Accounting and Business Review
Vol.4.No.1,(Jan. 1997) 91-103
OPTION VALUE OF REDEEMABILITY AND CLOSED-END FUND
DISCOUNT PUZZLE
Gongmeng CHEN
Joseph K. CHEUNG
Department of Accountancy
The Hong Kong Polytechnic University
This study examines the nature of the discount associated with closed-end mutual fund shares. It
observes that the lack of a redemption option in closed-end fund shares is the fundamental reason
for the discount. Starting with Margrabe's (1978) exchange option theory, the paper develops a
valuation model for the discount. From this model, it is shown that the discount is directly related
to (a) the volatility of the differential returns between the market share price of a closed-end fund
and its net asset value per share and (b) the expected duration of the closed-end fund. Simulation
results based on the model indicate that it is in keeping with existing empirical evidence. In
particular, the average discount is 14.6 percent, with a range from 6 to 24 percent, on the
assumptions that the volatility vary from 0.05 to 0.15 and that the fund's expected duration vary
from 20 to 50 years.
For more than two decades, the existence of discounts on closed-end mutual funds has
been one of the major puzzles in finance. This paper aims to explain this phenomenon by
examining the lack of redeemability of closed-end fund shares.
A mutual fund is a limited company whose earnings are dividends, interests, and
capital gains from a portfolio of financial securities. The fund's market value, referred to as its net
asset value (NAV), is calculated as the value of its portfolio minus its short-term liabilities. The
two typical organizational forms for mutual funds are open-end and closed-end. In the former case,
investors apply to the fund to purchase new shares or to redeem shares already held. The number
of shares varies from day to day. In the case of a closed-end fund, in contrast, the number of shares
is fixed. This makes a closed-end fund resemble a typical limited company which must go through
an underwriting process in order to issue new shares. An implication from this is that the market
value of closed-end fund shares is determined by the supply and demand for that particular share,
which may or may not be identical to the fund's NAV at the time. In fact, most closed-end funds
are sold below its NAV by about 10 to 20 percent, a discount that is the focus of this paper. ~
Recent evidence of discounts found in closed-end funds has been documented by Weiss
(1989), Forsyth (1995), Groves (1995), Laderman (I995), Malkiel (1995) and Taub (1996).
These studies indicate several empirical regularities. First, closed-end funds on average sell at a
discount. Second, the discounts usually last for a long period of time while premiums, if they exist,
last for a much shorter period of time. Third, premiums are normally found only in the case of
initial public offerings.
Academic interest in the discount puzzle started in the 1970s. This line of work, which
includes studies by Boudreaux (1973), Roendfelt and Turtle (1973), Sharpe and Sosin (1974),
Mendelson (1977), Malkiel (1977) and Thompson (1978), puts forth three different explanations.
~.
We appreciate the comments of Larry H.P. Lang, Larry J. Merville, J.B. Kim, Bob Whaley, and Yexiao Xu.
However, all remaining errors are our responsibility.
Accounting and Business Review
Vol.4.No.1,(Jan. 1997) 91-103
One is agency cost in that the discount reflects the difference between the fund's performance and
the fees charged by the fund manager. The second explanation is the illiquidity of the assets in a
closed-end fund. The third explanation is the tax liabilities related to a closed-end fund. The
inadequacy of these explanations for the discount puzzle has been addressed by Lee, Schleifer
and Thaler (1991).
More recently, De Long et al. (1990) offer a noise trader hypothesis to explain the
discount. They maintain that the discounts on closed-end mutual funds are much like the
fluctuations of prices in that both are determined by the changing sentiment Of individual
investors. Discounts are high when investors are pessimistic about future returns, and low when
they are optimistic. On average, discounts exist because the unpredictability of investor sentiment
impounds a risk to holding shares in a closed-end fund in addition to the risk inherent in the fund's
portfolio. In keeping with such a view, Lee et al. (1991) find that discounts on closed-end funds
are a proxy for changes in individual investor sentiment. However, the noise trader theory is not
without its controversy. Chen, Kan, and Miller (1993), for example, have raised serious questions
about its validity.
This study aims to provide an alternative explanation for the discount phenomenon, on the
basis that the discount in a closed-end fund is equivalent to the value of an option which, if it
existed, would give the fund's shareholders a right to redeem their shares at the NAV. Because
closed-end fund shares are not redeemable,: such an option does not exist, thus making closed-end
fund shares worth less than their NAV. The rest of the article is arranged as follows. Section 2
develops the valuation model for the discount in a closed-end fund. Section 3 examines the key
properties about the discount valuation model. Section 4 presents some simulated results based on
the valuation model, and section 5 concludes the paper.
