Fix wording

efficient_frontier.ipynb

@@ -10,7 +10,7 @@
     "\n",
     "![frontier image](https://upload.wikimedia.org/wikipedia/commons/e/e1/Markowitz_frontier.jpg)  \n",
     "\n",
-    "The following code implements this concept in Python, and is based on a [blog post](http://blog.quantopian.com/markowitz-portfolio-optimization-2/) from Quantopian.  Their original code returns an optimal portfolio for any given level of risk when given a list of historic returns for a group of assets.  I then extend the concept of an efficient frontier to the area charitable giving, and refactor the code to take a list of charitable returns (in DALYs) and a covariance matrix as an input.  The final output is a philanthropic efficient frontier.  \n",
+    "The following code implements this concept in Python, and is based on a [blog post](http://blog.quantopian.com/markowitz-portfolio-optimization-2/) from Quantopian.  Their original code returns an optimal portfolio for any given level of risk when given a list of historic returns for a group of assets.  I then extend the concept of an efficient frontier to the area charitable giving, and refactor the code to take a list of charitable returns (in DALY/\\$) and a covariance matrix as an input.  The final output is a philanthropic efficient frontier.  \n",
     "\n",
     "The following is a rough sketch of my concept, feel free to critique and contribute!\n",
     "\n",
@@ -604,7 +604,7 @@
     "\n",
     "- Using the standard deviation assumes a normal distribution.  There are versions of MPT that use other distributions that might be useful here.  \n",
     "\n",
-    "- There is very little reason to be risk averse when it comes to philanthropy because we shouldn't care if the returns accrue to us.  Someone else benefits, so why not take the riskiest option with the highest expected value? \n",
+    "- There is very little reason to be risk averse when it comes to philanthropy because the returns don't accrue to us.  Someone else benefits, so why not take the riskiest option with the highest expected value? \n",
     "\n",
     "A more complete list of criticisms to Modern Portfolio Theory is available [here](https://en.wikipedia.org/wiki/Modern_portfolio_theory#Criticisms)."
    ]
@@ -617,7 +617,7 @@
    "source": [
     "# Conclusion\n",
     "\n",
-    "This approach might be a useful way to come up with an optimal portfolio for charitable giving.  This allows us to not only combine DALY/\\$ measures, but also account for the uncertainty involved in imperfect estimates.  The next step for this analysis is to find a real world dataset that incorporates both DALY/\\$ estimates along with sensitivity analyses.  I theorize that this dataset whould show that there are returns to risk taking in charitable giving, which would be demonstrated by an upward sloping efficient frontier.  \n",
+    "This approach might be a useful way to come up with an optimal portfolio for charitable giving.  This allows us to not only combine DALY/\\$ measures, but also account for the uncertainty involved in imperfect estimates.  The next step for this analysis is to find a real world dataset that incorporates both DALY/\\$ estimates along with their variances.  I think it's likely this dataset whould show that there are returns to risk taking in charitable giving.  \n",
     "\n",
     "The DALY is an imperfect measure, as there are some things that are very hard to quantify, but it is still a worthwhile exercise to try to come up with back of the envelope calculations to partially inform the decision making process.  Note that this approach isn't necessarily tied to the DALY/\\$, as any cross-intervention measure of social return could be used.  In some cases back of the envelope calculations could actually harm our judgement by giving us a false sense of certainty.  One way to guard against this risk is to only use this as a part of the decision making process, rather than relying solely on this type of analysis.   \n"
    ]
@@ -630,7 +630,7 @@
     "\n",
     "### Real Datasets\n",
     "\n",
-    "Although the above example shows a theoretical upward slope between risk and return, it should be possible to use a few existing datasets to find if this relationship holds in reality.  \n",
+    "Although the above example shows a theoretical relationship between risk and return, it should be possible to use a few existing datasets to find if this relationship holds in reality.  \n",
     "\n",
     "Here are some potential data sets to test this theory:\n",
     "\n",