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EQUILIBRIUM PRICING OF SPECIAL BEARER BONDS

By

Jayanth Rama Varma

Working Paper No. 817

August 1989

Indian Institute of Management, Ahmedabad

Abstract

Special Bearer Bonds provide immunity to investors in respect of black money invested

in them. This paper derives equilibrium prices of these bonds in a continuous time

framework using the mixed Wiener-Poisson process. The Capital Asset Pricing Model

(CAPM) is modified to take into account the risk of tax raids for black money investors.

The pricing of all other assets relative to each other is shown to be unaffected by the

presence of black money. This result extends the CAPM to capital markets like India

where black money is widespread. Other applications include estimating the magnitude

of black money.

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Equilibrium Pricing of Special Bearer Bonds

1. Introduction

Special Bearer Bonds were made available for sale from the 2d February 1981 (vide

Notification No. F 4(1)-W & M/81 dt. 15/1/81, 128 ITR 114). There were no application

forms to be filled up for buying the bonds which are repayable to bearer. The bonds of a

face value of Rs.10000 are redeemable after 10 years for Rs.12000. The premium on

redemption is exempt from income tax, and the bonds themselves are exempt from

wealth tax and gift tax. The most important provision is, however, the immunity

conferred by Section 3 of the Special Bearer Bonds (Immunities and Exemptions) Act

1981 (7 of 1981) :

3. Immunities (1) Notwithstanding anything contained in any other law for the time being

in force,

(a) no person who has subscribed to or has otherwise acquired Special Bearer

Bonds shall be required to disclose, for any purpose whatsoever, the nature and

source of acquisition of such bonds;

(b) no inquiry or investigation shall be commenced against any person under any

such law on the ground that such person has subscribed to or has otherwise

acquired Special Bearer bonds; and

(c) the fact that a person has subscribed to or has otherwise acquired Special Bearer

Bonds shall not be taken into account and shall be inadmissible as evidence in

any proceeding relating to any offence or the imposition of any penalty under

any such law. S. 3(2) provides that the above immunity shall not extend to

proceedings relating to theft, robbery, misappropriation of property, criminal

breach of trust, cheating, corruption and similar offences; the immunity does not

also cover civil liabilities (other than tax liabilities).

These provisions make the Special Bearer Bond (black bond for short) an attractive

investment to those who fear tax raids or prosecutions. An active secondary market has

also developed as the bonds, being payable to bearer, are transferable by delivery, thereby

offering complete anonymity to buyer and seller alike.

The question that arises is how does this instrument get integrated into the capital market,

and how is its price determined in this market.

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2. The Model

To simplify the algebraic manipulation, we shall work in a continuous time framework.

We assume that for the white investor (i.e., an investor who has no black money) the

asset returns are generated by a Wiener process :

dPj

––– = ?j dt + dzj

Pj

(1)

The zj are Wiener processes with drift zero and instantaneous covariances ? = [?ij]. Thus

the white investor sees an instantaneous return vector ? and instantaneous variance

matrix ?. We assume that there is a risk free asset (a white bond) which gives a return of

RFW.

The black investor sees things slightly differently: as long as there is no raid or

investigation, the returns evolve according to a Wiener process as above; but if there is a

raid leading to a detection of black money, taxes and penalties would be imposed leading

to a negative return. We assume that the raids follow a Poisson process with parameter ?,

and that conditional on a raid having taken place, the loss suffered by the investor is a

fraction h of his wealth (excluding black bonds). The fraction h, which we shall call the

grayness ratio, would vary from person to person depending on the fraction of wealth

which is unaccounted for and the skill with which this black wealth has been concealed;

it would also depend on the rates of tax and penalties leviable. We assume that the

grayness ratio is always less than one; typical values would probably be in the range 0.1

to 0.5. (We can express h as f1f2f3 where f1 is the fraction of black wealth to total wealth,

f2 is the fraction of black wealth which will be detected during a raid and f3 is the taxes

and penalties as a fraction of the detected black wealth. Though f3 could conceivably

exceed one, the product f1f2f3 must be less than one; else the individual will have to file

for bankruptcy.) Under these assumptions, the returns accruing to the black investor

would follow a mixed Wiener-Poisson process of the form:

dPjB

–––– = ?j dt + dzj - h dq

PjB

(2)

where q is a Poisson process with intensity ?. The Weiner and Poisson processes are

independent of each other.

