# Risk and Return: Past and Prologue
## Rates of Return
**Holding-period return (HPR)**: Rate of return over a given investment period.

$$\text{HPR} = \frac{\text{Ending price - Beginning price + Cash dividend}} {\text{Beginning price}}$$

for this formula we have assumed that the dividend is paid at the end of the holding period.

**Dividend yield**: The percentage return from dividends

$$\text{Dividend yield} = \frac{\text{Cash dividend}} {\text{Beginning price}}$$

And the left is the **capital gains**.
### Multi-period average return
|                                             | 1st <br/> quarter | 2nd <br/> quarter | 3rd <br/> quarter | 4th <br/> quarter |
|:-------------------------------------------:|:-----------------:|:-----------------:|:-----------------:|:-----------------:|
| Assets under management at start of quarter |       $1.0$       |       $1.2$       |       $2.0$       |       $0.8$       |
|           Holding period of return          |       $10.0$      |       $25.0$      |      $-20.0$      |       $20.0$      |
|       Total assets before net inflows       |       $1.1$       |       $1.5$       |       $1.6$       |       $0.96$      |
|                  Net inflow                 |       $0.1$       |       $0.5$       |       $-0.8$      |       $0.6$       |
|  Assets under management at end of quarter  |       $1.2$       |       $2.0$       |       $0.8$       |       $1.56$      |

for this table we assume that the net inflows happen at the end of quarter, 

$$\text{HPR} = \displaystyle \frac{\text{Total assets before net inflows - Assets under management at start of quarter}} {\text{ Assets under management at start of quarter}}$$

$$\text{Assets under management at end of quarter} = \text{Assets under management at end of quarter} + {\text{Net inflow}}$$

But how to find the yearly return rate?

**ARITHMETIC AVERAGE**: The sum of returns in each period divided by the number of periods.

> $$\frac{10\% + 25\% -20\% + 20\%} {4} = 8.5\%$$

**GEOMETRIC AVERAGE**: The single per-period return that gives the same *cumulative* performance as the sequence of actual returns. (time-weighted average return)

>$$(1 + 10\%)(1 + 25\%)(1 - 20\%)(1 + 20\%) = (1 + r_G)^4 \Rightarrow r_G = 7.19\%$$

**DOLLAR-WEIGHTED RETURN**: The *internal rate of return* (**IRR**) on an investment.

> |               |    0   | 1st <br/> quarter | 2nd <br/> quarter | 3rd <br/> quarter |  4th <br/> quarter |
|:-------------:|:------:|:-----------------:|:-----------------:|:-----------------:|:------------------:|
| Net cash flow | $-1.0$ |       $-0.1$      |       $-0.5$      |       $0.8$       | $-0.6+1.56 = 0.96$ |
>
>$$0 = -1.0 + \frac{-0.1} {1 + \text{IRR}} + \frac{-0.5} {(1 + \text{IRR})^2} + \frac{0.8} {1 + \text{IRR})^3} + \frac{0.96} {(1 + \text{IRR})^4} \Rightarrow \text{IRR} = 3.38\%$$

***
>**e.g.** A fund begin with $\$\$10$
>
>|             | 1st <br/> month | 2nd <br/> month | 3rd <br/> month |
|:-----------:|:---------------:|:---------------:|:---------------:|
| Net inflows |      $-0.1$     |      $-0.5$     |      $0.8$      |
|   HPR (%)   |       $2$       |       $8$       |       $-4$      |
>
>We can use this to construct a full table
>
>|             | 1st <br/> month | 2nd <br/> month | 3rd <br/> month |
|:-----------:|:---------------:|:---------------:|:---------------:|
| Assests under management at begin of month| $10.0$ | $13.2$| $19.256$ |
| Net inflows during month |      $3$     |      $5$     |      $0$      |
| Profit this month   |       $0.2$       |      $1.056$      |       $-0.77$      |
| Assests under management at end of month| $13.2$ | $19.256$ | $18.486$ |
| Net cash flow | $-3.0$ | $-5.0$ | $+18.486$ |

### Conventions for Annualizing Rates of Return
**Annual percentage return (APR)**: $\text{Per-period rate} \times \text{Periods per year}$

**Effective annual rate (EAR)**: defined as following

$$1 + \text{EAR} = \left( 1 + \frac{\text{APR}} {n}\right)^n$$

So that when $n$ becomes very large, we are actually assume that a **continuously compounded rates**, we have $1 + \text{EAR} = e^{\text{APR}} \Rightarrow \text{APR} = \ln (1 + \text{EAR}):= r_{cc}$

## Inflation and The Real Rate of Interest
**Consumer Price Index (CPI)**: measures the cost of purchasing a representative bundle of goods, the “consumption basket” of a typical urban family of four.

**Inflation rate**: $i$, the rate at which prices are rising, measured as the rate of increase of the CPI. 

Then if we let $r$ be the real interest rate, we have 

$$1 + r = \frac{1 + R} {1 + i} \Rightarrow r \approx R - i$$

here $R$ is the nominal interest rate, the growth rate of money.

**nominal interest rate**: The interest rate in terms of nominal (not adjusted for purchasing power) dollars.

**real interest rate**: The excess of the interest rate over the inflation rate. The growth rate of purchasing power derived from an investment.

