[R-SIG-Finance] Simulated expected return for a hedge fund
G See
gsee000 at gmail.com
Sun Dec 25 05:31:50 CET 2011
David,
I attempted to work the problem myself, and have concluded that it is
a poor question. :-P
The question says that you will sell for a profit of _at least_
$100,000, if the value of the stock
rises to at least $1,100,000 or sell for a loss if the value falls
below $950000.
I interpret that to mean that on the first day that the stock value is
at or above 1.1 million, I sell
at the prevailing price (which may, and probably will, be higher than
1.1 million). Likewise, I think
it says that I will sell on the first day that the value of the stock
is below $950,000. But, if the
stock gets to $950,000, I'm out of capital. It is unlikely that the
value of the stock will be
exactly $950,000, which means that I would be losing more than 100
percent of my capita. I
don't know how to convert an N-day return that is less than negative
100 percent to either a daily
or annual return. So, I just divided it by the number of days. i.e.
if you lose 110% over 10 days
then you're losing 11% per day on average. (But, I'm aware that that
doesn't make sense)
My attempt is below; I came up with an "answer" of -0.02197322
niter = 10000 ## number of iterations
set.seed(2009)
returns <- rep(0, niter)
for (i in 1:niter) {
r = rnorm(100, mean = .05/253, sd = .23/sqrt(253))
SV <- 1e6 * exp(cumsum(r)) #StockValue
daily.ret <- if (min(SV) > 950000 && max(SV) < 1100000) {
#between
SellPrice <- tail(SV, 1) # Sell on last day (100 returns in the future)
ret <- (SellPrice - 1000000) / 50000
ret / 100 # average daily return
} else {
# Sell at first value that is less than 950000 or greater than 1100000
SellPrice <- head(SV[SV < 950000 | SV > 1100000], 1)
days.held <- head(which(SV == SellPrice), 1) # first place
that StockValue equals SellPrice
#if (SellPrice < 950000) SellPrice <- 950000
ret <- (SellPrice - 1000000) / 50000
ret / days.held
}
returns[i] <- daily.ret
}
mean(returns)
# -0.02197322
HTH,
Garrett
On Sat, Dec 24, 2011 at 5:51 PM, Afshartous, David
<d.afshartous at vanderbilt.edu> wrote:
>
>
> Thanks Garrett. I fixed that typo and also put the conditional returns in terms of daily returns and the answer now seems reasonable. The changes to the code are:
>
> ret.above = 1/above.ending.day ## daily return
> ret.below = -1/below.ending.day
>
> ret.between = net/100
>
> which now yields:
> exp.return = mean(above)*mean(ret.above) + mean(below)*mean(ret.below) + mean(between) * mean(ret.between)
> > exp.return
> [1] -0.03455923
>
> Interesting, since the expected profit is positive:
>
> exp.profit = mean(above)*100000 - mean(below)*50000 + mean(between)*(mean(between.ending.value) - 1000000)
> > exp.profit
> [1] 7530.114
>
>
>
>
>
>
>
> ________________________________________
> From: G See [gsee000 at gmail.com]
> Sent: Saturday, December 24, 2011 1:27 PM
> To: Afshartous, David
> Cc: r-sig-finance at r-project.org
> Subject: Re: [R-SIG-Finance] Simulated expected return for a hedge fund
>
> I'm not at a computer, but it looks like if the account value falls from 50,000 to zero, you are counting that as a 0% return instead of -100%?
>
> On Dec 24, 2011 12:58 PM, "G See" <gsee000 at gmail.com<mailto:gsee000 at gmail.com>> wrote:
>
> Sorry, please disregard. I didn't look at your code closely.
>
> On Dec 24, 2011 10:55 AM, "G See" <gsee000 at gmail.com<mailto:gsee000 at gmail.com>> wrote:
> Hi David,
>
> Here is my hint:
>
> 1> set.seed(2009)
> 1> r = rnorm(100, mean = .05/253, sd = .23/sqrt(253))
> 1> logPrice = log(1e6) + cumsum(r) ## 1e6 = initial price
> 1> logPrice
> [1] 13.80340 13.80755 13.80516 13.80164 13.82105 13.79453 13.79750 13.80408 13.78316 13.77246 13.77529
> [12] 13.79100 13.78908 13.79427 13.79268 13.76960 13.78917 13.79897 13.80548 13.80673 13.83131 13.82956
> [23] 13.82829 13.83970 13.84860 13.88746 13.90191 13.89439 13.88666 13.89621 13.92104 13.91894 13.92787
> [34] 13.91217 13.90804 13.91452 13.92865 13.95498 13.98419 13.98820 13.99140 14.00165 13.97519 13.96530
> [45] 13.95029 13.95800 13.94906 13.94807 13.91299 13.90617 13.90202 13.90779 13.89517 13.88682 13.89177
> [56] 13.88537 13.89815 13.87475 13.86916 13.88294 13.87048 13.86705 13.87571 13.86564 13.85744 13.87504
> [67] 13.88990 13.88075 13.87261 13.86023 13.85191 13.85510 13.84572 13.84649 13.85447 13.83746 13.82143
> [78] 13.80111 13.78505 13.79430 13.81151 13.81456 13.80200 13.79527 13.80288 13.76598 13.79024 13.81018
> [89] 13.80240 13.77962 13.78402 13.78493 13.80514 13.79839 13.77328 13.78443 13.78491 13.77488 13.75898
> [100] 13.74589
>
> That's probably not what you intended ;-)
>
> Garrett
>
> On Fri, Dec 23, 2011 at 2:43 PM, Afshartous, David <d.afshartous at vanderbilt.edu<mailto:d.afshartous at vanderbilt.edu>> wrote:
>
> All,
> Below is an R-lab problem from Ch2 of Statistics and Data Analysis for Financial Engineering (Ruppert, 2010). The R code runs okay but the answer seems strange. Any hints much appreciated.
