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<section id="title-slide" class="quarto-title-block center">
<h1 class="title">Analyze Retirement Plans with an Economic Scenario Generator</h1>
<div class="quarto-title-authors">
<div class="quarto-title-author">
<div class="quarto-title-author-name">
Evan Miller, Brian Hartman, Jean-Francois Begin (Simon Fraser University)
</div>
</div>
</div>
</section>
<section class="slide level2">
<!-- ## Planning for Retirement {background-color="#FFFAEE"} -->
<!-- ::: {.fragment .semi-fade-out} -->
<!-- **Pre-retirement:** -->
<!-- - How much money do I need to retire? -->
<!-- - How long will it take to save that much? -->
<!-- - When should I retire? -->
<!-- ::: -->
<!-- **Post-retirement:** -->
<!-- - How much can I spend in retirement? -->
<!-- - How long will my money last? -->
<!-- - Where should I put my assets? -->
</section>
<section id="retirement-longevity-risk" class="slide level2" data-background-color="#FFFAEE">
<h2>Retirement & Longevity Risk</h2>
<p>Goal: <strong>Minimize longevity risk</strong></p>
<ul>
<li>Longevity Risk: The probability of running out of assets in retirement</li>
</ul>
<p>Financial Unknowns:</p>
<div class="fragment fade-up">
<ul>
<li>Cost of Living (inflation)</li>
<li>Equity Returns</li>
<li>Medical Costs</li>
<li>End of Life Costs</li>
<li>Age of Death</li>
</ul>
</div>
<!-- ## How is this risk quantified? {background-color="#FFFAEE"} -->
<!-- :::::: {.fragment .fade-out} -->
<!-- #### Deterministic approach: -->
<!-- ::::: columns -->
<!-- ::: {.column width="40%"} -->
<!-- Inflation: 2% -->
<!-- Stock Returns: 7%, 12% -->
<!-- ::: -->
<!-- ::: {.column width="60%"} -->
<!-- Withdrawal Rate: 4% (\$40,000) -->
<!-- ::: -->
<!-- ```{r, fig.width=10, fig.height=4} -->
<!-- library(tidyverse) -->
<!-- library(scales) -->
<!-- # deterministic approach -->
<!-- inflation <- 0.02 -->
<!-- stock_returns <- 0.12 -->
<!-- stock_returns2 <- .07 -->
<!-- short_term_rate <- 0.03 -->
<!-- withdrawal_amount <- 1000000 * .04 -->
<!-- # retirement savings -->
<!-- savings <- 1000000 -->
<!-- # retirement period -->
<!-- retirement_years <- 35 -->
<!-- savings_by_year <- numeric(retirement_years) -->
<!-- savings_by_year[1] <- savings -->
<!-- savings_by_year2 <- numeric(retirement_years) -->
<!-- savings_by_year2[1] <- savings -->
<!-- # simulate retirement -->
<!-- for (i in 2:retirement_years) { -->
<!-- savings_by_year[i] <- savings_by_year[i-1] - withdrawal_amount*(1+inflation) + (stock_returns * .2*savings_by_year[i-1]) -->
<!-- savings_by_year2[i] <- savings_by_year2[i-1] - withdrawal_amount*(1+inflation) + (stock_returns2 * .2*savings_by_year2[i-1]) -->
<!-- if(savings_by_year[i] < 0) { -->
<!-- savings_by_year[i] = 0 -->
<!-- } -->
<!-- if(savings_by_year2[i] < 0) { -->
<!-- savings_by_year2[i] = 0 -->
<!-- } -->
<!-- } -->
<!-- # set negative values to 0 -->
<!-- df <- data.frame(year = 1:retirement_years, -->
<!-- savings1 = savings_by_year, -->
<!-- savings2 = savings_by_year2) -->
<!-- ggplot(df) + -->
<!-- geom_line(aes(x = year, y = savings1, color = '12%')) + -->
<!-- geom_line(aes(x = year, y = savings2, color = '7%')) + -->
<!-- labs(x = 'Years into Retirement', y = 'Savings ($)', color = 'Stock Returns', -->
<!-- title = 'Retirement Portfolio with Different Stock Returns') + -->
