presented at an earlier TIVS
meeting went most of the way in deriving these formulae without realizing
that they are in all cases fully equivalent. D. E. Farrar in an unpublished
paper did just that up to a small mathematical mistake.
You will realize that the last part of the second equation gives the
value of the somewhat more simple warrant. The exercise price, y, is not
As well as R. H. M. Associates, New York,
a stochastic variable and we can write
Expected value of the warrant EPW = / (x-y) h(x) dx
This is known since Bachelier's work at the turn of the century. Part
of the notation here is taken from a thesis by Case Sprenkle on warrants.
But back to the convertible bond. Inspection of our second form of the
equation for the convertible bond shows that while the straight debt value
is not affected by the stock price movements x, the option will fluctuate
with the straight debt value y, which in turn is a function of fluctuating
bond yields. We see the effect of this clearer if we substitute for h(x,y)
a specific joint density function, e.g., the lognormal one. I do not want
to use my time defending this choice, much less do I want to become involved
in the Mandelbrot-Cootner arguments to whether the Pareto-Levy distribution
is more appropriate but rather refer you to the articles collected by Paul
H. Cootner in a book entitled "The Random Character of Stock Market Prices."
If we, then, make the above-mentioned substitution we get:
EPW = 11 l"'^-^^^ 2 Inx-n Iny-u Iny-u 2
CO CO - - -^â€” [ ( ^)-2R + ( â– ) J
^ ^ 2^^7f- e 1-R ^ ^ y dlnxdlny
o y X y
is the correlation coefficient between x and y.
This integral is exceedingly cumbersome to evaluate and, before we
take the trouble, we want to make sure that the effort is well spent.
In other words, we would like to know that, by letting the value of y
vary, we are significantly improving our estimation of EPW. To test this,
we shall observe empirically the effect of changes in bond yields, y, on
the value of the conversion option, EPW. We first test, assuming perfect
correlation between y and x, the value of stock for which the bond is
convertible, then for zero correlation which is closer to our estimate
of the actual correlation between the two. If changes in the value of
y do not have a significant effect on the estimate of EPW, then we could
safely fix y at its expected value, n , with o =0, and return to our simple
model of options,
EPW = / (x-y)f(x)dx
Let me simply state the results:
If we have perfect positive (negative) correlation, neglect of bond
price variability results in an overestimate (underestimate) of the con-
version option by 367. ' and we would not be justified in neglecting it.
Now these calculations assumed perfect correlation between stock price and
bond yield. In reality the correlation is much closer to zero than to one
as we shall see in a moment.
ihe coefficient of correlation between the inverse of Standard and
Poor's stock price index and Standard and Poor's index of yields of AAA
bonds in the year 19A6-61 was R=-.840. Trend removal reduces this figure
This is for yearly standard deviation of stock price of 147,, i.e.,
low stock price volatility. The standard deviation of bond prices is
ca . 3.47. per year - which is about average.
It may be surprising at first blush that a positive correlation
of bond prices and stock prices should decrease the value of the option.
I have given the strict proof elsewhere, let me here just state some
considerations to make this plausible to you:
If bonds and stocks move together, there is little to be gained by
switching (converting) from one to the other. For negative correlation
the reasoning is the converse of this .
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We mention, as mildly interesting aside, that even for zero corre-
lation between stock prices and bond yields, bond yield variability has
some effect. A somewhat stronger statement can be proved: if the ex-
pected bond price is equal to the expected stock price and both are normally
distributed with finite variance, the value of the conversion option in-
creases with the bond yield variance. However, numerical computations
have shown that for values of stock price volatility and bond yield
volatility as actually prevail, such influence is negligible.
Next, we checked where between zero and perfect correlation the
actual one is to be found.
