This document provides a single-page overview on the different factors that play a role in personal finance. It covers inflation, investments, taxes, and their interaction on a high level, setting a starting point for further research.

The value of money is determined by its purchasing power, the amount of goods or services you can purchase per given amount.
The price of those goods and services can vary, and the long-term increase of overall prices is defined as *inflation*, and its decrease as *deflation*.
During inflation, the value of a fixed quantity of money will decrease, since prices increase.

$$
V = \frac{M}{P} \\
\begin{align}
\text{with } V =& \text{ the value} \\
M =& \text{ the amount of money} \\
P =& \text{ the price} \\
\end{align}
$$

Inflation can be measured through change of the *consumer price index*,
which samples the price of representative items a consumer would buy, such as clothing and food.
Inflation is often reported as an annualized percentage change, for example an inflation rate of 2% corresponds with a 2% yearly increase in overall prices.

When planning finances, it is common to calculate a forecast of the future value of assets, a *projection*.
Due to how inflation affects the value of money over time, it is an important factor to take into account.
The inflation-adjusted value of assets is called the *real value* as opposed to the *nominal value* which does not take inflation into account.
The same naming applies to interest rates and growth rates.

When projecting the value of an uninvested amount of money, its nominal value without inflation will remain constant over time. However, adjusting for inflation, the real value of the uninvested money will decrease by the inverse of the inflation rate each year, as the value of money in goods and services is inversely proportional to their price.

$$
V_t = \frac{M}{P_t} = \frac{M}{(1 + i)^t P_0} \\
\begin{align}
\text{with } t & = \text{the time in years} \\
i & = \text{the inflation rate} \\
\end{align}
$$

For example, assuming an inflation rate of 2% (which is the target inflation rate as mentioned below), a fixed amount of money will reach 82% of its original value within 10 years.

Inflation is affected by a number of different related factors, such as the economic activity, the total money supply and the overall demand for goods and services. Since the above elements play such a large economic role, they are indirectly controlled through monetary policy by a monetary authority, namely by the Federal Reserve in the US. This control is achieved by affecting one of the elements, namely the total money supply of the economy, through the banks.

Banks have a *reserve requirement* and/or *capital requirement*,
which obligates them to satisfy a minimum ratio of reserves versus deposits and capital versus assets respectively.
These minimums affect how much money a bank can lend out.
To aid in satisfying the requirements, banks have access to *interbank lending*,
where banks can lend money from each other at an interest rate called the *interbank rate*, or the *federal funds rate* in the US.

When monetary authorities aim to control the inflation rate or the economic state, they influence the total money supply by targeting a specific interbank rate.
This interbank rate is achieved commonly through *open market operations*, the buying and selling of government securities to banks.
Buying securities from banks will increase their amount of money, and therefore lower its demand and the interbank rate.
In addition, the increased amount of money will allow banks to give out more loans, increasing the total money supply.
While selling securities to banks, the opposite will happen.

The interbank rate at any given time will rapidly affect other short-term interest rates, such as short-term loans to businesses and households. Longer term interest rates are affected by the expectations of the future interbank rate. These expectations take into account the economic outlook and communications from the monetary authority about their long-term policy.

The subsequent interest rates affect the way businesses and households spend money. A decrease in interest rates will cause businesses to expand, a rise in employment rate and a higher overall demand. When this growth reaches its capacity and falls behind the increase in total money supply, the inflation rate will start to increase. To keep inflation stable, the interbank rate will then be increased. Overall, monetary policy has to balance the economic growth and the growth of the total money supply, decreasing the total money supply when the inflation rate or the utilization of the economic capacity is too high and vice versa.

Monetary policy usually does not target an inflation rate of 0% but instead targets a positive inflation rate of around 2%. This provides some margin since it becomes increasingly difficult to control the inflation rate through monetary policy rates during deflationary periods, as deflation increases inflation-adjusted interest rates and nominal interest rates can't be lowered below 0%. In addition, wages have downward rigidity, and a positive inflation can correct for overpriced wages.