<!-- scale_color_manual(values = c('12%' = 'black', '7%' = 'red')) + -->
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<!-- ``` -->
<!-- ::::: -->
<!-- :::::: -->
</section>
<section id="how-is-this-risk-quantified" class="slide level2" data-background-color="#FFFAEE">
<h2>How is this risk quantified?</h2>
<h4 id="simulation-from-the-past">Simulation from the past:</h4>
<ul>
<li>Apply a specific “retirement plan” to a year from the past
<ol type="1">
<li>Pick a “plan” (i.e. 4% Withdrawal Rate, 50% equity porfolio)</li>
<li>Apply that plan to a year from the past</li>
<li>Simulate cash-flows using past data</li>
<li>Iterate over “all” years</li>
</ol></li>
</ul>
</section>
<section id="our-approach" class="slide level2" data-background-color="#FFFAEE">
<h2>Our Approach:</h2>
<div class="fragment">
<ol type="1">
<li>Build a stochastic model for different economic variables</li>
</ol>
</div>
<div class="fragment">
<ol start="2" type="1">
<li>Combine individual models to represent “an economy” (Economic Scenario Generator)</li>
</ol>
</div>
<div class="fragment">
<ol start="3" type="1">
<li>Simulate future economic scenarios</li>
</ol>
</div>
<div class="fragment">
<ol start="4" type="1">
<li>Apply retirement plans to these scenarios and analyze outcomes</li>
</ol>
</div>
</section>
<section id="economic-scenario-generator-esg" class="slide level2" data-background-color="#FFFAEE">
<h2>Economic Scenario Generator (ESG)</h2>
<aside class="notes">
<p>Model of an economic environment used to simulate financial markets and economic variables Used in pricing financial products such as insurance and financial derivatives</p>
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<img data-src="documents/www/ESG_diagram.png" class="quarto-figure quarto-figure-center r-stretch"></section>
<section id="fitted-esg" class="slide level2 smaller" data-background-color="#FFFAEE">
<h2>Fitted ESG</h2>
<p>Model Selection Criteria:</p>
<ol type="1">
<li>Behavior of simulated draws</li>
<li>Model fit</li>
<li>Model parsimony</li>
</ol>
<p>Data:</p>
<ul>
<li>Inflation and Interest Rates: FRED</li>
<li>Equity Returns: S&P 500 from <code>tidyquant</code> package</li>
<li>Mortality: SOA Mortality Table</li>
</ul>
<p><img data-src="documents/www/ESG_structure_models_transparent.png" class="absolute" style="bottom: 70px; right: -200px; width: 700px; "></p>
</section>
<section id="wage-inflation-model-selection" class="slide level2 smaller" data-background-color="#FFFAEE">
<h2>Wage Inflation Model Selection</h2>
<p><img data-src="documents/www/eci_diagram.png" class="absolute" style="top: 0px; right: -100px; width: 400px; "></p>
<ol type="1">
<li>Explore Relationship between CPI and ECI</li>
</ol>
<p><img data-src="documents/www/eci_cpi_plot.png"></p>
<ul>
<li>Lagging: <span class="math inline">\(Cor(\texttt{ECI}_t, \texttt{CPI}_{t-1}) = .557\)</span></li>
</ul>
</section>
<section id="wage-inflation-model-selection-1" class="slide level2 smaller" data-background-color="#FFFAEE">
<h2>Wage Inflation Model Selection</h2>
<p><img data-src="documents/www/eci_diagram.png" class="absolute" style="top: 0px; right: -100px; width: 400px; "></p>
<ol start="2" type="1">
<li><p>Test Different Models</p>
<ul>
<li>Differing lags (for both ECI and CPI)</li>
<li>Differing data subsets (1990 vs. 2010)</li>
<li>Include Differencing?</li>
