## Chapter 7 graphs and equations

damianow
Posts: 3
Joined: Sun Oct 23, 2011 1:53 am
For anyone that has a copy of the book around, if you could follow me:
Figure 7.12 is equation 7.10 graphed (N as a function of P0)

Figure 7.13 is equation 7.15 graphed (M as a function of r)
But what equation represents figure 7.14? It is also M as a function of r. I don't understand what variable is replaced in equation 7.15 to get to this graph, and how you incorporate equation 7.16. I am an engineer graduate, and a few friends of mine sat around and tried to figure this out. Both figures 7.13 and 7.14 are different, but are graphing the same parameters.
Does it have to do with P0? P0 should represent the size of the fund at the end of the period worked such that it should last until the end of your life (or N+M=80). Should I replace P0 in equation 7.15 with N in equation 7.10? How does the constraint equation 7.16 get used?

jacob
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Fig 7.14 is eqs 7.10 and 7.15 with the constraint of 7.16. This is one equation (just "7.10"+"7.15"=80) with one parameter, i, one independent, r, and one unknown P0/p which must be solved for. Once you have P0/p, you insert it back into 7.15 which gives you M. Then you have your plot.
I found P0/p numerically.
The difference between figs 7.13 and 7.14 is that fig 7.13 assumes P0/p=30 for all values. This constraint is replaced by 80=N+M in fig 7.14.

damianow
Posts: 3
Joined: Sun Oct 23, 2011 1:53 am
I used Goal Seek in Excel to manipulate 80=(eq 7.10)+(eq 7.15) to determine a P0/p for any given r and i. I plugged in this P0/p back into equation 7.15 to determine M. For validation, I plugged the same P0/p back into 7.10 to determine N. What I should find is that N and M at this point should still add up to 80, but it did not.
Given r=0.4, i=0.1, and N+M=80; P0/p = 10.9, M = 10.2 and N = 50.1. Please advise. Thanks Jacob!

damianow
Posts: 3
Joined: Sun Oct 23, 2011 1:53 am
Ignore last post, I figured it out. Thanks a lot Jacob!

Fish
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Joined: Sun Jun 12, 2016 9:09 am

### Re: Chapter 7 graphs and equations

The chapter 7 equations don't render well in Kindle version of the ERE book. It's not hard to figure out what was intended, but for convenience I'm writing some of the more useful results for future reference. I matched the formatting of the paper ERE book... obtained from the library after a 5-month wait.

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P_N = P_0(1+i)^N - p\frac{(1+i)((1+i)^N - 1)}{i} \tag{7.8} \label{eq:7.8}

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N = \log\bigg[\frac{1}{1-\frac{P_0}{p}\frac{i}{1+i}}\bigg]\Big/\log(1+i) \tag{7.10} \label{eq:7.10}

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P_0/p = \frac{r}{1-r} \frac{(1+i)^M - 1}{i} \tag{7.14} \label{eq:7.14}

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M = \log\bigg(1+i \frac{P_0}{p}\frac{1-r}{r}\bigg) \Big/ \log(1+i) \tag{7.15} \label{eq:7.15}