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Musharakah Mutanaqisah PartnershipA Mathematical Derivation
Note
P = Price of asset, e.g. a home B0 = Financiers contribution
into the partnershipC 0 = Customers contribution into the
partnership
Therefore, 00 C B P +=
R = Periodic rental, eg. monthly A = Additional periodic payment
by customer to redeem the financiers equity faster
M = R + A , is therefore, the total periodic payment
et iC = the customers equity !o"nership# of the asset in period
i
et the proportion of customers equity in period i, P
C r i
i =
Therefore,
00 C C =
A Rr C C ++= 00$ A Rr C C ++= $$% A Rr C C ++= %%&
.
.
. A Rr C C
nnn ++= $$
Therefore,
00 C C =
A Rr C C ++= 00$ Ar r RC A Rr A Rr C C %#! $00$00% +++=++++=
Ar r r RC A Rr A Rr A Rr C C ! %$00&$00&
++++=++++++=...
nAr r r r RC C nn ++++++= #....! $%$00nAC C C C
P
RC C
nn ++++++= #....! $%$00 'ince P
C r i
i =
et P
R x = , then
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AC x A xC C C ++=++= 000$ #$!
( ) A xC x x A A xC C C xC C #%!#%$!% 0%0000% ++++=+++++=
( )( ) A x xC x x x
A A A xC C C C A xC C C xC C
#&&!#&&$!&%
%0
&%
00000000&
++++++=+++++++++++=
.
.
.Therefore,
AC xC ++= 0$ #$!
A xC xC #%!#$! 0%
% +++=
A x xC xC #&&!#$!%
0&
& ++++= A x x xC xC #()(!#$! %&0
(( +++++=
.
.
.
A x
xC xC
nn
n
+++= $#$!#$! 0 !$#
and, of course, the proportion of the customers equity in the n
th period is P
C r n
n =
*e"riting equation !$#, the number of periods ta+en by the
customer to fully o"n the house is gi en by,
"here P C n = ,
x
AC
x
A P
x
A x x
AC x
A x
A x
xC x P
n
n
nn
++=+
++=
+++=
0
0
0
#$!
$#$!
$#$!#$!
#$ln!
lnln 0
x
x
AC
x
A P
n+
+
+
= !%#
%
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-nce the rental, R, has been determined and the customer has
decided on the period of partnership, i.e. then, then the periodic
amount the customer has to top up additionally is gi en by
$#$!
#$! 0
++
= nn
x
C x P x A
!
and, the formula for determining the periodic payment is gi en
by
[ ] [ ]
[ ]
$#$!#$!
$#$!#$!
'ince $#$!
#$!#$!
$#$!#$!#$!
$#$!#$!$#$!
0
0
0
0
0
++=
++=
=++++=
++++=
++++=
+=
n
n
n
n
n
nn
n
nn
n
nn
x
B x x
x
C P x x
R xP x
C x x R R P x x
x
C x x xP R x R
x
C x P x x R
A R M
$#$!#$! 0
++= n
n
x
B x x M !(#
"hich, interestingly, is similar to the normal annuity formula
used for computing the payment incon entional loan calculations.
Hence, mathematically, the normal annuity formula can also be
usedfor Mushrakah Mutankisah calculations, but the periodic
interest rate is replaced by the rental
rate, P
R x = . ndeed then, the periodic rate of return for Mushrakah
Mutankisah Partnership is solely
determined by the rental rate, P
R x = . Therefore, the
nternal *ate of *eturn ! **# to ban+ P
R= !/#
Also,
Total payment made to financier = Mn !)#
Total profit to financier = 0 B Mn ! #
&
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Note: 'ince the rate of return ! **# for the financier is solely
determined by the rental rate, P
R x = ,
irrespective of the initial capital pro ided by the financier !
0 B # and1or the duration of the partnership ! n #, the financier
may be tempt to finance only homes with high rental values 2"hile
it is in the interest of the customers to negotiate for lo"
rentals. At the e3treme, if the rentalis nil, then the Mushrakah
Mutankisah financing "ill become similar to Qard al-Hassan .
(
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Worked Example
Price of 4ouse = *5%00,000*ent = *5$,000 per month
Conventional Mortgage & BBA Musharakah MutanaqisahAP* =
$067o"npayment = *5%0,000 !$06#oan -* Financing = *5$80,000
!906#7uration = %0 :ears monthly payments
n con entional mortgage the Present ;alue ofAnnuities formula is
used