VALUATION OF CLOSED-END FUND DISCOUNTS
Consider two mutual funds that have an identical NAV per share and that one of them is
open-end while the other is closed-end. When a share from each fund are compared, they are
identical except that the share from the open- end fund gives its owner a right to. redeem at the
fund's per-unit NAV while the share from the closed-end fund does not. We observe that the
redemption right is an American-style exchange option, which accounts for the difference in value
between the two shares. 1
Margrabe (1978) has derived a pricing formula for a European-style exchange option and has
shown that such a formula can be applied to an American-style exchange option as well. To apply
Margrabe's pricing formula to the redemption option inherent in an open-end mutual fund, let the
following variables be defined:
σ 1 τ = standard deviation of return on the closed-end fund;
σ 2 = standard deviation of return on the NAV;
ρ = correlation between the closed-end fund and its NAV;
σ = standard deviation of the differential return of the closed-end fund
1
and its
An American-style option is one that can be exercised at any time up to the option's expiration
date. A European-style option is one that can only be exercised at the option's expiration date.
Accounting and Business Review
Vol.4.No.1,(Jan. 1997) 91-103
NAV;
τ = expected duration of the closed-end fund; taken to be the fund's
existence
period, from the present to the time when it is liquidated or
when it becomes
an open-end fund;
POE = per-share price of the open-end fund;
PCE. = per-share price of the closed-end fund;
Navps = per share NAV for either the closed-end fund or the open-end
fund 2 ;
VE = value of an option to exchange one share of a closed-end fund for
an
amount of cash at the NAV;
δ = discount of the closed-end fund given by dividing the fund's NAV
the difference between the fund's NAV and its market value.
Based on the above definitions, the following three relations hold:
δ=
Navps − PCE
P
= 1 − CE ,
Navps
Navps
into
(1)
POE=Navps,
POE=PCE +VE.
(2)
Applying Margrabe's (1978) formula to VE, we have the value of the redemption option as:
VE = PCE N (d1 ) − NavpsN (d 2 )
Ln(
where, d1=
PCE
σ 2τ
)+
Navps
2
σ τ
, d2=d1 −σ τ ,
(4)
(5)
σ 2 = σ 12 + σ 22 − 2 ρσ 1σ 2 .
Combining equations (2), (3) and (4), we arrive at
Navps – PCE = PCE N(d1) - Navps N(d2).
Rearranging (6), we have
1
Ln(1 − δ ) + σ 2τ
2
d1 =
, and
σ τ
d 2 = d1 − σ τ .
(6)
(7)
Finally, substituting (1) into (7) and rearranging terms, we have the following
valuation equation for the discount ~ of a closed-end fund shares from its NAV
δ=
N (d1 ) − N (d 2 )
1 + N (d1 )
where; by substituting (1) into (5),
2
The closed-end fund and the open-end fund mentioned in this analysis are assumed to be
identical expect for the redeemability of open-end fund shares.
(8)
Accounting and Business Review
Vol.4.No.1,(Jan. 1997) 91-103
1
Ln(1 − δ ) + σ 2τ
2
d1 =
, and
σ τ
d 2 = d1 − σ τ .
(9)
According to equations (8) and (9), the discount 5 is a function of the standard deviation and
the expected duration of a closed-end fund. Although there are no closed-form solutions to the
discount in equations (8), it is possible to approximate the theoretical range of the discount 8 and
to examine how the
standard deviation and the remaining time period of the closed-end fund affect the magnitude of
the discount. 3
SOME PROPERTIES ABOUT THE VALUATION MODEL
and
τ
From equations (8) and (9), we note that two major variables, σ , the standard deviation,
, the expected duration of the closed-end fund, taken to be the expected existence period
from present to open-ending or liquidating, affect the discount. The relations of these two
variables with the discount have important implications for our understanding the nature of
closed-end fund discounts.
Proposition 1: The standard deviation of the differential return of a closed-end fund and its NAV,
σ ,is positively related to the magnitude of the discount.
That is,
∂δ
∂δ
>0
> 0.
∂τ
∂σ
Proposition 2: The expected duration of the closed-end fund, τ , is positively related to the
magnitude of the discount 4 . That is,
∂δ
> 0.
