# Math Help - Annual Percentage Growth Rate

1. ## Annual Percentage Growth Rate

The value of an asset rose from £110,000 to £694,694 in 26 years.

Calculate the average annual percentage growth rate of the value of the asset assuming:

(a) growth in annual discrete 'jumps'

(b) continuous growth

Any help would be greatly appreciated.

2. Originally Posted by Archimedes
The value of an asset rose from £110,000 to £694,694 in 26 years.

Calculate the average annual percentage growth rate of the value of the asset assuming:

(a) growth in annual discrete 'jumps'
The value grows from $v$ to $(1+r/100)v$ in $1$ year, so after $26$ years will have grown from $v$ to $(1+r/100)^{26} v$, where r is the percentage annual growth rate. So using the numbers in the question we have:

$694694 = (1+r/100)^{26} 110000$

So after some algebra we have:

$r=\left(\left(\frac{694694}{110000}\right)^{1/26}-1\right)\times 100$

RonL

3. Originally Posted by Archimedes
The value of an asset rose from £110,000 to £694,694 in 26 years.

Calculate the average annual percentage growth rate of the value of the asset assuming:

(a) growth in annual discrete 'jumps'

(b) continuous growth
With continuous growth at $r\%$ annual rate we have:

$
\frac{dv}{dt}=\frac{r}{100} \ v
$

where time is measured in years. So:

$v=e^{(r/100)t}v_0$

or in our case:

$
694694=e^{(r/100)26}\ 110000
$
,

and after some algebra:

$
r=\ln\left( \frac{694694}{110000} \right) \times \frac{100}{26}
$

RonL

4. So after some algebra we have:

,

and after some algebra:

There's a slight typo on CaptainBlack's part. 100000 in both cases should have been 110,000.
Equivalently,
(a)
P = £110,000
F = £694,694
j = nominal rate compounded annually [average annual percentage growth rate of the value of the asset assuming: (a) growth in annual discrete 'jumps']
m = 1 (as in annually)
t = 26 years
$
F = P\left( {1 + \frac{j}
{m}} \right)^{tm} \Leftrightarrow j = m\left[ {\left( {\frac{F}
{P}} \right)^{\frac{1}
{{tm}}} - 1} \right] \approx .0734569906

$

(b)
P = £110,000
F = £694,694
j = nominal rate compounded continuously [average annual percentage growth rate of the value of the asset assuming: (b) continuous growth]
t = 26 years
$
F = Pe^{jt} \Leftrightarrow j = \frac{{\ln \left( {\frac{F}
{P}} \right)}}
{t} \approx .07088427289

$

5. Originally Posted by jonah
There's a slight typo on CaptainBlack's part. 100000 in both cases should have been 110,000.
Equivalently,
(a)
P = £110,000
F = £694,694
Deliberate error to see if everyone was awake

RonL