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, o£. 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 —
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 ». The expected value of the bond thus
w y OB
E(P) "■ / [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) ■= / [ / 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 - - -^— [ ( ^)-2R + ( ■ ) 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 - / — 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, »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 — ~— — —
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 .
- 17 -
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 « / (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. .- — .,
[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.
We have pointed out earlier that simple operating with the lognormal
distribution, i.e., neglecting the truncation appears unsatisfactory to
us, furthermore, the strong skewness imparted to the distribution by the
truncation allows us to test whether skewness is a significant factor in
the mind of the investor or whether for him the distribution is adequately
described by its first two moments.
- 19 -
We have used the word truncated for the above distribution . This,
however, is not correct. In the case of a truncated distribution, we
take the area under the lopped off tail and distribute it over the re-
mainder of the distribution proportionately to the density there, here
we concentrate it at y: every stock price below y for us is exactly
the same as a stock price y. The censored distribution does not fit
our bill either, it assumes no knowledge of the distribution below the
point of censorship, y, and thus is not defined in its moments at all.
We have not found anywhere our distribution and Its moments and call
it for lack of a better name 'truncated distribution of Class Two.'
The. method of moment computation is similar to that of the truncated
one. The J-th moment around the origin is given by the Stteltjes
' V, - / X^dP
- y* Hy\[i,oh + f x^dA(x I n.a^)
- y^ A(y I ^i.oh + e^^^^J ^ [1-A(y ( n+ja^a2) ]
A (y I ^,0^) is the cumulative lognormal distribution of y with
mean |i and variance a .
The first moment around the truncation point y instead of the origin
gives the expected value EPW, which we will call mean, higher moments,
namely, variance, skewness, and curtosis are computed around the mean with
the aid of the Taylor expansion.
The formula makes it clear that ^,o , x and y define the momenta
completely, there is no need to go back to the original data to compute
them. On the other hand, any three moments contain all information about
the distribution. Now consider
PK = e m
In Pk = -iT + In m
where m , the first moment around the truncation point is the expected
value (EPW) as set down earlier.
If we regiress In PK on In m and T we get a poor fit and i the coef-
ficient of T will have a large error relative to its size; this much we
know already from the attempt to work with a simple rate of return.
If, then, the valuation of the option depends on risk, and risk as
felt by the investor is well described by the moments we worked out in
the last few pages then
In PK " const + c. In m + Co 1" "9 ^ ^^3 ^" '"3
+ c. In m, + c.RATING + c,ISS + c T
1445 o 7
where i. ' ''
PK price of one option to acquire one share. It is calculated as price
see Section 4.4
of the conversion privilege divided by lOO/PC, the number of shares that
can be acquired through conversion of a $100 bond if the conversion price
Is PC. The price of the conversion privilege, In turn, Is the price of
the bond the first time It Is quoted on the market after the flotation,
minus the estimated straight debt value at that time.
m zero-th moment of the truncated lognormal distribution of the outcomes .
This Is the cumulative probability that the stock price will exceed at
the horizon T the straight debt value per share (which in turn is roughly
equal to the conversion price) .
m first moment around the truncation point, i.e., expected value of the
option at the horizon T, mean
m second moment around the mean m , variance
m. third moment around the mean m , skewness
m, fourth moment around the mean m , kurtosis (excess)
n m In m. , In ra , In m, are the natural logarithms of the above moments
^TING quality rating (Aa, A, Baa, Etc.) by Moody's Investors Service, Inc.,
a proxy measure for risk
ISS date of issue, introduced in order to detect any change In either
a. investor attitude towards the convertible bond or
b. terms of the conversion privilege
over time i
should give a much better regression and even the coefficient of T will
be accompanied by a much smaller standard error relative to its size,
its meaning now being the rate of return demanded after accounting for
risk. ' •
See also below
We shall also regress
In PK - Cj^ + C2r + c^lny + c^ a + c Inx + c, RATING
+ c^ISS -f Cq''
r expected future stock price growth rate; we use alternatively
r weighted pre-issue stock price growth rate
r "market" rate of return
r a compound of the preceding two
r actual (post-issue) stock price growth rate
in order to investigate how the investor estimates future stock
price growth and whether his estimate is close to growth actually
o, a standard deviation and variance of the logarithms of stock prices
in the Last fifty-three weeks before the flotation of the bond
In X logarithm of the stock price observed when the bond is quoted the
first time after flotation, a possible proxy variable for risk,
with higher priced stocks presumably being less risky
In y logarithm of the straight debt value divided by the share price
at the time of issue. (Since x is set to 1 as explained below,
In y/x =ln y) . The variable is a measure of the distance of the
stock price at the time of the bond issue to the conversion price.
in order to see how well we can predict or explain PK without the relatively
complicated model developed earlier and the moments derived in one of the
last sections. If we can get just as good an explanation from a few easily
obtainable simple variables, there is no point in searching for elaborate
models. We have tried to make this naive model as good as possible.
