DMDPS2.Rmd

---
title: 'DMD Problem Set #2'
author: "Harold Kyle"
output:
  pdf_document:
    fig_height: 7
    fig_width: 8
    toc: yes
---

#Introduction
This is my time using R, so please excuse the roughness of this. 

I am using R Markdown to combine inline text with R. 

Code is online here:

[R code](https://gist.github.com/haroldkyle/6cb7ab44162c352b5208)

[R Markdown code](https://gist.github.com/haroldkyle/432e6f4eb714ed03191c)



# 2.16
```{r}
sailboats <- c(2,3,4,5,6,7)
prob <- c(0.15,0.20,0.30,0.25,0.05,0.05)
```
## a)
### Mean:
```{r}
sum(sailboats*prob)
```
### Standard Deviation:
```{r}
sd(sailboats*prob)
```

## b)

### Mean:
```{r}
bmean <- sum(sailboats*prob)*4800+30000
print(bmean)
```
### Standard Deviation:
Fixed costs do not affect variation or standard deviation.
```{r}
bsd <- sd(sailboats*prob*4800)
print(bsd)
```

## c)
### Mean:
Increase by the difference of the fixed costs
```{r}
cmean <- bmean + 53000-30000
print(cmean)
```

### Standard Deviation:
Fixed costs do not affect variation, so this doesn't change from b.
```{r}
csd <- bsd
print(csd)
```

double check:
```{r}
sum(sailboats*prob)*4800+53000 == cmean
sd(sailboats*prob*4800) == csd
```

# 2.22

*Large store (L)*

E(L) = 147.8 

Std(L) = 51 

Var(L) = Std(D)^2
```{r}
51^2
```
	a (price) = $17

*Discount store (D)*

E(D) = 63.2
  
Std(D) = 37
	
Var(D) = Std(D)^2
```{r}
37^2
```
b (price) = $9

Corr(L,D) = 0.7

mean revenue is the sum of the means
E[L+D] = E[L] + E[D]

## Revenue:

Revenue[L,D] = aE[L] + bE[D]
```{r}
(147.8*17)+(63.2*9) 
```

## Variance:

Var(L+D) = a^2\*Var(L) + b^2\*Var(D) + 2ab\*Std(L)\*Std(D)*Corr(L,D)
```{r}
17^2*2601 + 9^2*1369 + 2*17*9*51*37*0.7
```

## Standard Deviation:

Std(L+D) = sqrt( Var(L+D) )
```{r}
sqrt( 1266773.4 )
```

# 2.24

First I will load some libraries and define some helper functions...

```{r, echo=FALSE}
# install.packages("ggplot2")
# install.packages("reshape")
# install.packages("gridExtra")

require(ggplot2)
require(reshape2)
library(gridExtra)
```

```{r}
### ASSUMPTIONS ###

aReturn <- 0.15
bReturn <- 0.20
aStd <- 0.05
bStd <- 0.06

# correlations that we want to test 
corrs = seq(from=-0.6, to=0.6, by=0.3)


### FUNCTIONS ###

# dmd "namespace"
dmd = new.env()

dmd$ValidateRatio <- function(ratio){
  # Ratios of asset a to asset b need to be pairs of numbers that add to 1 (100%) 
  if( length(ratio) != 2 || ratio[1] + ratio[2] != 1 ){
    stop("Ratios need to be pairs of values that add to 1.")
  }
}

dmd$GetRatioLabel <- function(ratio){
  # concatenate ratio values to create a label
  # Args:
  #   ratio between asset a and asset b, list of two values whose sum is 1  b<-ratio[2]
  dmd$ValidateRatio(ratio)
  a<-ratio[1]
  b<-ratio[2]
  paste(round(a,2),":",round(b,2))
}

dmd$GetExpectedReturn <- function(ratio, aReturn, bReturn){
  # finds the mean (expectation) for a given asset ratio
  # Args:
  #   ratio: Ratio between asset a and asset b, list with two values whose sum is 1
  #   aReturn: Expected return on asset a
  #   bReturn: Expected return on asset b
  dmd$ValidateRatio(ratio)
  a<-ratio[1]
  b<-ratio[2]
  a*aReturn+b*bReturn
}

dmd$GetCombinedStd<-function(ratio, aReturn, bReturn, aStd, bStd, corr){
  # find the standard deviation for a given asset ratio combination and correlation
  # Args:
  #   ratio: Ratio between asset a and asset b, list with two values whose sum is 1
  #   aReturn: Expected return on asset a
  #   bReturn: Expected return on asset b
  #   aStd: Standard deviation of asset a
  #   bStd: Standard deviation of asset b
  #   corr: Correlation between asset a and asset b
  dmd$ValidateRatio(ratio)
  a<-ratio[1]
  b<-ratio[2]
  
  # standard deviation is square root of the combined variances
  sqrt( a^2*aStd^2 + b^2*bStd^2 + 2*a*b*aStd*bStd*corr )
}
```

## a) 

### Mean:
```{r}
dmd$GetExpectedReturn( c(0.5,0.5), aReturn, bReturn ) 
```
### Standard Deviation:
```{r}
dmd$GetCombinedStd( c(0.5,0.5), aReturn, bReturn, aStd, bStd, 0.3)
```

## b) 

### Mean:
The mean will remain the same because it is not affected by changes in correlation.

### Standard Deviation:
```{r}
dmd$GetCombinedStd( c(0.5,0.5), aReturn, bReturn, aStd, bStd, 0.6)
dmd$GetCombinedStd( c(0.5,0.5), aReturn, bReturn, aStd, bStd, -0.6)
dmd$GetCombinedStd( c(0.5,0.5), aReturn, bReturn, aStd, bStd, -0.3)
dmd$GetCombinedStd( c(0.5,0.5), aReturn, bReturn, aStd, bStd, -0)
```

## c) and d)

```{r}

### DATA ###

# set up ratios of asset A to asset B.
aamt=seq(from=0,to=1,by=0.05)

# As percentages, these ratios will always combine to equal 1
bamt=1-aamt

# merge ratio values into a single vector
assets = cbind(aamt, bamt)

# add labels for ratios (to display alongside data and in graph)
ratioLabels <- apply( assets, 1, dmd$GetRatioLabel )

# create a data frame combinedData to contain data, with ratio labels as the first column
combinedData <- data.frame( label = ratioLabels )

# find the expected return for each asset blend
expectedReturns = apply(assets, 1, dmd$GetExpectedReturn, aReturn, bReturn)

# add expected returns to the combinedData data frame as the "mean" column
combinedData <- cbind(combinedData, mean = expectedReturns)

# iterate through correlations to test each at every asset ratio
for(corr in corrs){
  
  # add data for each correlation to our data frame
  combinedData <- cbind(combinedData, apply(assets, 1, dmd$GetCombinedStd, aReturn, bReturn, aStd, bStd, corr))
  
  # label the column with the correlation value
  idx = length(combinedData)
  colnames(combinedData)[idx] <- paste("Std @",corr, " Corr")
}
```

```{r}
### TABLE OF DATA ###

grid.table(combinedData,gp=gpar(fontsize=7, lwd=2))
```

```{r}
 
### GRAPHING ###

# melt data for graphing
melted = melt(combinedData, id.vars="label")

# ...and graph it!
ggplot() + geom_line(data=melted, aes(label,value,group=variable,color=variable)) + theme(axis.text.x = element_text(hjust=1, angle=45)) + xlab("Ratio of asset a to asset b")
```