Calculating Apparent Magnitude of The Moon

[luuurey]

Hello everyone,
I've been wondering if I can calculate the apparent magnitude of The Moon since I know the apparent magnitude of The Sun(say -27mag) and reflection coefficient of Moon's surface(say 0.12).
We know the equation

$\Delta m = -2.5 log_{10}(I/I_0)$

Where I is the flux of the light that comes from source and I_0 is flux of a reference source. I know the magnitude of The Sun, so I can say:

$-27 = -2.5 log_{10}(I_S/I_0)$

Where I_S is the flux that I can measure on the sunlight. I know that the flux of the light from The Moon is just reflected sunlight and hence$I_M=0.12 I_S$, where I_M is the flux of light that comes from The Moon. So I get:

$\Delta m_M = -2.5 log_{10}(I_M/I_0)= -2.5 log_{10}(0.12 I_S/I_0)= -2.5 log_{10}(I_S/I_0)-2.5 log_{10}(0.12) = -27 -(-2)=-25$

I'm sure that the apparent magnitude of The Moon is definitely not 25mag. It's around -13mag. What's wrong with my idea? Where did I make mistake?

Thank you for your answer. I will be very thankful for your help!

 

[phyzguy]

You have assumed that all of the sun's light reflected by the moon makes it to the Earth. Actually, the light is reflected in all directions from the moon's surface, and only a small fraction of the light intercepts the Earth. You need to take this into account.