Next we combine the variables from both regression equations to see
whether the second set contains elements we have not taken care of auto-
matically by our moments.
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One adjustment remains to be explained: all stock prices x are
normalized to one in the computation of the moments and everywhere else
(except in In x , of course) . PK is likewise divided by x . Since the
unadjusted option price PK presumably varies with the price of the optioned
stock, x^, and the moments definitely vary with x , not making the adjustment
builds positive correlation (as well as heteroscedasticity) into the re-
gression. After the adjustment, all variables are dimensionless or have
only time as dimension.
We have room only for a summary of the results:
In the regression without moments r , In x , In y, a explain about
467. of thi variance of In PK; adding RATING, DVYLD, and time of issue,
ISS, boosits that figure to 58%. All variables are significant except
a and RATING.
This is as far as analysis without moments carries us in explaining
the value' an investor puts on the option. Considering that we are dealing
with a cross-sectional study, with the members being taken over a 15 year
interval, the correlation is high and very highly significant. We may
feel somewhat uneasy when we notice that the proxy variable for risk a
has a positive sign, from a risk-averting investor we expect the opposite;
furthermore, a is totally insignificant.
The behavior of O , however, is no riddle: in PK or In PK it has
two conflicting roles: it increases expected value, but also heightens
risk. DVYLD has a negative coef f icient--the investor apparently reasons
that the higher the dividend, the lower retained earnings and expected
- 24 -
price Increase, r^ and DVYLD are highly (negatively) intercorrelated.
Turning to the regression with the logarithms of the moments, we
notice that the expected value as computed in our model taken alone ex-
plains the actual price paid for the option better than rate of return,
stock price, ratio of conversion price to stock price, stock price taken
together (80% vs. 597.). Adding the second through fourth moments adds
37* to explained variance.
The logarithms of the moments alternate in sign. This would seem
to indicate risk aversion, preference for skewness and dislike of kurtosis.
All variables are significant at the .01 level. Some caution in reading
the results seems to be in order. The logarithms of the moments are high
collinearlty causes instability of the coefficients. If the moments them-
selves are added, unexplained variance is almost halved (R = .89) . Ex-
pected value, variance, etc. are now represented twice with the result
that each appears once with a positive and once with a negative coefficient.
Adding r , o i etc . , increases R only from .89 to .93, whereas adding the
moments and their logarithms to these variables increases explained variance
from 587.1 to 927..
The Role of the Date of Issue
A question of interest is whether the bonds or the Investors' attitude
towards them has changed over the 15 years time span from 1948 to 1963.
Somewhat surprisingly, we got very much the same results in both
regressions using PK as dependent variable (and the moments themselves
as independent variables) .
- 25 -
In the zero-order correlations time of issue is highly significant
(t=-6.38); the more advanced the years, the higher the price paid for
the option and the lower the required rate of return. When, however, we
re-examine its t- coefficient as the moments are introduced, we discover
that it drops below the 10% level of significance and when In y, etc.
are added it drops below the 207= level (t=-.34). This combination of
findings plus the discovery that there exists a modest but highly signifi-
cant correlation (R= .58) between the ISS and the logarithms of the moment
of expected value, which is not affected by the rate of return required by
the investor, suggests to us a different explanation. While the investors'
attitude may or may not have changed over the years, the corporations' attitude
towards the bond has changed. Not only are there more bonds issued in later
years, particularly the second half of the Fifties, but the corporations
have also made the conversion privilege more attractive, more valuable.
This is confirmed by the tendency to issue bonds of poorer quality (positive
correlation RATING and ISS, R = .38) which nonetheless are positively corre-
lated with the logarithm of expected value (R = .12). There is a strongly
positive correlation ( R = .54) between time of issue and price of the option
in per cent of the total package named convertible bond. One method corpo-
rations have chosen to make the conversion privilege more attractive is to
lower the conversion price in relation to the stock price at the time of
issue; we thus find a negative correlation between In y and ISS (R = .23) .