</ul></li>
</ol>
<p><img data-src="documents/www/eci_model_compare.png"></p>
</section>
<section id="wage-inflation-model-selection-2" class="slide level2 smaller" data-background-color="#FFFAEE">
<h2>Wage Inflation Model Selection</h2>
<p><img data-src="documents/www/eci_diagram.png" class="absolute" style="top: 0px; right: -100px; width: 400px; "></p>
<ol start="3" type="1">
<li>Simplifying the Model</li>
</ol>
<p><img data-src="documents/www/eci_timeseries.png"></p>
</section>
<section id="wage-inflation-model-selection-3" class="slide level2 smaller" data-background-color="#FFFAEE">
<h2>Wage Inflation Model Selection</h2>
<p><img data-src="documents/www/eci_diagram.png" class="absolute" style="top: 0px; right: -100px; width: 400px; "></p>
<ol start="3" type="1">
<li>Simplifying the Model
<ul>
<li>Yearly instead of Quarterly</li>
<li>CPI instead of lagged CPI as predictor</li>
<li>AR(1)</li>
</ul></li>
</ol>
<p><img data-src="documents/www/eci_final_draws.png"></p>
</section>
<section id="inflation-models" class="slide level2 smaller" data-background-color="#FFFAEE">
<h2>Inflation Models</h2>
<ol type="1">
<li>General Inflation: ARMA(1,1)</li>
</ol>
<p><span class="math display">\[\log{ \left( \frac{\text{CPI}_t}{\text{CPI}_{t-1}} \right) } = q_t = \mu + \phi q_{t-1} + \theta\epsilon_{t-1} + \epsilon_t, \quad \text{where } \epsilon_t \sim N(0, \sigma^2).\]</span> 2. Medical Inflation: ARMA(1,1)</p>
<p><span class="math display">\[\log{ \left( \frac{\text{MedCPI}_t}{\text{MedCPI}_{t-1}} \right) } = m_t = \mu + \phi m_{t-1} + \theta\epsilon_{t-1} + \epsilon_t, \quad \text{where } \epsilon_t \sim N(0, \sigma^2).\]</span> 3. Wage Inflation: AR(1) w/ CPI as predictor</p>
<p><span class="math display">\[\log{\left(\frac{\text{ECI}_t}{\text{ECI}_{t-1}}\right)} = w_t = \mu + q_{t} + \phi w_{t-1} + \epsilon_t, \quad \text{where } \epsilon_t \sim N(0, \sigma^2).\]</span></p>
</section>
<section id="interest-rates" class="slide level2 smaller" data-background-color="#FFFAEE">
<h2>Interest Rates</h2>
<ol type="1">
<li>Interest Rate Transformation:</li>
</ol>
<p>For <span class="math inline">\(i = \{3/12,1,2,3,5,7,10,20,30\}\)</span>:</p>
<p><span class="math display">\[
\tilde{r}_{i,t} = \begin{cases}
r_{i,t} & \text{if }r_{i,t} > \overline{r} \\
\overline{r} - \overline{r}\log(\overline{r}) + \overline{r}\log(r_{i,t}) & \text{if } r_{i,t} \le \overline{r}
\end{cases} ,
\]</span> where <span class="math inline">\(\overline{r} = .005\)</span> (Bégin 2021)</p>
<ol start="2" type="1">
<li>Short Term Rates: ARMA(1,1) & GARCH(1,1)</li>
</ol>
<p><span class="math display">\[\tilde{r}_{3/12,t} = \mu + q_{t} + \phi \tilde{r}_{3/12,t-1} + \theta\epsilon_{t-1} + \epsilon_t, \quad \text{where } \epsilon_t \sim \mathcal{N}(0,\sigma_t^2),\]</span></p>
<div class="r-stack">
<p>and</p>
</div>
<p><span class="math display">\[\sigma_t^2 = \omega + \alpha \epsilon_{t-1}^2 + \beta \sigma_{t-1}^2,\]</span></p>
</section>
<section id="yield-curves" class="slide level2 smaller" data-background-color="#FFFAEE">
<h2>Yield Curves</h2>
<ol type="1">
<li>Functional Form</li>
</ol>
<p><span class="math display">\[\begin{align*}
\text{Level}: &\, \tilde{r}_{3/12,t} \\
\text{Slope}: &\,\tilde{r}_{30,t} - \tilde{r}_{3/12,t}, \\
\text{Curvature}:&\, \tilde{r}_{3/12,t} + \tilde{r}_{30,t} - 2\tilde{r}_{10,t}.