Investments are assets purchased to provide some future return.
Those returns can be variable and uncertainty on them is known as *risk*.

One of the aspects about risk is implied by the *efficient market hypothesis*.
This hypothesis states that the price of an asset reflects all available information, including risk and expected return,
and is therefore correctly priced.
This implies the *risk-return trade-off*, where any higher return rate has an accordingly higher risk.
Otherwise, the asset would be underpriced, violating the efficient market hypothesis.

Risk can also be divided up in two components: *systematic risk* and *unsystematic* or *idiosyncratic risk*.
Systematic risk is a global risk that can't be diversified against.
It represents the risk of the entire market of every possible asset, influenced by macroeconomic factors such as economic growth, or employment rates.
Idiosyncratic risk on the other hand is risk associated with a singular investment.
For example, one company or sector may risk underperforming, but this risk does not extend to the entire market.
This risk can therefore be removed through diversification.
As it can be removed, idiosyncratic risk is not compensated for by a higher return rate.

As assets are traded, their price will fluctuate, taking into account new information.
The *random walk hypothesis* describes these price changes or returns as random, in that they are unpredictable,
as all available information has already been taken into account.
As such, the returns are sometimes modeled through a statistical random distribution (often the normal or log-normal distribution).
This modeling can help reason about the expected return and total risk, and allows for computing probabilities of future returns.

A common measure for the total risk (systematic and idiosyncractic) of returns is the *standard deviation*,
which corresponds roughly to the expected deviation from the mean.
When assuming a normal distribution, the standard deviation can be used to calculate confidence intervals,
as 68% of the values sampled from such a distribution fall within 1 standard deviation of the mean,
95% within 2 standard deviations,
and 99.7% within 3 standard deviations.

There are a number of monetary contracts or *financial instruments* that are used as investments.
Some are *securities*, such as stocks and bonds, which are freely tradeable by the holder, as opposed to instruments such as loans and deposits.

Stocks represent the ownership of a company through a proportional number of shares, that are traded at a certain price.
Shareholders can also receive part of the company's uninvested profits through payments known as *dividends*.

Stocks are one of the instruments that carry the highest risk, especially when investing in a small number of individual stocks. The ideosyncratic risk of each stock is associated with the company, and can be seen as company risk.

Bonds are long-term debt instruments.
A bond buyer effectively borrows money to the issuer, who pays the holder interest (called *coupons*) until a specific date, the *maturity date*,
after which the bond is redeemed, and the debt or *principal* is repayed.

Bonds are subject to a number of different risks:

- Interest rate risk: If interest rates rise above the interest rate of the bond, the price of the bond will fall due to opportunity cost. This risk increases as the maturity date becomes further in the future.
- Default risk: Borrowers may default on the payment. This risk depends on the credit rating of the issuer. For example, US government bonds have negligible default risk.

Local governments issue municipal bonds to finance public projects. Their main advantage is that their interest is exempt from federal taxes, and can possibly be exempt from state taxes too if the bond originated from the same state the taxpayer resides in.

Treasury Inflation Protected Securities are a specific kind of bond issued by the US government where the principal and coupons are adjusted for inflation. As expectations for inflation are already included in the pricing of bonds, this security protects against unexpected inflation.

Index funds are holdings of securities that seek to track an index, a weighted group of securities specified by some rules. For example, the S&P 500 is a selection of company stocks chosen to be representative of the US economy, weighted by market capitalization. Index funds provide an easy way to diversify, mitigating idiosyncractic risk.

Index funds are implemented either through an *exchange-traded fund (ETF)* or a *mutual fund*.
Exchange-traded funds operate like stocks, are listed on stock exchanges and are maintained through arbitrage,
exchanging ETF shares for equivalent amounts of its components and vice versa.
Mutual funds on the other hand are a separate instrument.
Both charge a fee called an expense ratio, this fee is deducted from the underlying value of a fund's share.
ETF's have a number of advantages over mutual funds: lower expense ratios, higher tax efficiency, more liquidity.

