A1
A
PPENDIX
B
Finding the Rate of Interest (
i
)
Without Preprogramming
B.1
Finding the Conversion Rate
i
For Simple Annuities
A.
Finding the conversion rate
i
without preprogramming
when the future value of a simple annuity is known
Finding the periodic rate of interest when FV
n
, PMT, and
n
are known is a rather
awkward problem. Substituting the known values in Formula 11.1 or the general
solution of the formula for
i
results in an exponential equation that is difficult to
solve.
For example, if FV
n
5
25 000.00, PMT
5
500.00, and
n
5
32, the substitution in
Formula 11.1 gives
25 000.00
5
500.00
1
}
[(1
1
i
)
i
32
2
1]
}
2
50.00
5
}
[(1
1
i
)
i
32
2
1]
}
50.00
i
5
(1
1
i
)
32
2
1
(1
1
i
)
32
2
50.00
i
2
1
5
0
However, instead of trying to solve the equation, the value of
i
can be found by
using an approximation method based on trial and error. A value for
i
is arbitrar
ily selected and the compounding factor
}
(1
1
i
i
)
n
2
1
}
is computed. The numerical
value of the factor indicates whether the rate to be found is more or less than the
selected rate. The process of choosing a rate, computing the compounding factor,
and comparing this factor with the actual compounding factor is repeated until an
approximation sufficiently close to the actual rate is obtained.
To avoid writing the accumulation factor repeatedly, the factor
}
(1
1
i
i
)
n
2
1
}
is
represented by the symbol
s
n
B .
i
(read “
s
angle
n
at
i
”).
}
(1
1
i
i
)
n
2
1
} 5
s
n
B .
i
FV
n
5
PMT
s
n
B .
i
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EXAMPLE B.1A
Find the nominal annual rate of interest at which $200.00 deposited at the end of
each quarter for fifteen years will amount to $20 000.00.
SOLUTION
FV
n
5
20 000.00;
PMT
5
200.00;
n
5
15(4)
5
60
s
n
B .
i
5
}
(1
1
i
)
i
60
2
1
} 5
}
20
20
0
0
00
} 5
100.00
We want to find the quarterly rate
i
for which the accumulation factor
s
60
BB
.
i
5
100.0000. The first selection of
i
should allow for a reasonable range within
which the nominal annual rate might be expected to fall. For most practical
purposes, this range is 6% to 20%. Thus, we will try 12% as our initial selection.
Try
i
5
}
12
4
%
}
5
3%
5
0.03.
s
60
.
3%
5
}
(1.0
0
3
.
)
0
60
3
2
1
} 5
163.05344
Since
s
60
.
3%
5
163.05344 is greater than
s
60
.
i
5
100.00, the selected rate of 3% is
greater than the actual rate
i
. The actual nominal annual rate must be less than 12%.
We must now choose a second selection for
i
. This selection should allow for
the relationship between the value of the actual compounding factor and the
factor computed for the first selected value of
i
. It should preferably result in a
compounding factor smaller than 100.00. A considerable drop in rate is
indicated since
s
60
.
3%
is considerably greater than
s
60
.
i
.
Try a nominal rate of 6% where
i
5
1.5%
5
0.015.
s
60
.
1.5%
5
96.214651
This means
s
60
.
1.5%
,
s
60
.
i
, that is,
i
.
1.5%.
By now we know that 1.5%
,
i
,
3%. The nominal annual rate lies between
6% and 12%. Furthermore, because of the closeness of
s
60
.
1.5%
to
s
60
.
i
, the actual
nominal annual rate must be much closer to 6% than to 12%.
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 Fall '05
 TonyCirusolo
 Math, Nominal Interest Rate, nominal annual rate, Conversion rate

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