I know that after reading page 376 in the Equity book all of you have had a hard time sleeping. How the heck do they move from:
V0 = (E1+B0-B1)/(1+r) + (E2+B1-B2)/(1+r)^2 + …
to:
V0 = B0 + (E1+r*B0)/(1+r) + …
And the worst question of all, what happens to B3?
Rest easy here comes the derivation for the first term:
V0 = (E1+B0-B1)/(1+r) = B0/(1+r) + (E1-B1)/(1+r) = [B0 + B0(1+r) - B0(1+r)]/(1+r) + (E1-B1)/(1+r)
= B0 + [B0 -B0(1+r)]/(1+r) + (E1-B1)/(1+r) = B0 -r*B0/(1+r) + (E1-B1)/(1+r) = B0 + [E1-r*B0]/(1+r) -B1/(1+r)
The -B1/(1+r) term is then extended with (1+r) : -B1(1+r)/(1+r)^2 which gives for the next term:
[E2+B1-B1(1+r)-B2]/(1+r)^2 = [E2-r*B1]/(1+r)^2 -B2/(1+r)^2
The -B2/(1+r)^2 term is then extended with (1+r) and so on and so on..
Which means that the last term does not dissapear, so the formula is mistated, but it normally works out because like in the example 4, the last year its a liqudidation which means last bookvalue (B3 in this example is zero).
Feels much better now doesn’t it?
V0 = (E1+B0-B1)/(1+r) + (E2+B1-B2)/(1+r)^2 + …
to:
V0 = B0 + (E1+r*B0)/(1+r) + …
And the worst question of all, what happens to B3?
Rest easy here comes the derivation for the first term:
V0 = (E1+B0-B1)/(1+r) = B0/(1+r) + (E1-B1)/(1+r) = [B0 + B0(1+r) - B0(1+r)]/(1+r) + (E1-B1)/(1+r)
= B0 + [B0 -B0(1+r)]/(1+r) + (E1-B1)/(1+r) = B0 -r*B0/(1+r) + (E1-B1)/(1+r) = B0 + [E1-r*B0]/(1+r) -B1/(1+r)
The -B1/(1+r) term is then extended with (1+r) : -B1(1+r)/(1+r)^2 which gives for the next term:
[E2+B1-B1(1+r)-B2]/(1+r)^2 = [E2-r*B1]/(1+r)^2 -B2/(1+r)^2
The -B2/(1+r)^2 term is then extended with (1+r) and so on and so on..
Which means that the last term does not dissapear, so the formula is mistated, but it normally works out because like in the example 4, the last year its a liqudidation which means last bookvalue (B3 in this example is zero).
Feels much better now doesn’t it?