The money market is a collection of short-term and low-risk debt instruments.
It includes deposits and treasury bills, which are effectively risk-free investments since they are guaranteed by the government
(through the Federal Deposit Insurance Corporation (FDIC) for deposits or directly through treasury bills), and their interest risk is very low due to being short-term.
This *risk-free interest rate* sets the lower bound for investments, and higher rates of returns come with associated risk.
It is strongly influenced by the interbank or federal funds rate mentioned above, and their values will be close together.

Deposits are made at a bank to accounts such as checkings or savings accounts. This is viewed as debt by the bank for which it pays interest. Deposits can be withdrawn at any time.

Certificates of deposit (CD's), also known as time deposits, have a fixed term over which they gain interest. They can't be withdrawn without penalty until maturity. As certificates of deposit also can't usually be traded, they carry additional liquidity risk which is compensated for, proportional to their term length.

Treasury bills return a given amount of money at their maturity date. They are sold at a discount by the government, and the difference is gained by the investor as interest. Treasury bills are securities, and as such, can be freely traded.

Investments are chosen in such a way to maximize the return and minimize the total risk appropriate for the goals and timeline of the investor.
This group of selected investments constructed for the investor is called a *portfolio*.

*Modern portfolio theory* describes a method to construct such an optimal portfolio.
As seen above, systematic risk to the entire market is compensated while idiosyncratic risk is not because it can be removed through diversification.
In addition, due to the efficient market hypothesis, the systematic risk will be compensated at the same amount per unit of risk.
Therefore, a portfolio with only systematic risk and no unsystematic risk achieves the optimal return per unit of risk.
By the definition of systematic risk, this is the *market portfolio*,
and consists of all possible assets in the world, weighted by their total value.
The market portfolio would then be the only necessary portfolio to invest in,
and money could be borrowed or invested at the risk-free rate in combination with the market portfolio to achieve the desired risk for the investor.

However, the market portfolio is not practical to construct, and not everyone has access to borrowing at the risk-free rate. Therefore, portfolio's are instead optimized based on a given number of asset classes to achieve the maximum return for a given level of risk, which is equal to minimizing the idiosyncratic risk. This construction follows a number of steps.

**1. Asset class selection**

Different asset classes are chosen based on their representativeness of the total market, and their possibility for diversification.
They might include US stocks, emerging market stocks, bonds, commodities.
For each of the asset classes, statistics need to be computed to estimate the expected return and total risk of the composed portfolio.
Assuming the asset class returns are normally distributed, the necessary statistics are the mean, standard deviation and pairwise correlation of the returns.
The correlation of the returns between two asset classes give a measure of how much diversification is possible between the two.

**2. Investment vehicle selection**

Appropriate securities are selected to represent each asset class. Often, there exist well-diversified ETF's for each asset class.

**3. Portfolio optimization**

The relative weight of the components are calculated to achieve the maximum return for the desired amount of risk.
The expected return and standard deviation of the resulting portfolio is calculated based on the above statistics given normally distributed returns.
Any correlation less than 1 allows the standard deviation i.e. the total risk to be less than the weighted average of the two, while the expected return equals the weighted average.

Income is subject to both federal and state taxes, with federal taxes being the dominant factor. They need to be reported yearly by filing a tax return. During the year, advance payments are either withheld when working for an employer or made through estimated tax payments. Withheld payments are reported on a W-2 form together with the income earned from that employer. The final tax payment or refund will be made together with the yearly tax return. Taxes other than income taxes include property taxes, estate taxes and gift taxes.

Taxable income is calculated as all income minus certain deductions. Ordinary taxable income, such as wages, interest, is taxed at the following federal tax rates. The tax rates are based on brackets, which means you will pay the tax rate for a specific bracket only for income within that bracket's range.

