---
title: "Project 1: Saving for a Car"
author: "Your Name"
format:
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    code-fold: false
execute:
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  message: false
---

## The question

In one or two sentences, explain the decision this model will help make.

## Model and assumptions

Briefly describe how each quantity changes during one month:

- new-car price;
- current-car trade-in value; and
- savings balance.

State two assumptions made by this model.

1. 
2. 

## Base-case simulation

```{r}
# Time in years; one update per month
start_time <- 0
end_time <- 5
delta_t <- 1 / 12
time <- seq(start_time, end_time, by = delta_t)
month <- 0:(length(time) - 1)

# Parameters and initial conditions
annual_inflation_rate <- 0.05
annual_interest_rate <- 0.04
annual_depreciation_rate <- 0.08
monthly_contribution <- 600

new_car_initial <- 30000
trade_in_initial <- 13000
savings_initial <- 6000

# Create the three stock vectors.
# TODO

# Store the three initial conditions.
# TODO

# Update all three stocks one month at a time.
for (i in 2:length(time)) {
  # TODO
}

# Combine the results and calculate the cash needed.
# TODO
```

## Verification

Calculate the new-car price, trade-in value, and savings after the first month by hand. Show your calculation, then display the first two rows of your simulation and explain whether they agree.

```{r}
# Display the first two rows here.
# TODO
```

## Result

Find the first month when savings are at least as large as the cash needed.

```{r}
# Use the first affordable position to display the crossing month and the
# preceding month. Remember that a vector position is not a month number.
# affordable_positions <- which(results$savings >= results$cash_needed)
# TODO
```

In a few sentences, report the first affordable month and explain how you confirmed it was the first.

## Graph

Create one graph showing savings and cash needed over time.

```{r}
# Your graph goes here. Include a title, labeled axes, two distinguishable
# lines, and a legend.
# TODO
```

What does the graph show? Describe where the two lines cross.

## One experiment

Change the monthly contribution from $600 to either $400 or $800.

Before running the modified model, predict whether the buyer will afford the car earlier or later and explain why.

**Prediction:** Replace this sentence with your prediction.

```{r}
# Duplicate or adapt your simulation for the new monthly contribution.
# A function is not required.
# TODO
```

Report the new purchase month. Was your prediction correct? Explain the difference from the base case.

## Conclusion

Write a short recommendation to the buyer. Include:

- the base-case purchase month;
- what the experiment revealed; and
- two important limitations of the model.