The instantaneous returns are now given by ?j - h?; the instantaneous covariances by ?ij +

h2?. Letting e denote a vector of ones, the vector of instantaneous returns for the black

investor can be written as ? - h?e; the instantaneous variance matrix is given by ? +

h2?ee'. The white bond is no longer risk free; its variance is h2?, and its covariance with

other risky assets is also h2?; its mean return is RFW - h?. The risk free asset is the black

bond which offers a return of RFB. Under equilibrium, no white investor would hold a

black bond, but black investors may hold the white bond.

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We shall assume in all our analysis that investors have quadratic utility functions, or

equivalently evaluate portfolio choices in a mean variance framework. Under this

assumption, equilibrium returns on various assets must obey the well known Capital

Asset Pricing Model (CAPM) developed by Tobin(1958), Sharpe(1964), Lintner(1965)

and Mossin(196), and extended to continuous time by Merton(1973).

3. Equilibrium: A Simple Case

The simplifying assumption that we make in this section is that all investors are black;

this means that the prices of all assets including the white bond are determined by the

black investors. We also assume that all investors have the same grayness ratio h. This

means that all investors see the same mean vector and covariance matrix of returns; of

course, these are not the same as what a white investor would see, but there are no white

investors. Under this condition, we have a traditional CAPM relationship between the

means and betas as seen by the black investors. We will have :

EB(Rj) - RFB = EB(RMB - RFB)CovB(Rj, RMB) / VarB(RM) = ? CovB(Rj, RMB) (3)

where ? = EB(RMB - RFB) / VarB(RM)

We use the subscript B with all the expectations, covariances and variances to emphasize

that the stochastic processes to be used are those of the black investor; we write RMB

because the universe of risky assets for the black investor includes the white bond which

is not part of the risky market portfolio as seen by a hypothetical white investor. We shall

presently relate the quantities in Eqn. 3 to the corresponding quantities as seen by a

hypothetical white investor.

Let c be the fraction of the black risky portfolio invested in assets other than the white

bond (or equivalently, the fraction of white risky assets to all white assets); and let the

subscript j denote any portfolio which does not contain black bonds. We then have :

RMB = c RM + (1-c) RFW (4)

EB(Rj) = E(Rj) - h? (5)

CovB(Rj,RMB) = c Cov(Rj,RM) + h2? (6)

VarB(RMB) = c2 Var(RM) + h2? (7)

CovB(Rj,RFW) = h2? (8)

? = [c RM + (1-c) RFW - h? - RFB] / [c2 Var(RM) + h2?] (9)

RFW = RFB + h? + ? h2 ? (10)

We can rewrite Eqns (9) and (10) as

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RFW - h? - RFB

? = –––––––––––––– = ––––––––––––––––––––––––––––––––

h2? c2Var(RM) + h2?

c[ E(RM) - RFW] + RFW - h? - RFB

(11)

If x/y = (x+a)/(y+b) with b ? 0, then a/b = x/y. Hence, we have

RFW - h? - RFB c[ E(RM) - RFW] E(RM) - RFW

? = –––––––––––––– = ––––––––––––––– = –––––––––––

h2? c2Var(RM) c Var(RM)

(12)

and equation (10) becomes :

RFB = RFW - h? - h2? [E(RM) - RFW] / c Var(RM) (13)

Eqns. (13) expresses the dependence of the equilibrium black bond return on h and ? in

terms of parameters familiar to our hypothetical white investor. To derive the pricing for

other assets, we apply Eqn. (3) to the portfolio consisting of portfolio j fully financed by

white borrowing (i.e. shorting the white bond). Using equations (5) and (6) and the fact

that Cov(X-Y,Z) = Cov(X,Z) - Cov(Y,Z), we get :

E(Rj) - RFW = c Cov(Rj, RM) EB(RMB - RFB) / VarB(RMB) (14)

In this equation, portfolio j can be the white risky market portfolio; substituting RM for Rj

and rearranging, we get :

EB(RMB - RFB) = [E(RM) - RFW] VarB(RMB)/ c Var(RM) (15)

Substituting this value of EB(RMB - RFB) into Eqn. (14) gives

Cov(Rj, RM)

E(Rj) - RFW = –––––––––––– E(RM - RFW)

Var(RM)

(16)

valid for any portfolio which does not contain black bonds. This is exactly the CAPM

equation that a hypothetical white investor would write down if he were to completely

ignore the existence of black bonds and black money, and compute all returns and betas

in purely white terms using the white bond as the risk free asset. Here then is a market in

which there is a black CAPM equation (Eqn. 3) which uses the black bond as the risk free

asset and relates the returns as seen by the black investor to the betas as computed by

him; this equation is valid for the black bond also. There is also a white CAPM equation

(Eqn. 16) which uses the white bond as the risk free asset and relates the returns as seen

by the white investor to the betas as seen by him ; this equation does not apply to the

black bond.