∂τ
The implication from Proposition 1 is that a higher discount is required of a closed-end
fund whose return is more volatile. Proposition 2 implies that the value of a closed-end fund
approaches that of an open-end fund as the closed-end period gets shorter. For example, suppose
we have a closed-end fund which is to become open-ended in one day. Then, the fund shares
will be sold at a very small discount, if any, from its NAV. Although a closed-end fund does not
have any life limit, it usually has an expected life. In practice, no fund remains forever. A
closed-end fund can be terminated by becoming open-ended or liquidated. These situations will
occur if doing so is favorable to the fund shareholders 5 .
Proposition 2 is also in keeping with empirical evidence found by previous researchers.
Brauer (1984) and Brickley and Schallheim (1985) find that when closed-end funds become
open-ended or are liquidated, the discounts narrow.
They also report that a small discount
Since d1 is a function of δ , there is no closed-form solution to (8) for the discount δ , However,
it will be shown in the following section that there exists a unique solution for δ .
4
All proofs are presented in Appendiex A.
5
Of course, the fund managers may resist either open-ending or liquidation, which is actually part
of agency cost.
3
Accounting and Business Review
Vol.4.No.1,(Jan. 1997) 91-103
persists until the final termination or open-ending. Further, Brauer (1988) provides evidence that
the information contained in a discount is an incomplete prediction of the fund's likelihood of
being open-ended profitably. He shows that abnormal returns can be earned by portfolio
formation based on future open-ending potential.
From the derived model, we can also examine the theoretical range of the discount.
Additionally, although there is no analytical solution to the model, we can check the existence of a
unique numerical solution. Propositions 3 and 4 address these issues.
Proposition 3: The theoretical discount of a closed-end fund from its NAV ranges from 0 to 50
percent. That is, 0 < δ < 0.5.
Proposition 4: Given any two feasible values
unique solution to equation (8).
δ ∈ (0, 1) and τ ∈ (0, ∞ ), there exists one
NUMERICAL SIMULATIONS
Although there is a unique solution to (8), the equation cannot be solved analytically.
This section offers some insights via numerical simulation. Given any two values of the volatility,
cs, and the expected remaining period, τ , we can solve for δ with a trial and error method. It is
reasonable to believe that the return earned by a closed-end fund is highly correlated with the
return on its NAV. Consequently, the combined volatility must be close to
| σ1 − σ 2 | .
Table 1 shows the theoretical discount of a closed-end fund from its NAV. given the
range of volatility values from 0.05 to 0.15 and the expected duration of the fund from 20 years to
50 years. The error of the numerical solution is controlled within 0.0001. It is interesting to note
that actual discounts observed in the real world are quite consistent with the theoretical discounts
derived. The discount averaging across volatility from 0.05 to 0.15 is 11.4% for a 20-year
expected duration, and 17.2% for a 50-year expected duration. The overall average discount is
14.6% averaging across volatility from 0.05 to 0.15 and with an expected duration from 20 to 50
years.
Table 1
The Simulated Discount Rates of a Closed-End Fund with the volatility ranging from 0.05 to
0.15 and the remaining period ranging from 20 years to 50 years.
Volatility of Closed-End Fund
No.of
years
0.05
0.07
0.08
0.09
0.10
0.11
0.12
0.13
0.14
0.15
Average
20
6.0%
8.3%
9.4%
10.5%
11.5%
12.6%
13.6%
14.6%
15.6%
16.6%
11.4%
22
6.3%
8.6%
9.8%
10.9%
12.0%
13.1%
14.2%
15.3%
16.3%
17.3%
11.9%
24
6.5%
9.0%
10.2%
11.4%
12.5%
13.7%
14.8%
15.9%
16.9%
18.0%
12.4%
26
6.8%
9.3%
10.6%
11.4%
13.0%
14.2%
15.3%
16.5%
17.6%
18.6%
12.9%
28
7.0%
9.7%
11.0%
12.2%
13.4%
14.7%
15.8%
17.0%
18.1%
19.2%
13.3%
30
7.3%
10.0%
11.3%
12.6%
13.9%
15.1%
16.3%
17.5%
18.7%
19.8%
13.7%
32
7.5%
10.3%
11.7%
13.0%
14.3%
15.6%
16.8%
18.0%
19.2%
20.3%
14.1%
34
7.7%
10.6%
12.0%
13.3%
14.7%
16.0%
17.2%
18.5%
19.7%
20.9%
14.5%
36
7.9%
10.9%
12.3%
13.7%
15.1%
16.4%
17.7%
18.9%
20.2%
21.4%
14.9%
38
8.1%
11.2%
12.6%
14.0%
15.4%
16.8%
18.1%
19.4%
20.6%
21.8%
15.2%
Accounting and Business Review
Vol.4.No.1,(Jan. 1997) 91-103
40
8.3%
11.4%
12.9%
14.4%
15.8%
17.2%
18.5%
19.8%
21.1%
22.3%
15.6%
42
8.5%
11.7%
13.2%
14.7%
16.1%
17.5%
18.9%
20.2%
21.5%
22.8%
15.9%
44
8.7%
11.9%
13.5%
15.0%
16.5%
17.9%
19.3%
20.6%
21.9%
23.2%
16.3%
46
8.9%
12.2%
13.8%
15.3%
16.8%
18.2%
19.6%