As a first step, we ran a correlation and regression analysis of
the yields of bonds rated AAA, AA, A, BBB by Standard and Poor and time
against the inverse of the price index for 425 industrial stocks . Ob-
servations are taken monthly from January 1946 to December 1963. As
we all know, bond yields and stock prices all rose sharply during that
period. But since we cannot expect bond yields to rise simply as they
have risen in the past, we eliminated the time trend from the fluctuations
of bond yields and stock prices. The principal results are:
First, that common stock yield differentials have an extremely low
positive correlation with all other variables - including preferred stock
yield differentials (the correlation with AA, A, BBB bonds is actually
Second, that all zero order regressions have insignificant coefficients;
And third, although the various bond yield differentials are signifi-
cantly correlated, the correlation is smaller than expected. The coefficient
of determination, I.e., the percentage of variance of one bond which is ex-
plained by that of another, typically is only 60% or so;
Finally, standard deviations fall m onotonlcallv from AAA to BBB bonds. ^
The fall is even more pronounced if we normalize by dividing standard devi-
ation by average yield In each case. If AAA bonds on the average have a
longer life than BBB bonds, AAA bonds fluctuate more in price than BBB bonds
(or AA or A bonds), even more than indicated by the difference in the fluctu-
ation of yields .
Having found that for zero correlation between bond yield variability
and stock prices, bond yield variability has no significant influence on
If data from 1946 to 1963 are considered, A bonds are an exception,
Professor Cootner has suggested a possible explanation of this
phenomenon. The variability of bond prices (or yields) is due to two
causes, first, variation of the pure rate of interest, second, change
in the risk of default (or cessation of interest payments) due to change
in prospects or the company or change of the investor's view thereof.
Yield variation among the very high grade corporate bonds would be due
mainly to the first cause, while to explain yield variation of low grade
bonds, the second cause would have to be added. If bond yields fall (or
bond prices rise) in a recession and rise during a boom, while the risk
of default increases during a recession and falls during a boom, Chen
for a low grade bond whenever one factor pushes up bond prices, the other
would push them down. For high grade bonds the countervailing influence
is much weaker. This then leads to larger bond price variance for high
grade than for low grade bonds (especially if recessions are slight).
This hypothesis would seem to be empirically verifiable. If true, it
means that investors in high grade bonds satisfy the yield differential
between high grade and low grade bonds exclusively for greater safety,
not for greater bond price stability.
We obtain similar results by holding time constant in the un-detrended
un-dlf ferenced regression analysis.
the expected value of the conversion option, having found furthermore that
no significant correlation exists, vc happily simplify the complete ex-
pression for the conversion option to
EPW - / (x-y) / (x) dx
In the case of lognormally distributed prices we write
EPW - / â€” i; (x-y) e "^ ^~ >
y v27r o *
The Investor's Horizon
Thus' far we have dealt with the expected value of the option at a
given tirafe t from now, EPW(t) . If the investor requires a race of return
i on his investment, the present worth, PW, or price he is willing to pay
at most for the option is
PW = e'^'^EPW
If the investor at time is told he can make his choice of converting
at any time T between now and expiration of the conversion privilege and
if he is also told (unrealistically) that he must say, now, at what T he
will decide to convert, then the value of the conversion option, PW , is
PW - max e'^''" EPW
,dPW, Â»n . d PW
-iT -il dEPW
dPW . - ie ^ EPW + e ^-^:r
Hje shall discuss what determines i later on. We might mention
factors such as the differential interest rate-dividends, risk associated
with the stock and the like.
Assuming i is not predicted to change in a systematic way.
dEPV;/EPW ^ ^
This is the time-worn prescription that the value of an asset is equal
to the discounted value of the asset at the moment when its growth rate
per dollar invested,
is equal to the required rate of return.
If the option were held longer than to T, the return would fall below
the required one. Therefore, nothing that happens beyond that point
matters. The example usually adduced is that of the growing tree.
Expected Value and Investor Horizon
We now have two equations
PW = e'^^EPW (T)
dEPW/EPW ^ .
for our two unknowns PW and T,
Now we know, that when the investor buys his convertible bond he
is not told to decide when he will make his choice whether to convert
or not. He is free to do so any time up to the expiration date of the
conversion feature and he has available for his decision the knowledge
of what happened up to the point of conversion.
We claim, however, that to neglect this will not change the results
Neglecting Risk: A Single Rate of Return
To bring out the importance of risk and its various measures in
explaining conversion option prices, we neglect risk as a first step,
in order to have a standard of reference. Thus we make the assumption
that the investor is interested only in the expected gain accounting
properly for the horizon. In this case, every bond should yield the
same rate of return.
PKj^ = e "^"^k EPW [rj^.aj^.\l + \ i = l,2....,n
= PW, + u, , where
k : running variable of the observations (bonds)
PK : observed price of the conversion option of bond k in our cross-
section shortly after their flotation
i : required rate of return (to be estimated)
T : length of horizon of bond k
EPW : expected value of bond R at horizon T given the expected growth
rate R and volatility a, of the underlying stock
u : error term
We estimate 1 by the maximun likelihood method as explained in some detail
below. If this naive model is a good explanation of investor behavior then
the variance of u is small relative to the variance of PK.