The black and the white security analysts can certainly live in perfect harmony in this

world without even being aware of each other's existence. But we still have to populate

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this world with white investors, and let the black investors assume various shades of

gray; it is to this that we turn in the next section. Those readers to whom our derivation of

the white CAPM equation looked like a piece of legerdemain can also have the pleasure

(if such be it) of arriving at this result from first principles through the route of matrix

algebra.

3. Equilibrium in the General Model

We now remove the assumption that all investors have the same grayness ratio. In

particular, the grayness ratio of some investors could be zero; we allow white investors

into the economy.

It is easily verified (see, for example, Merton(1973) that, under quadratic utility, the first

order condition for utility maximization for investor k with wealth Sk, facing a mean

vector ?k, variance matrix ?k and risk free return rk is :

?k Skdk = gk (?k - rke) (17)

where gk is the reciprocal of the investor's Arrow Pratt measure of risk aversion and dkj is

the proportion of wealth invested in asset j (the investment in the risk free asset is 1-dk'e).

The Arrow Pratt measure of risk aversion (Pratt, 1964) is equal to -U''(W)/U'(W) where

U is the utility function for wealth.

In our case, ?-hk?e and ? + (hk)2?ee' are the mean vector and variance matrix for the

white risky assets as seen by investor k with grayness ratio hk. In addition, the white risky

asset must also be treated as a risky asset with mean return RFW - h? and covariance (hk)2?

with all risky assets. Thus Eqn. (17) takes the following form :

?

?? ?

?? ? 0?

?? ? + (hk)2?ee' ? Sk ?

?? 0 0 ?

??

?

?

?

?

?

?

?pk

?

?

?

?

?

= gk ?

?

? ? - hk?e - RFBe

? RFW - hk? -RFB

?

?

?

?

?

?

(18)

?

?

?

?bk

?

where we write dk as (pk' , bk)'.

We can expand Eqn. (18) into two equations :

[ ? pk + (hk)2?ee'dk ] Sk = gk [? - hk?e - RFBe] (19)

(hk)2?e'dk Sk = gk [RFW - hk? - RFB] (20)

To facilitate aggregation of the above equations over k we define :

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h2 = [ ? (hk)2 e'dk Sk ] / [ ? e'dk Sk ]

k

h = [ ? hk gk ] / [ ? gk ]

k k

d = [ ? dk Sk ] / [ ? Sk ]

k k

? = [ ? e'dk Sk ] / [ ? Sk ] = e'd

k k

p = [ ? pk Sk ] / [ ? Sk ]

k k

? = [ ? e'pk Sk ] / [ ? Sk ] = e'p

k k

g = ? gk

k

S = ? Sk

k

Aggregation of Eqns. (19) and (20) gives :

(21)

k

(22)

(23)

(24)

(25)

(26)

(27)

(28)

?p + h2??e = (g/S) [? - h?e - RFBe] (29)

h2?? = (g/S) [RFW - h? - RFB] (30)

Substituting Eqn (30) into Eqn (29) gives

?p = (g/S) [? - RFWe] (31)

Since p/? is the weight of the white risky market portfolio, we can compute the

covariances as seen by a white investor :

Cov(R, RM) = ?p/? = [g/S?] [? - RFWe] (32)

giving the CAPM equation

? - RFWe = ? Cov(R, RM) (33)

or, in component (or portfolio) form,

E(Rj) - RFW = ? Cov(Rj, RM) (34)

where

? = ?S/g (35)

Since Eqn (34) holds for the market portfolio also, we have:

? = [E(RM) - RFW] / Var(RM) (36)

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so that the usual form of the CAPM equation obtains :

E(Rj) - RFW = ?j [E(RM) - RFW] (37)

Substituting Eqn.(35) into Eqn (30) we get

h2d? = (?/?) [RFW - h? - RFB] (38)

On using Eqn (36), this becomes

RFW = RFB + h? + h2? [E(RM) - RFW] / c Var(RM) (39)

where c = ?/? is the fraction of all white assets invested in risky white assets (i.e. assets

other than the white bond).