21.0%
22.3%
23.6%
16.6%
48
9.1%
12.4%
14.0%
15.6%
17.1%
18.6%
20.0%
21.4%
22.7%
24.0%
16.9%
50
9.3%
12.7%
14.3%
15.9%
17.4%
18.9%
20.3%
21.7%
23.1%
24.4%
17.2%
Average
7.7%
10.6%
12.0%
13.4%
14.7%
16.0%
17.3%
18.5%
19.7%
20.9%
14.6%
CONCLUDING REMARKS
This study observes that the fundamental reason for the discount in a closed-end fund is the lack
of redeemability. We develop a model to value the discount. It is found that the volatility of the
differential returns between the market share price of a closed-end fund and its net asset value per
share has a positive effect on the discount's magnitude. It is also found that the longer the
expected duration of a closed-end fund, the larger is the discount. Based on this theory, the upper
limit of the discount is 50 percent. There is no closed-form solution to the model. The numerical
simulations show that the discount varies between 6 and 24 percent when the volatility goes from
0.05 to 0.15 and with an expected duration from 20 years to 50 years. The average discount is
14.6 percent. Our finding is largely in keeping with extant empirical evidence.
From the standpoint of mutual fund investors, one of the implications of this study is that
closed-end funds selling at a premium should be avoided. It is shown here that, in equilibrium,
closed-end funds should sell at a discount. So a premium can be viewed as a temporary deviation
from the equilibrium price 6 . Another implication for investors is that they should choose
closed-end funds that both sell at a deep discount and have a high potential of open-ending, since
the discount will disappear when a closed-end fund becomes open-ended.
From the standpoint of fund managers, they should realize that a closed- end fund's
selling price is directly related to the perceived potential that the fund will become open-ended.
Such potential would be viewed as high if management's resistance to future liquidating or
open-ending can be minimized. Based on this, fund managers should ensure that special
open-ending or liquidating clauses are embodied in the closed-end fund contracts.
Finally, a study like this would not be complete without noting the limitations. Several of
them stem from the assumptions in the option model in general, which are well known, and in the
Margrabe exchange, option model in particular. For instance, one might question whether the
per-share price of the closed-end fund, the underlying asset of the option in our model, and the
per- share net asset value of the fund, the stochastic exercise price, truly follow a joint diffusion
process. In addition, the notion that the closed-end fund has an "expiration" date as suggested by
the fund's having an expected duration is highly stylized. The paper should be put in the context
of these limitations.
More controversial, perhaps, is the implicit assumption in our analysis that the ability of
6
A possible reason for this deviation to occur is the participation of noise traders, as suggested by
the evidence that premiums are negatively correlated with future returns (see Pontiff (1995)).
Other reasons for a premium to occur, mostly during initial public offerings, is the aggressive
selling by retail brokers and the desirable economic conditions chosen by the fund issuer.
Accounting and Business Review
Vol.4.No.1,(Jan. 1997) 91-103
closed-end fund investors to sell their shares in the market does not constitute an option. One
might argue that although closed-end fund shares cannot be redeemed at the net asset value, they
can nonetheless be "exchanged" for the shares' resale value. Thus, closed-end fund investors also
hold an exchange option. Strictly speaking, therefore, our analysis should have compared the
values of two options: one held by open-end fund investors and another held by closed-end fund
investors where the two funds are identical otherwise. When comparing the two options, one will
find their major difference lies in the exercise price. In the open-end fund case, the exercise price
is the fund's net asset value at each point in time, the maximum value of the fund at that particular
point. In the closed-end fund case, the exercise price is the fund's selling price. But the new buyer
of the closed-fund share cannot be sure he can resell his shares at the net asset value in the future,
so he is not likely to pay maximum price for his purchase. Hence, one can make the case that the
exchange option associated with the open-end fund has a larger exercise price than that
associated with the closed-end fund. Given two otherwise identical call options, the one with a
larger exercise price is worth more.
