# Annual Percentage Growth Rate

• Jul 31st 2008, 06:19 PM
Archimedes
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.
• Jul 31st 2008, 09:55 PM
CaptainBlack
Quote:

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

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

So after some algebra we have:

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

RonL
• Jul 31st 2008, 10:03 PM
CaptainBlack
Quote:

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 $\displaystyle r\%$ annual rate we have:

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

where time is measured in years. So:

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

or in our case:

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

and after some algebra:

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

RonL
• Aug 1st 2008, 01:39 AM
jonah
Quote:
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
$\displaystyle F = P\left( {1 + \frac{j} {m}} \right)^{tm} \Leftrightarrow j = m\left[ {\left( {\frac{F} {P}} \right)^{\frac{1} {{tm}}} - 1} \right] \approx .0734569906$
(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
$\displaystyle F = Pe^{jt} \Leftrightarrow j = \frac{{\ln \left( {\frac{F} {P}} \right)}} {t} \approx .07088427289$
• Aug 1st 2008, 06:10 AM
CaptainBlack
Quote:

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 (Wondering)

RonL