\end{align*}\]</span></p>
<ol start="2" type="1">
<li>Modeling Slope and Curvature: VAR(1)</li>
</ol>
<p><span class="math display">\[\boldsymbol{F}_t = \boldsymbol{\mu} + \boldsymbol{A}\,\boldsymbol{F}_{t-1} + \boldsymbol{b}\,\tilde{r}_{3/12,t} + \boldsymbol{\epsilon}_t, \quad \text{where } \boldsymbol{\epsilon}_t \sim N_2(\boldsymbol{0}, \boldsymbol{\Sigma}),\]</span></p>
<ol start="3" type="1">
<li>Yield Curve Construction: Factor Loadings</li>
</ol>
<p><span class="math display">\[\tilde{r}_{i} = \boldsymbol{x'}\boldsymbol{\beta}_i + \epsilon_{i}, \quad i = \{1,2,3,5,7,20\}, \quad \text{where } \epsilon_i \sim N(0, \sigma^2_i).\]</span></p>
<p><span class="math display">\[ \boldsymbol{x'} = [\texttt{level}, \texttt{slope},\texttt{curvature}]\]</span></p>
</section>
<section id="yield-curves-1" class="slide level2 smaller" data-background-color="#FFFAEE">
<h2>Yield Curves</h2>
<video src="documents/www/yield_curve.mp4" loop="" muted="" autoplay="" playsinline="" style="
position: absolute;
top: 0px;
right: 100px;
width: 500px;
border: 2px solid #dddddd;
border-radius: 12px;
box-shadow: 0 8px 20px rgba(0,0,0,0.25);
background: white;">
</video>
<div style="position: absolute; bottom: 0; left: 50%; transform: translateX(-50%); width: 90%;">
<p><img src="documents/www/sample_yield_curve.png" style="width: 100%; height: auto;"></p>
</div>
</section>
<section id="equity-returns" class="slide level2 smaller" data-background-color="#FFFAEE">
<h2>Equity Returns</h2>
<ul>
<li>Broad Based Index Fund: S&P 500</li>
<li>Regime Switching GARCH(1,1):</li>
</ul>
<p><span class="math display">\[y_t = \mu + q_t\gamma_1 + r_{3/12,t}\gamma_2 + \epsilon_t, \quad \text{where } \epsilon_t \sim \mathcal{N}(0,\sigma_{t}^2),\]</span></p>
<div class="r-stack">
<p>and</p>
</div>
<p><span class="math display">\[\sigma_{t}^2 = \omega_{R_t} + \alpha_{R_t} \epsilon_{t-1}^2 + \beta_{R_t} \sigma_{t-1}^2\]</span></p>
<div style="display: flex; justify-content: center; align-items: center; gap: 30px;">
<!-- Image -->
<p>Transition Matrix:</p>
<!-- LaTeX equation, slightly bigger -->
<div>
<p><span class="math display">\[
\Theta =
\begin{pmatrix}
\theta_{11} & \theta_{12} \\
\theta_{21} & \theta_{22}
\end{pmatrix}
\]</span></p>
</div>
</div>
</section>
<section id="simulation-steps" class="slide level2" data-background-color="#FFFAEE">
<h2>Simulation Steps:</h2>
<ol type="1">
<li>Generate <span class="math inline">\(n\)</span> simulations from the ESG <span class="math inline">\(x\)</span> years into the future</li>
<li>Draw <span class="math inline">\(n\)</span> sample death ages from mortality table</li>
<li>Apply simulations to a financial plan</li>
<li>Calculate success criteria</li>
</ol>
</section>
<section id="case-study" class="slide level2">
<h2>Case Study</h2>
<p><strong>Primary Question:</strong> For a given amount of spending, how much should be annuitized?</p>
<p><strong>Key Parameters:</strong></p>
<ol type="1">
<li>Withdrawal Rate: annual spending as a percentage of savings at retirement</li>
<li>Annuity Proportion: percentage of savings at retirement to annuitize</li>
</ol>
<p><strong>Key Result:</strong> Probability of out-living assets</p>
</section>
<section id="withdrawal-rates-and-annuity-pricing" class="slide level2 smaller" data-background-color="#FFFAEE">
<h2>Withdrawal Rates and Annuity Pricing</h2>