This completes the analysis of the capital market in terms of white parameters.

We can also obtain a black version of the CAPM equation by aggregating Eqn (18) as

follows :

?B d = (g/S)(?B - RFWe) (40)

where

?

? ? 0 ?

?

? 0 0 ?

?

?

?B = ?

?

? ? - h?e ?

? RFW – h? ?

?

?

?B =

? + h2?ee'

? (41)

?

?

Since h is a weighted average of hk and h2 is a weighted average of (hk)2, ?B and ?B can

be interpreted as the mean vector and variance matrix applicable to the average black

investor (except that h2 need not equal h2). Now, d/? is the weight of the market portfolio

including white bonds; we can, therefore, derive a black CAPM equation as follows :

CovB(R, RMB) = ?Bd/? = [g/S?] [?B - RFBe] (42)

?B - RFBe = ?B CovB(R, RMB) (43)

E(Rj) - RFB = ?B CovB(Rj, RMB) (44)

where

?B = ?S/g (45)

Since Eqn (44) holds for the black market portfolio also, we have:

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?B = [E(RMB) - RFB] / VarB(RMB) (46)

so that the black CAPM equation holds :

E(Rj) - RFB = ?jB [E(RMB) - RFB] (47)

In equations (42) - (47), the vector R is extended to include the white bond also; further, j

may be any portfolio whatsoever including black bonds.

We have thus derived the white CAPM equation (Eqn 37) and the black CAPM equation

(Eqn 47) both of which explain the risk return relationship in the capital market. These

correspond to and generalize Eqns (16) and (3) which we obtained in the simple model

earlier. Once again the white CAPM equation does not apply to black bonds; however,

Eqn. (39) which generalizes Eqn. (13) expresses the dependence of the equilibrium black

bond return on h, h2 and ? in terms of white parameters. The main differences between

the results in this model and the earlier simpler model are:

(a) h is now a weighted average of the hk, with the weights being the reciprocals of the

Arrow-Pratt risk aversion coefficients;

(b) h2 is a weighted average of the (hk)2 with the weights being the wealth invested in

white assets; and

(c) h2 need not equal h2.

Of course, h and h2 do not enter the white CAPM equation (Eqn. 37) at all.

Those readers who still finds it surprising that the white CAPM equation should hold in a

market which has black investors (or even has only black investors) may find it useful to

reflect on the following matrix identity :

?

??

??? 0 ?

??

??0 0 ?

??

?

?

?

?–1

?

?

??-1

?

?(hk)2(?-1e)'

?

(hk)2??-1e

(1/(hk)2?) + e'?-1e

?

?

?

?

?

=

?+ (hk)2?ee' ?

?

?

?

?

which follows from the formulas for partitioned inverses and for updating inverses after a

rank one correction. The block of the inverse matrix corresponding to white risky assets

continues to be ?-1 regardless of the existence of black money risks. Moreover, the

correction in the mean returns hk? is the same for all white assets. These facts imply that

black money should not affect the pricing of white risky assets inter se. Since the white

risk free asset can be regarded as the limiting case of a white risky asset, the pricing of

this relative to the other white risky assets should also be unaffected by the presence of

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black money. This indicates that the white CAPM equation should hold for all white

assets. The only asset to which this argument does not apply is the black bond.

4. Conclusions and Implications

One important conclusion of this paper is that the presence of black money investors

(who face the risk of tax penalties in addition to the normal investment risks) does not

affect the pricing of white assets at all. This implies that the ordinary CAPM can continue

to be used in all matters where black assets are not involved even if the white assets being

considered are known to attract a lot of black investors. This provides justification for

using the CAPM in corporate finance and portfolio management in a capital market like

India where black money is widespread.

A simple relationship was shown to hold between the return on the black bond and that

on the white bond :

RFW = RFB + h? + h2? [E(RM) - RFW] / c Var(RM) (36)

where c is the fraction of all white assets invested in risky white assets (i.e. assets other

than the white bond), h and h2 (÷ h2) represent the prevalence of black money in the

economy, and ? represents the intensity of the Government's tax enforcement policies

(frequency of raids).