Second, corporations issuing convertible bonds had a significantly better past
growth record (positive correlation between r and ISS is R = .15). Third, stock
- 26 -
price volatility o grew with ISS (R = .30) making for greater expected
value. It clinches our case to find that in the regression for the re-
quired rate of return, 1, the time of issue, ISS is represented with a
positive though insignificant coefficient once the coefficients of variance
and of other variables are present.
The Rate of Return Demanded by the Investor
PK - e m^
InPK T IT + In m
Rates of return demanded by the Investor 1 were computed as the rate
that discounted expected value of the horizon T such as to equal the
price PK paid for the option.
If 1 Is a function of risk as expressed by variance, skewness, etc.
m m m
InPK = (1 +c, —2 + c, —3/2 + c, — 2 + . . .) T + m
o Z m, J m_ J ra^ 1
I i z /
where m /m.
coefficient of variation squared
coefficient of skewness
coefficient of skewness
We also found that for a sample of 62 bonds issued between 1948
and 1956, the average option was priced at $10 per $100 bond vs. $18
for a sample of 63 bonds issued between 1959 and 1963.
- 27 -
ig : rate of return demanded on a riskless investment
+... : other factors one might like to introduce
Rearranging the above equation slightly
InPK -In m. m m m
i = ^ = i^ + -f2 + c- -^3/2 + c, -^2
*■ o z m^^ 3 m- 4 m
To introduce the moments themselves or their logarithms would be
undesirable when considering the required rate of return which is dimension-
free except for time. We have, therefore, computed magnitude -free measures
of variance, skewness and kurtosis.
Starting again with the regression without the moments we find the
rate of stock price growth is one of the variables with the greatest
influence on the required rate of return i, the higher the one, the higher
the other; this is well in keeping with earlier remarks that the investor
may extrapolate past growth but that he is aware of the fact that the
higher the growth rate of the past, the less reliable the extrapolation.
If we use r which is only based on past performance, this finding is
brought out even more strongly. The magnitude of the stock price at cho
time of issue; of the convertible bonds turns out to make a significant
contribution in explaining the variance of the dependent variable, too.
The correlation Is a negative one. Bigger, more stable corporations have
higher priced shares than the more risky, smaller corporations. And, one
does assoclatle risk with penny-stocks. Unlike in the regression for
In PK, a is la highly significant indicator of risk, since it no longer
has to do double duty as a variable standing for high expected value and
- 28 -
high risk. Trying to explain 1, the investor's discount rate, with the
coefficients of variation, skewness and kurtosis alone is disappointing.
Only 17% of total variance in the required rate of return are explained
but taken together with r^ and a^ we get a substantial improvement not
only over the coefficients of variation taken by themselves but also r
0, etc., considered alone (477.). Although the number of independent
variables has increased, the size of the coefficients relative to their
standard errors has gone u£ over what they were in the separate regressions.
The coefficient of skewness reduces unexplained variance highly signi-
ficantly, so does the coefficient of kurtosis, which when introduced also
increases the t-coefficient of the coefficient of skewness.
All the Variables have the 'correct', i.e., the predicted sign--the
required rate tof return Increases with a , coefficient of variance, kurtosis
and decreases with skewness .
As these variables are highly intercorrelated, our confidence in
the results is' not as great as it would be without colltnearity. It is
comforting to notice, however, that when variables are added that, in
our opinion, measure the same thing (e.g., range of the outcomes) which
a variable already in the regression also measures, they tend to reduce
the t-coef ficiient of the variables introduced earlier. When, however,
to such a variable another is added, which is also collinear with it
but in our opinion measures something different (e.g. skewness) something
towards which according to our hypothesis the investor has the opposite
attitude<--then the t-coefficient of the variable introduced first is in-
creased. If our hypothesis that the investor likes skewness and dislikes
variance is correct, this is exactly what we expect. Suppose we introduce
a first and then the coefficient of variance. The second variable will
contribute some new information, but otherwise duplicate information already
introduced through the first one. If the latter aspect is important, the
t-coefficient of the first variable may well fall, especially, if the
second variable specifies the information in a superior fashion. Since
loosely speaking there is only so much 'significance' to distribute, the
t-coefficient of the first must fall of needs if the second variable is
assigned a high one. Now let us change the picture. Suppose we introduce
first a measure for variance then one of skewness. The two measures are
intercorrelated. Now assume that the investor's utility is increased
through skewness but decreased through variance. When variance is introduced
alone then due to collinearity it stands for both variance and skewness with
low significance as the result. When, however, both variables are present,
there is one for each job; to measure variance and to measure skewness and
both emerge as what they are: highly significant.