If u is normally distributed then the probability of getting the
observed u, is
k 1 n
- ^ L 2
p(u ,u , ...u ) =
o 2 k=l u,
^ 2 " iaj.lir)'''^ du^du^.-.du^
As pointed out by Sprenkle we cannot really expect u to be distributed
normally since PK > , or u > PW is not admissible. But if our model is
good, we expect u to be small, in which case the truncation at the left
end does not matter.
It is a different question whether bias is introduced into the estimate
of i by other imperfections. Heteroscedasticity may be expected but does
not impart a bias to i though impairing the efficiency of the estimation
This is, in essence, the approach taken by Sprenkle, although as
mentioned above his observations are not different warrants, but the
same warrant taken at different points in time.
^Ibid, p. 228.
procedure. Serial correlation would seem to be a small danger in a cross-
sectional sample, if the issues are not ordered with respect to an omitted
variable. The likelihood function is n
, ^ - ^^
We can demonstrate that there must be a minimum, but there is no guarantee
that i > (see Figure 2) .
To find the maximum likelihood i we differentiate log L with respect
to the desired parameter i and set it equal to zero in order to find the
mexim of the likelihood L. We can demonstrate that the maximum exists.
The process employed was one of iteration by Newton's method.
The results such a naive approach gives can be described as dismal.
The variance of the actual conversion premia (around their mean) is 69.96
($ ) , the variance of PW, the estimated conversion premia has a minimum
of 73.00 (using r = r ), i.e., the mean of all conversion premia is a
better estimate of the conversion premium than the one derived with the
complicated process outlined above, or our model explains nothing.
This is so for various estimates of the growth rate as expected by
At this point a word is in order about the parameters r, the stock
price growth and a, the stock price volatility we assume that the investor
expects to prevail when he buys the conversion option.
We cannot inspect the mind of the inspectors as it was when they
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bought the options nor can we hope that the same estimation procedure
was employed for every bond or by every investor. We, therefore, ex-
perimented with various estimates.
For the more critical expected rate of growth we mention three
estimates plus a compound of two of these:
r weights the growth rates in the last six years before the issuance
of convertible bonds with 1,2,.., 6, the highest weight being assigned to
the most recent year;
r is derived from the assumption that all stocks can be expected to
yield the same return to their owner and the more is taken out in dividends
the less will be received in the form of price appreciation;
r is the actual growth of the stock between the date of issue and
four years later (or retirement or June 15, 1964, whatever comes first).
r is a compound of r and r
c w m
a is the variance of the logarithms of stock price ratios, we tried
two estimates, one based on the stock prices in the last fifty-three weeks
before the bond sale, the other on seven yearly stock prices before the
1 r - r
1 -J. w m
r = r + r â€” ~â€” â€” â€”
c m w r
If r is close to r the formula gives r ; the larger the difference r -r ,
w m w cm
the smaller the fraction of it that is added to r up to a maximum of 2r .
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flotation, the first estimate proved to be definitely superior and results
mentioned in the following are based on it.
We now turn to the main purpose of the chapter, namely, to devise
various measures of risk and to test their effectiveness in explaining
observed discount rates (rates of return demanded by the investor) and
option prices .
Value of the Option. Required Rate of Return, and Risk
As we expected beforehand, differences between option prices could
not be explained with the differences in expected value at the horizon
using the same rate of discount for all options.
We shall, therefore, try to explain Loth option prices and rates
of return by the investor by introducing measures of risk. Before talking
about the results, let me introduce as a third and final model a more
Unless the investor is risk-neutral, he will look not only at the
expected value of outcome, i.e., at
EPW Â« / (x-y) f (x) dx
properly discounted to the present, but also at the shape of the distri-
bution of outcomes, which in turn can be represented by the various moments
The distribution of outcomes has this shape
eo. .- â€” .,
[stock prices ]
and the cumulative distribution the one belov
It is constructed from the lognormal distribution of stock prices
(dotted line) by adding everything that is to the left of the exercise
price y (below which conversion is unprofitable) to the distribution at
point y and following the lognormal distribution from there.