Perp Twist
You are asked to find the value (present value) today, , of an asset generating a cash flow of starting at time . The cash flow is expected to grow by a constant every year in perpetuity, starting at . The relevant rate of return is .
Underneath is a table of the first 3 years:
HINT: You only need to use the perpetuity formula:
The answer is 30000.
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1 solution
Usually whenever there's growth in a perpetuity it's expressed as a percentage g , which is subtracted from r in the perpetuity function as seen in the Gordons Growth model :
P V t = r − g C F t + 1
However in this case the growth is not a steady percentage, but instead a constant amount - so we need to use a different trick.
It's easier to solve the problem if you visualize it in a table like so:
However it does need rearrangement to gain the insight needed the solve the problem:
Above I've given all the cash flows a separate row, from this we now see that each year when we get the extra 200, it's as if a new perpetuity is starting. All rows can then be valued using the perpetuity formula, for example the 1000 row is just P V 0 = 0 . 1 1 . 0 0 0 :
Doing so we can see that all the individual present values of the rows containing the 200 in growth actually forms a perpetuity of 2000 starting at t = 1 . To finish it off we just need to use the perpetuity formula and add this to the present value of the 1000 perpetuity:
P V 0 = 1 0 . 0 0 0 + 0 . 1 2 . 0 0 0 = 3 0 . 0 0 0