- 29 -
Date of issue ISS is insignificant and positive. Thus, discrediting
the hypothesis that the investor has become more familiar with the con-
vertible bond and is satisfied with a low rate of return. It is interesting
to notice that variable In x^, the price of the stock at time of issue, is
insignificant once other risk measuring variables are introduced, though
highly significant in the zero-order correlation. This means the investor
does not associate (demonstrably) a high priced share with low risk cet . par ,
or per se .
Anticipation of Objections
Several objections may have come to your mind.
Let me anticipate three of them.
1. One may ask whether in view of high intercorrelation between some
of the 'independent' variables might not bias the coefficients, or, much
worse, whether the signs of the coefficients of which we made so much are
not built-in by this method of attack. The answer is that some bias quite
likely may exist that, however, the change of signs from the coefficient
of variance to that of skewness and back to that of kurtosis can be shown
not to be built in. If there is an influence dragging the size of the
coefficient of variance in one direction, it would be even stronger in
the same direction for skewness. The same observation can be made with
respect to the pair--skewness and kurtosis.
2. The second anticipated objection concerns the horizon. The
horizon, it will be remembered, was found in a diagram of expected value
as a function of time by tilting a straight line around the price actually
paid for the option (as pivot) until it touched the expected value curve.
- 30 -
The slope of the curve gave the rate of return demanded by the investor,
the abscissa at the point of contact, the horizon. Moments then were
calculated at that horizon and regressed of the option price. It can
be maintained that the dependent variable, the option price, is used
in the computation of the independent variables and that correlation
is to be expected. To remove that objection we employed a two-step
procedure. First, we did the moment calculation and regression analysis
with the same average horizon for all bonds, then we compute the option
price as predicted by the regression equation. These option prices are
then used to compute individual horizons and the moments associated with
them. In a second step, the actual option prices are then regressed on
those moments and other variables. (The procedure is somewhat reminiscent
of two-stage least squares.)
The result is simple to state: correlation is reduced but all con-
clusions are unaffected, in particular, the coefficients still have the
'correct,' i.e., the predicted sign.
3. Third, it may be supposed that the high return on convertible
bonds is incompatible with portfolio efficiency. Does there exist a
portfolio of straight debt and conversion option yielding more than
straight debt while not exceeding it in variance? While no attempt has
been made to refute this for every bond individually, we have checked
that using the average figures for the options and stocks concerned, the
high return on conversion options is consistent with portfolio efficiency.
Our final point takes up the question of how well the investor at
- 31 -
the time of purchase of the option was able to foresee future development
of stock prices. We can present the results only in very condensed form:
Extrapolation of past growth used as the estimate of future growth in
computing the moments used for the regression analysis explains option
prices far better than moments based on the actual post-issue growth
rate, i.e., on perfect foresight. Regressing the rates directly alone
or together with other variables shows the same pattern: the higher
preissue price growth, the higher the option price, the higher post-
issue growth, the lower the option price. Furthermore, correlation is
higher between price and pre-issue growth than between price and post-
issue growth. Pre- and post-issue growth are negatively correlated.
All this suggests that the investor, on the average, had incorrect ex-
pectations. The picture, however, is more complicated: the required
rate of return is more highly correlated with post-issue growth rate
than with the pre-issue rate, furthermore, the correlation is positive
also for the latter one, indicating to us that though the investor was
not able to predict which stocks would rise, he was quite well able to
identify the risky ones.
Our findings throw a curious oblique light on John F. Muth's
"Rational Expectations" hypothesis. To quote Muth
1. Averages of expectations in an industry are more
accurate than naive models and as accurate as
elaborate equation systems, although there are
considerable cross-sectional differences of opinion.
2. Reported expectations generally underestimate
- 32 -
the extent of changes that actually take place.
While Muth's statement No. 1 may hold for firms as he claims, it is not
borne out in our case for individual investors. On the other hand, our
investor seems to have a good feeling for stocks where the extent of
the change is likely to be great and classifies such stocks as risky,
but we cannot quite maintain that this is in contradiction to statement
No. 2 without further analysis. In any case, the investor does not seem
to be able to forecast the change of direction of stock price growth.
^John F. Muth, "Rational Expectations and the Theory of Price
Movements," Econometrica (29, No. 3), July 1961, p. 316.