There is different, beneficial taxation for certain types of income, namely:

*Capital gains*: the profit made from the sale of an asset. Only capital gains from assets held longer than 1 year,*long-term capital gains*, are taxed at a lower rate.*Qualified dividends*: dividends gained from US affiliated stock that was held for a number of days around the dividend date.

In 2013, an additional 3.8% net investment income tax was instated for high income tax payers on investment income such as interest, dividends, and rent.

Alternative minimum tax (AMT) is an alternative tax calculation that follows different tax rates and has a different set of possible deductions. One has to pay the higher amount of regular income tax or alternative minimum tax. Often, AMT comes into play at higher income levels.

The US provides tax-advantaged, investable accounts to save for retirement and health-care expenses. There are a number of possible different tax advantages.

Growth or income within a tax-advantaged account is tax-free until withdrawal:
no income tax is paid on dividends or interest and no capitals gains tax is paid when selling investments.
This removes *tax drag*, increasing the effective compounding expected returns.

Some accounts allow for deducting contributions from taxable income,
effectively allowing *pre-tax* contributions (as opposed to *after-tax*), lowering taxable income.
When the subsequent withdrawals are taxed as ordinary income, this can be viewed as *tax deferral*.
Tax deferral is beneficial when one's income falls in a higher tax bracket at the time of contribution than in retirement,
but can be counterproductive in the reverse case.

Other accounts such as Roth accounts only allow after-tax contributions. As these don't provide tax deferral, they are often chosen when the expected tax bracket in retirement will be higher than at the time of contribution, only taking advantage of the removal of tax drag.

If withdrawals are made outside of the intended scope of the account, a penalty is often paid in addition to ordinary income taxes on the withdrawal. However, company matches in 401(k) accounts might offset this penalty.

A number of tax-advantaged account types with a brief selection of their properties are the following:

Eligibility:

Income limits for tax-deductible contributions
($63,000 to $73,000 for single filers in 2018)

Contributions:

Pre-tax

Contribution limits:

$5,500 (age 49 and below) - $6,500 (age 50 and above) across all IRA accounts.

Withdrawals:

- Taxed as income at the time of withdrawal.
- Penalty-free after the age of 59½.

Withdrawal penalty in addition to ordinary income taxes:

10%

Eligibility:

Income limits for contributions
($120,000 to $135,000 for single filers in 2018)

Contributions:

After-tax

Contribution limits:

$5,500 (age 49 and below) - $6,500 (age 50 and above) across all IRA accounts.

Withdrawals:

- Contributions: at any time.
- Earnings: tax and penalty-free after the age of 59½.

Withdrawal penalty in addition to ordinary income taxes:

10%

Eligibility:

Employer-based

Contributions:

- Pre-tax
- Possible employer match where the employer contributes a fraction of the employee's contributions

Contribution limits:

- $18,500 for employee contributions across all 401(k) plans.
- $6,000 in additional catch-up contributions for employees age 50 and above.
- $55,000 for employer and employee contributions.

Withdrawals:

- Taxed as income at the time of withdrawal.
- Penalty-free after the age of 59½.

Withdrawal penalty in addition to ordinary income taxes:

10%

Eligibility:

Employer-based

Contributions:

- After-tax
- Possible employer match where the employer contributes a fraction of the employee's contributions

Contribution limits:

- $18,500 for employee contributions across all 401(k) plans.
- $6,000 in additional catch-up contributions for employees age 50 and above.
- $55,000 for employer and employee contributions.

Withdrawals:

- Contributions: at any time.
- Earnings: tax and penalty-free after the age of 59½.

Withdrawal penalty in addition to ordinary income taxes:

10%

Eligibility:

Based on coverage by a high-deductible health plan (HDHP).

Contributions:

Pre-tax

Contribution limits:

$3,450 for individuals.

Withdrawals:

At any time for documented medical expenses such as healthcare costs before meeting the health insurance deductible, eyeglasses, dental care.

Withdrawal penalty in addition to ordinary income taxes:

20%

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