Thus the option associated with the open-end fund is larger
in value, so the value of an open-end fund dominates that of its closed-end counterpart. This result
is equally consistent with the observation that closed-end funds tend to sell at a discount when
compared with open-end funds. Thus, it does not change the central finding of the paper.
However, it does overstate the option value embedded in an open-end fund share and should be
noted as a limitation of the paper.
APPENDIX A
Proof for proposition 1:
Rearranging (8) as
δ + (δ − 1) N (d1 ) + N (d 2 ) = 0
Take a first-order partial derivative of (10) with respect to
(10)
σ . We have
∂d
∂d
∂δ ∂δ
N ( d1 ) + (δ − 1) n( d1 ) 1 + n(d 2 ) 2 = 0,
+
∂σ ∂σ
∂σ
∂σ
(11)
where n( ) is a density function of standard normal distribution.
It can be shown that
n(d 2 )
= 1 − δ . Thus, equation (11) becomes
n( d1 )
n( d 2 ) ∂
τ n( d 2 )
∂δ
>0.
(d1 − d 2 ) =
=
1 + N ( d1 )
∂σ 1 + N ( d1 ) ∂σ
(12)
Proof for Proposition 2:
Take the first-order partial derivative of (10) with respect to
τ and follow a similar
process as in proposition 1. We have
∂δ
n( d 2 )
>0
=
∂τ 2 τ [1 + N (d1 )]
(13)
Accounting and Business Review
Vol.4.No.1,(Jan. 1997) 91-103
Proof for Proposition 3:
It is known that
σ ∈ [0, 1], τ ∈[0, ∞ ]. From proposition 1 and 2, we also know
that the discount, δ , is a monotonically increasing function of the volatility, σ , and of the
remaining time period, τ . Therefore, the minimum value of δ is reached at σ = 0 and τ = 0,
and the maximum value of δ is at σ =1 and τ = ∞ .
Based on (10), when σ = 0 and τ = 0, we have d2 = d1 and, thus,
N(d2)=N(d1).
(14)
Substituting (14) into equation (8), we get δ = 0.
At the other extreme, when σ =1 and τ = ∞ , we have
7
1
Ln(1 − δ ) + σ 2τ
2
= ∞,
d1 =
σ τ
1
Ln(1 − δ ) − σ 2τ
2
= −∞.
d2 =
σ τ
These correspond to
N(d1)=1, and N(d2)=0.
Substituting (15) into equation (8) reaches
(15)
δ = 0.5.
Proof for Proposition 4:
Let f( δ ) be equal to the left side of equation (10). That is,
f( δ ) = δ + ( δ - 1) N(d1 )+ N(d2).
(16)
Given any two feasible values σ and τ , it is necessary to show that there exists one and only
one value of δ which makes f( δ ) equal to zero. Proposition 3 shows that 0 ≤ δ ≤ 0.5.
Therefore, the proof is done if it can be shown that f( δ ) is a monotonically increasing and
concave function of δ , and f(0) < 0 and f(0.5) >0.
Take the first- and the second-order derivatives of f( δ ) with respect to δ , respectively.
We have
f ' (δ ) = 1 + N（d1）+(δ − 1)n(d1 )
∂d1
∂d
+ n( d 2 ) 2
∂δ
∂δ
∂
(d1 − d 2 )
∂δ
= 1 + N（d1）>0, and
= 1 + N（d1）-n(d 2 )
7
It can be shown that
δ
= 0 can be achieved at either
σ
= 0 or
τ
= 0. The intuition behind this
is that when the standard deviation of the difference between the returns on closed-end fund and its
net asset value is zero, there is no discrepancy between the market price of the closed-end fund and its
fundamental value, net asset value; when the closed-end fund is to be open-ended or liquidated
immediately, the discount of the closed-end fund from its net asset value disappears.
Accounting and Business Review
Vol.4.No.1,(Jan. 1997) 91-103
f ''(δ ) = −
n(d1 )
< 0.
σ (1 − δ ) τ
(17)
In addition,
1
1
f (0) = N (− σ τ ) − N ( σ τ )
2
2
1
= −[2 N ( σ τ ) − 1] < 0, and
2
f (0.5) = 0.5 − 0.5 N (d1 ) + N (d 2 )
= 0.5[1 − N (d1 )] + N (d 2 ) > 0.
Thus, there exists a unique solution of
(18)
δ to equation (10).
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