notes

October 30 - November 12

Meeting Notes:
1. get monthly data for each measure or keep everything the same frequency
2. look at vector autogressives (multivariate AR model)
3. look at correlation between residuals from each ARIMA model

Notes:
ECI is only quarterly. How do we address this?
Residuals from each model don't seem to be correlated.
Medical Inflation forecast seems to have a downward trend, this doesn't feel right...

November 13 - November 19

Meeting Notes:
1. Get Draws of each time series
2. Try ARIMA(1,0,1) for each model and compare AIC
3. Model Monthly CPI and use quarterly CPI to model ECI
4. Get draws from CPI and use those draws to get draws from ECI model

Notes:
Why is a differencing term of 1 favored when modelling log difference of CPI?
What is the intuition behind ECI and Med Inflation being related to CPI? 
  Lag of CPI has higher correlation than CPI
  The models AIC values indicate adding CPI doesn't really help
  
November 20 - November 26

Meeting Notes:
1. Fit models just after the rounding stops (2010ish)
2. Simulate draws from each model and compare Intervals and Mean
3. Recalculate log_dif_cpi for quarterly data and re-check correlation
4. Include lagged and unlagged cpi as covariates (or linear model on difference between cpi and eci)

Notes:
CPI:
Massive outlier for log_dif_cpi in December 2008. CPI drops from 211 to 203 from 09/08 to 12/08
Fitting to data after 2010 shows AIC values lower for ARIMA(1,0,1)
Seasonality model (with differencing for CPI) shows growing confidence bands
When should we start using CPI data?

ECI
I am only using CPI data back to 2010 for the CPI model, but this only provides 58 observations 
  for the ECI and Medical Inflation model.
When using both lagged and unlagged CPI in ARIMA, code throws an error
Some weirdness going on when I fit with lagged values (because of limited data)

December 4 - December 10
Meeting Notes:
1. Run auto-arima using CPI data starting at 2010
2. Run auto-arima for ECI using lagged CPI as predictor
3. Explore correlation between residuals of independent models (lags as well)

Notes:
Decided to stick with AR(1,0,1) for CPI, using data back to 2010 AIC and BIC scores are best
Auto-arima on ECI using lagged CPI as predictor results in ARIMA(0,0,0)(0,1,1)
After comparing all models for ECI, ARIMA(1,1,1) with just lag CPI has lowest AIC and BIC
Explored correlation between residuals of independent models
Medical Inflation: 
 - doesn't seem to be any correlation between CPI and Medical inflation
 - AIC and BIC is lowest for for ARIMA(1,0,1)(0,0,2) but difference between ARIMA(1,0,1) is pretty marginal
 
Meeting Notes:
1. Decided on AR(1,1,1) w/ lag cpi for wage inflation
2. Run individual simulations and plot cpi & eci results to evaluate if they are related in the way we expect
3. Double check CPI is CPI without medical inflation
4. Explore correlation between wage inflation and medical inflation
5. Feel good about AR(1,0,1) for medical, but keep in mind exploration of med and wage
6. Create document documenting process of deciding inflation models

January 8 - January 14
Notes:
When running "simulate" does it use the last value in the data that the model is fit on to determine the starting point?
Documentation for simulate.ets in the forecast package is helpful here
Correlation between the simulations seems to be too low to me
The CPI model is returning log difference cpi for a time frame. 
To create quarterly log_difference_cpi to plug into the eci model I need to back transform the values to cpi and calculate quarterly log_dif_CPI
Not sure how to do this
Didn't find much correlation between wage inflation and medical inflation

January 15 - January 21
Meeting Notes:
1. Fix ECI simulation to have quarterly log difference CPI as predictor
2. Read ESG Primer on Interest Rates (short term interest rates)
4. Clean up inflation code

January 22 - January 28
Meeting Notes:
1. Test different date ranges of CPI to fit model on and compare simulated draws (1990)
2. Get 3-month interest rates and play around with it
3. Explore relationship between interest rate and cpi
4. Make sure dates for cpi and interest rates match up, or are close

