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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