> Cheers,
> David
>
> David Afshartous, Ph.D.
> Research Associate Professor
> PSTAT®: ASA Accredited Professional Statistician
> Department of Biostatistics
> Vanderbilt University Medical Center
>
>
> ## suppose Hedge fund owns $1,000,000 of stock and used $50,000 capital and
> ## $950,000 borrowed money for the purchase.
> ## Daily log returns on the stock have a mean of 0.05/year and SD of 0.23/year,
> ## which can be converted to per trading day by dividing by 253 and sqrt(253)
>
> ## the hedge fund will sell the stock for a profit of at least $100,000 if the value
> ## of the stock rises to at least $1,100,000 at the end of one of the first 100 trading days,
> ## sell it for a loss if the value falls below $950,000 at the end of one of the first
> ## 100 trading days, or sell after 100 trading days if the closing price has stayed between
> ## $950,000 and $1,1000,000.
>
> ## PROBLEM 7: What is the expected return? When answering this question, remember
> ## that only $50,000 was invested. Also, the units of return are time, e.g.,
> ## one can express a return as a daily return or a weekly return. Therefore,
> ## one must keep track of how long the hedge fund holds its position before selling
>
> The code below simulates whether value falls below 950,000, goes above 1,1000,000, or stays between. The amount lost/gained in the 1st/2nd scenario is a constant, whereas the amount lost/gained in the third scenario is random.
>
> ### simulation code; written for readability as opposed to speed
> niter = 10000 ## number of iterations
> below = rep(0, niter) ## set up storage
> above = rep(0, niter)
> between = rep(0, niter)
> between.ending.value = NULL
> below.ending.day = NULL
> above.ending.day = NULL
> set.seed(2009)
> for (i in 1:niter) {
> r = rnorm(100, mean = .05/253, sd = .23/sqrt(253))
> logPrice = log(1e6) + cumsum(r) ## 1e6 = initial price
> minLogP = min(logPrice) ## min price over next 45 days
> maxLogP = max(logPrice) ## min price over next 45 days
> crossBelowLogP.day = ifelse( minLogP < log(950000), min(which(logPrice < log(950000))), 101) ## 101 means doesn't cross
> crossAboveLogP.day = ifelse( maxLogP > log(1100000), min(which(logPrice > log(1100000))), 101)
> if ( (minLogP >= log(950000)) & (maxLogP <= log(1100000)) ) { ## Neither cross boundary
> between[i] = 1
> between.ending.value = append(between.ending.value, exp(logPrice[100]))
> }
> if ( (minLogP < log(950000)) & (maxLogP > log(1100000))) { ## Both cross
> ifelse (crossBelowLogP.day < crossAboveLogP.day, below[i] <- 1, above[i] <-1)
> ifelse (crossBelowLogP.day < crossAboveLogP.day, below.ending.day <- append(below.ending.day, crossBelowLogP.day),
> above.ending.day <- append(above.ending.day, crossAboveLogP.day))
> }
> if ( (minLogP < log(950000)) & (maxLogP <= log(1100000)) ) { ## Only cross below
> below[i] = 1
> below.ending.day = append(below.ending.day, crossBelowLogP.day)
> }
> if ( (minLogP >= log(950000)) & (maxLogP > log(1100000)) ) { ## Only cross above
> above[i] = 1
> above.ending.day = append(above.ending.day, crossAboveLogP.day)
> }
> }
> mean(above) ## this will give the probability of the value crossing the above threshold
>
> When annualizing returns, the day in which a boundary needs to be accounted for. The following code calculates the overall expected return by conditioning on each scenario:
>
> # cross above: always make constant 100,000 based on initial investment of 50,000
> # P_t/P_t-1 = 100,000/50,000 = 2 = gross return --> net return = 2 - 1 = 1
> ret.above = (1 + 1)^(365/(above.ending.day)) - 1 ### annualized
>
> # cross below: always lose constant 50,000 based on initial investment of 50,000
> # P_t/P_t-1 = 50,000/50,000 = 1 = gross return --> net return = 1 - 1 = 0
> ret.below = 0
>
> ## betwen: final value depends on realized value on 100th day
> # P_t/P_t-1 = between.ending.value/50,000 = --> net return = gross return - 1
>
> net = ifelse(between.ending.value < 1000000,
> (1000000 - between.ending.value)/50000 - 1, ## net return, gain
> (between.ending.value - 1000000)/50000 - 1) ## net return, loss
>
> ret.between = (1 + net)^(365/100) - 1
>
> exp.return = mean(above)*mean(ret.above) + mean(below)*mean(ret.below) + mean(between) * mean(ret.between)
> 8.669104e+50???
>
> _______________________________________________
> R-SIG-Finance at r-project.org<mailto:R-SIG-Finance at r-project.org> mailing list
> https://stat.ethz.ch/mailman/listinfo/r-sig-finance
> -- Subscriber-posting only. If you want to post, subscribe first.
> -- Also note that this is not the r-help list where general R questions should go.
>
>
More information about the R-SIG-Finance
mailing list