Notes:
Changing from 2010 to 1990 doesn't affect CPI draws at all
ECI draws have much wider bands when using data back to 1990
Also lagged_cpi has pretty much same correlation as cpi using 1990 data (using lagged vs unlagged doesn't affect draws)

Meeting Notes:
Correlations between raw inflation values and interest rate values
explore monthly, quarterly, semi annual, and annual inflation with interest rates
test different transformations on interest rates

Meeting Notes:
Look at change in interest vs log difference cpi
Start exploring AR model on interest rates

Meeting Notes:
Try auto_arima on interest rates model
Look into stock returns as a function of interest rates (start with basic AR model on log difference in returns)

Notes:
Leaning towards ARIMA(2,0,2) with log_dif_cpi as predictor for interest rate model

Meeting Notes:
Look into how auto.arima vs Arima function fits models
Look into MPE MAPE
Look into log transformation of rates, piecewise function
Dif_rate not homoskedastic, could benefit from heteroskedastic model
Look into GARCH model to address heteroskedasticity
Simulate with ARIMA(1,0,1)

Notes:
Options for fitting ARIMA models: MLE, CSS, CSS-MLE
Specifying CSS-MLE (default) for Arima and auto.arima give different fitted values
Problem: Single simulations show interest rates can drop below 0
Trying to implement JF's transformed rates
- how to we pick c_r
- how do we pick the order of GARCH model

Meeting Notes:
Verify if residuals are hetereoskedastic
rerun model with transformed series

Notes:
There doesn't seem to be much correlation between interest rates and inflation (especially when using transformed rates)
Readdress how we want to include inflation in the interest rate model

Meeting Notes:
1. Verify that sigma(garch_model) is getting standard deviation
2. Generate Simulated draws from interest rate model to see if we want to include inflation in model
3. Compare single simulations (look at if things are working how we expect comparing draws of all models)

Notes:
Still running into issue of interest rates dropping below 0, interest rates are
  also getting too high in the upper bounds of the simulated draws.

Meeting Notes:
1. Data going into GARCH model needs to be ts object
2. Look into how the model is parameterized (what is mu), why is mu .08 and why is our simulation mean around .05
3. Back-transform the rates
4. Start simulation at different time (see if the simulations move to average using different starting values)

Notes:
mu represents the constant mean of the time series. In the simulations we can see that the mean of the simulations revert back to the 
mu parameter fit in the model (.08)
Upper bounds feel way too high
JF's 2021 paper:
  what is the difference between observable dynamic variances and stochastic volatility type models
  pg 785: Explain how Correlation between errors of interest rate and inflation are incorporated into model
  
Meeting Notes:
1. Explore fit of fGarch package
2. Read asset return modelling in ESG primer

Notes:
https://cran.r-project.org/web/packages/rugarch/vignettes/Introduction_to_the_rugarch_package.pdf has model specification
It looks like the starting value of the series almost determines mu in rugarch
Implemented differencing model with rugarch
- time series still look very off
Tried implementing model using fGarch package (online it says fGarch isn't maintained anymore, ruGarch is the new, maintained packaged to use)
- plugging in fitted parameters doesn't produce valid simulated draws (all NaNs)

Meeting Notes:
subtract empirical mean from series, fit the model without a mu parameter, then add the mean back in
create issue and send to package maintainer

Notes:
Which days of stock market returns should we use (inflation numbers are monthly)
Regime Switching or GARCH model for stock returns

Notes:
External Regressors on Variance model?
Could we regress cpi and interest rate on stock returns and then fit residuals to Regime-Switching GARCH

Meeting Notes:
subtract off mean of stock returns
remove risk-free rate of return (short-term interest rate), then subtract mean
create projections to evaluate which model to choose
test out raw values of cpi and log_dif
create mean model -> model residuals with RS GARCH

Notes:
Rugarch package author responded, said the problem is the series is non-stationary
Am I adding back in mean and interest rate correctly? (line 293)

Meeting Notes:
Remove mean from interest rate, then fit model, then add back in to draws

Meeting Notes:
Go with RS GARCH model  
Write up: Introduction, Model Description

Read through interest rate/yield curve portion of JF's ESG paper 

Meeting Notes:
Try adding 3mo rate as regressor in var model
maybe correlation?
regress multiple interest rates on level, slope, and curve

Notes:
We are saying that the slope and curve don't have any relationship with the level since we are modeling short rate independent?
Should I have an intercept term in the linear models?