<ol type="1">
<li>Withdrawal Rate: <span class="math inline">\(r\)</span></li>
</ol>
<p><span class="math display">\[r = \frac{\texttt{Yearly Expenses}}{\texttt{Retirement Savings}}\]</span> <span class="math display">\[\texttt{Monthly Expenses} = c = \texttt{Retirement Savings} \times \frac{r}{12}.\]</span></p>
<ol start="2" type="1">
<li>Annuity Payout: <span class="math inline">\(b\)</span></li>
</ol>
<p><span class="math display">\[p_x
= (1+l) \sum_{t=1}^{T}
(1+r_{t/12})^{-t/12}\kappa_{x+\frac{t}{12}},\]</span></p>
<p><span class="math display">\[\texttt{Monthly Annuity Payout} = b = \frac{\alpha \times \texttt{Retirement Savings}}{p_x}\]</span></p>
</section>
<section id="retirement-plan-set-up" class="slide level2 smaller" data-background-color="#FFFAEE">
<h2>Retirement Plan Set-up</h2>
<div class="fragment">
<ul>
<li><strong>Stock Market withdrawal amount:</strong></li>
</ul>
<p><span class="math display">\[\text{CPI}_t c - b\]</span></p>
</div>
<div class="fragment">
<ul>
<li><strong>Portfolio Value at time</strong> <span class="math inline">\(t\)</span>:</li>
</ul>
<p><span class="math display">\[v_t = (v_{t-1} - (\text{CPI}_tc - b))y_t\]</span></p>
</div>
<div class="fragment">
<ul>
<li><strong>Final Set-up:</strong></li>
</ul>
<p><span class="math display">\[
v_t =
\Biggl(
v_{t-1} -
\Bigl(
\underbrace{(0.8 \, \text{CPI}_t + 0.2 \, \text{MedCPI}_t) \, c}_{\text{weighted spending}}
-
\underbrace{b}_{\text{annuity payout}}
\Bigr)
\Biggr)
\underbrace{y_t}_{\text{equity returns}}
\]</span></p>
</div>
<div class="fragment">
<ul>
<li><strong>Success Criteria</strong>: <span class="math inline">\(v_t > 0\)</span> at time of death</li>
</ul>
</div>
</section>
<section id="results-10000-simulations" class="slide level2" data-background-color="#FFFAEE">
<h2>Results (10,000 Simulations)</h2>
<img data-src="documents/www/sim_results.png" class="quarto-figure quarto-figure-center r-stretch"></section>
<section id="results-10000-simulations-1" class="slide level2" data-background-color="#FFFAEE">
<h2>Results (10,000 Simulations)</h2>
<img data-src="documents/www/sim_results_zoomed.png" class="quarto-figure quarto-figure-center r-stretch"><div class="fragment">
<ul>
<li><strong>4% Rule?</strong></li>
</ul>
</div>
</section>
<section id="looking-ahead" class="slide level2" data-background-color="#FFFAEE">
<h2>Looking Ahead</h2>
<div class="fragment fade-up" data-fragment-index="1">
<ul>
<li>Implement more realistic retirement plans</li>
</ul>
</div>
<div class="fragment fade-up" data-fragment-index="2">
<ul>
<li>Pre-retirement scenarios (Wage-inflation)</li>
</ul>
</div>
<div class="fragment fade-up" data-fragment-index="3">
<ul>
<li>Shiny App</li>
</ul>
</div>
<video class="fragment fade-up" data-fragment-index="3" src="documents/www/app_example.webm" loop="" muted="" playsinline="" onloadedmetadata="this.playbackRate = 1.75;" ontransitionend="this.play();" style="
position: absolute;
bottom: 40px;
right: 40px;
width: 750px;
border: 2px solid #dddddd;
border-radius: 12px;
box-shadow: 0 8px 20px rgba(0,0,0,0.25);
background: white;">
</video>
</section>
<section id="thank-you" class="slide level2">
<h2>Thank You</h2>
</section>
</div>
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<div class="footer footer-default">
</div>
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