Possible applications of this relationship include :

1. Black money investors could use this to decide on their policies relating to disposal

of their black wealth. They could use the equation to estimate likely prices of the

bond in future under alternative scenarios; they could also use current prices of the

bonds to assess the market's perception of the parameter ? (the likelihood of tax

raids) and use this as a crosscheck on their own judgment.

2. Researchers in economics and finance could use the prices of black bonds to estimate

the parameter h (prevalence of black money) or parameter ? (intensity of tax

enforcement) if the other parameter is known or can be independently estimated.

More importantly, we can make an estimate of the change in the prevalence of black

money (h) in any given period assuming that the tax enforcement parameter ? has not

changed during this period (or using an independent estimate of the change in ?);

alternatively, if an estimate of the change in the black money prevalence (h) is

available, the change in the tax enforcement parameter can be estimated.

3. The Government could perhaps use this to arrive at a fair price at which any future

issue of bearer bonds should be made. Typically, such issues are accompanied by an

unannounced change in the tax enforcement parameter ?; the issue of bearer bonds

itself (and the price at which it is issued) conveys information to the public about this

change. This would complicate matters considerably.

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APPENDIX

This appendix derives the means and variances as seen by a black investor in discrete

time, and indicates how, as the time interval is reduced, the continuous time version is

obtained.

If a random variable X is equal to a random variable X1 with probability p and to another

random variable X2 with probability 1-p, then we have

E(X) = p E(X1) + (1-p)E(X2)

E [(X)2] = p E [(X1)2] + (1-p) E [(X2)2]

[E(X)]2 = p2 [E(X1)]2 +(1-p)2 [E(X2)]2 + p(1-p)E(X1)E(X2)

Var(X) = p Var(X1) +(1-p) Var(X1) + p(1-p) [E(X1) - E(X2)]2

If X1 = (1-h)X2 then these simplify to

E(X) = (1-hp)E(X2)

Var(X) = [p(1-h)2 + (1-p)] Var(X2) + p(1-p)h2 [E(X2)]2

= [1 - p{1- (1-h)2}] Var(X2) + p(1-p)h2 [E(X2)]2

If the random variable Y equals the random variable Y1 when X equals X1, and equals the

random variable Y2 when X equals X2, then :

Cov(X,Y) = E(X,Y) - E(X)E(Y)

= p Cov(X1,Y1) + (1-p) Cov(X2,Y2) +

p(1-p)[E(X1)(E(Y1-EY2) + E(X2)(E(Y2-EY1)]

If X1 = (1-h)X2 and Y1 = (1-h)Y2 then this simplifies to

Cov(X,Y) = [1 - p{1- (1-h)2}] Cov(X2,Y2) + p(1-p)h2E(X2)E(Y2)

Consider a time interval t, and let the white investor's mean returns during this interval be

(1+rjt) and the covariances be ?ijt; let the probability of a raid during this time interval be

?t. For the black investor, the mean returns are given by

(1+rjt)(1-h?t)

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and the covariances are given by

?ijt[1 - ?t{1- (1-h)2}] + ?t(1-?t)h2(1+rit)(1+rj)t).

If we substitute these values into Eqn. 3, we can obtain analogs of Eqns. (4) to (13); but

the formulas are quite messy and difficult to use.

However, if t is small and we neglect terms of order t2, the means become [(1 + rjt) - h?t)

and the covariances become (?ijt + h2?t). In other words, the reduction of the means is

roughly h?t and the increase in the covariances is roughly h2?t. This agrees with the

continuous time formulation.

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REFERENCES

Lintner, J. (1965), "Valuation of Risk Assets and the Selection of Risky Investments in

Stock Portfolios and Capital Budgets", Review of Economics and Statistics, 47, 13-37.

Merton, R.C. (1973), "An Intertemporal Capital Asset Pricing Model", Econometrica,

41, 867-887.

Mossin, J. (1966), "Equilibrium in a Capital Asset Market", Econometrica, 34, 768-783.

Pratt, J. (1964), "Risk Aversion in the Small and in the Large", Econometrica, 32, 122-

36.

Sharpe, W.F. (1964), "Capital Asset Pricing : A Theory of Market Equilibrium Under

Conditions of Risk", Journal of Finance, 19, 425-442.

Tobin, J. (1958), "Liquidity Preference as Behaviour towards Risk", Review of

Economic Studies, 25, 65-85.