Meeting Notes:
Backtransform the rates to compare to actual data
try and fit VAR and Yield Curve models on untransformed rates

Meeting Notes:
Some negative values in yield curve predictions, try transforming all rates

Notes:
How many simulations should we run to feel good about the estimates of the tails


Meeting Notes:
add in loop case when value goes to 0
mortality: 

Meeting Notes:
functionality to start annuity at different ages

Notes:
Create functions to optimize over, where the functions represent different scenarios/goals
The main "parameter" we are interested in is probability of success. How do we quantify uncertainty in this parameter?
Is it a monte carlo standard error?
Around .067-.069 is area where things start changing
.07 and beyond all money in stock market has higher success rate

Meeting Notes:
investigate point where all lines cross: does it stay the same with different equity returns
adjust smoothing in plot (100% annuitization is either 1 or 0)
male vs. female
adjusting withdrawal rates

Lit Review Topics:
ESG
Economic Variable Modeling
Asset Allocation

Notes:
looking at scatterplot, 100% annuitization doesn't go from 0 directly to 1. There is some success_rate in between.
  This is due to different levels of inflation im assuming?
Where can I get data on medical expenses based on age
Writing the model specification for the yield curve: I'm including an intercept term in the model. Could I call this an "adjustment term"?

Meeting Notes:
What is the relationship between the coefficients across the different linear models for each interest rate? (show in a table)
look into behavior of yield curve models without intercept
intervals on the mean_value_at_death
apply percentage of expenses that go to medical to medical inflation

Notes:
Removing the intercepts, model fit (r squared) is almost the exact same. Quantiles of annuity prices are the exact same (.25,.5,.75)
According to: https://www.urban.org/sites/default/files/publication/51766/411130-understanding-expenditure-patterns-in-retirement.pdf
  married adults over 65 spend 20% on health care
  nonmarried adults spend 16%
implemented 18% of expenses to medical costs and success rates barely shifted lower (medical inflation tends to be a little higher than CPI)
created table of coefficients for the yield curve model

Ideas: risk of insurance insolvency
different mortality rates for the "types of people" who would have access to financial products

Notes:
Applying ECI to simulations (ECI tends to outpace CPI)
Not sure if I completely trust the ECI simulations
Annuity pricing:
- 2 knobs: buying an annuity in the future, age of annuity purchase
- 80 years in future
- ability to purchase between 50 and 80?

Meeting Notes:
linear interpolation between quarters (ECI)
look into how ECI model changes when using different "start date" for data

Meeting Notes:
model annual wage inflation

Notes:
how to include bond investments?
what kind of metric could be evaluated including wages?

Meeting Notes:
function in terms of savings rate
look into changing time until retirement

Meeting Notes:
explore how savings rate changes through simulation
simulate with a fixed savings rate

Meeting Notes:
plotting mean value at death against age at death should all be the same up to 60,then at 60 100% should be 0, 
then 0% annuity should stay where mean is

Notes:
Issue was I wasn't subtracting the annuity purchase price from the nest egg
Question: Would it be correct to add a single annuity payout to a portfolio in the same period that the annuity is purchased?
Idea: in the savings rate function, there's variation among withdrawal rates

Meeting Notes:
OG simulation: run for annuity proportion .9-1 by .01, and/or extend out withdrawal rate

Meeting Notes:
find numbers for how medical expenses change over time through retirement
how do we model medical expenses? look through literature

Meeting Notes:
look at how medical expenses spending changes through retirement
think about creating some randomness to medical expenses (normal distribution)

Meeting Notes:
Probability of long term care implementation from vangaurd article
Indication of health status at beginning of simulation