### The Equilibrium Nominal Rate of Interest
**Fisher equation**: the nominal rate increases one-for-one with increases in the expected inflation rate, $E(i)$.

$$R = i + E(i)$$

### Risk and Risk Premium
What HPRs are possible, and how likely are they? 

**scenario analysis**: Process of devising a list of possible economic scenarios and specifying the likelihood of each one, as well as the HPR that will be realized in each case.

**probability distribution**: List of possible outcomes with associated probabilities.

>**e.g.**
>
>

**expected return**: The mean value of the distribution of HPR. $E(r)$

$$E(r) = \sum _{s = 1} ^S p(s)r(s)$$

here $s$ stand for each scenario.

**variance**: The expected value of the squared deviation from the mean.

$$\text{Var}(r) \equiv \sigma^2 = \sum _{s = 1} ^S p(s)[r(s) - E(r)]^2$$

**standard deviation**: The square root of the variance.

$$\text{SD}(r) \equiv \sigma = \sqrt{\text{Var}(r)}$$

### The Normal Distribution
We can transform any normally distributed return, $r_i$, into a “standard deviation score,” with $0$ mean and $1$ variance.

$$sr_i = \frac{r_i - E(r_i)} {\sigma_i}$$

$Conclusion$
1. The return on a portfolio comprising two or more assets whose returns are normally distributed also will be normally distributed.
2. The normal distribution is completely described by its mean and standard deviation. No other statistic is needed to learn about the behavior of normally distributed returns.
3. The standard deviation is the appropriate measure of risk for a portfolio of assets with normally distributed returns. In this case, no other statistic can improve the risk assessment conveyed by the standard deviation of a portfolio.

***
**value at risk (VaR)**: Measure of downside risk. The worst loss that will be suffered with a given probability, often 5%: *How much would I lose in a fairly extreme outcome*

$\odot$
A loss-averse investor might desire to limit portfolio VaR.$\square$

And when normally distributed, $\text{VaR}$ can be calculated by following steps:
1. Consider the probability that the worst loss would happen: like $\alpha = 5\%$ 
2. use this as the quantile and the $E(r)$, $\sigma$ from the stock data to calculate the $\text{VaR}$
  1. Standard normal distribution, $\Phi$: $\text{VaR} = E(r) + [\Phi^{-1}(\alpha)]\sigma$
  2. Or directly get that from the Excel function `NORM.INV`: $\text{VaR} = NORM.INV(\alpha, E(r), \sigma)$, meaning that for $X \sim \text{N}(E(r),\sigma^2)$, $P[X \leq \alpha] = \text{VaR}$
  3. Discrete distribution, than sort them from low to high, count until reach the $\alpha$ much part. For example you have $1000$ scenario with returns ordered from high to low, and your $\alpha$ is $5\%$, then you can count from the bottom to the $50$, and use the corresponding return as the value of $\text{VaR}$.
  4. Sometimes the interpolation is needed. When you need to count $3.6$ observation from the bottom, it will be the thrid rate plus $0.6$ multiple the difference between the third rate and the forth rate.

>**e.g.**
>
>| Business <br/> conditions | Scenario <br/> $s$ | Probability <br/> $p$ | End-of-year value <br/> \$ million | Annual Dividend <br/> \$ million |
|:-------------------------:|:------------------:|:---------------------:|:--------------------------------------:|:------------------------------------:|
| High growth | 1 | $0.30$ | $35$ | $4.40$ |
| Normal growth | 2 | $0.45$ | $27$ | $4.00$ |
| No growth | 3 | $0.20$ | $15$ | $4.00$ |
| Recession | 4 | $0.05$ | $8$ | $2.00$ |
>
>The current value of a stock portfolio is $23$ million. 
>
>$(\text{a})$ What are the annual holding-period returns of the portfolio in each scenario? Calculate the expected holding-period return, the standard deviation of returns, and the 5% VaR. What is the VaR of a portfolio with normally distributed returns with the same mean and standard deviation as this stock?
>
>So first we can compute the advanced table.
>
>| $s$ | $p$ | End-of-year value <br/> million dollar | Annual Dividend <br/> million dollar | $\text{HPR}$ | $\text{HPR}\times p$ | Deviation <br/>$\text{HPR}-$mean | $\text{Deviation}^2\times p$ |
|:---:|:------:|:--------------------------------------:|:------------------------------------:|:--------------------------------------------------------------:|:--------------------:|:-----------------------------------------------:|:----------------------------:|
| 1 | $0.30$ | $35$ | $4.40$ | $0.713$ | 0.214 | $0.406$ | $0.049$ |
| 2 | $0.45$ | $27$ | $4.00$ | $0.348$ | 0.157 | $-0.040$ | $0.001$ |
| 3 | $0.20$ | $15$ | $4.00$ | $-0.174$ | -0.035 | $-0.481$ | $0.046$ |
| 4 | $0.05$ | $8$ | $2.00$ | $-0.565$ | -0.028 | $-0.873$ | $0.038$ |
| Sum |  |  |  | $\displaystyle \frac{V_{\text{end}}\,+ D - V_{\text{now}}} {V_{\text{now}}} $ | **0.307** |  | **0.135** |
>
>So the expected $\text{HPR}$ is $30.7\%$; the variance is $0.135$, and the standard deviation is $\sqrt{0.135} = 36.7\%$
>
>And since there're only $4$ scenarios available, we have our $5\text{% VaR}$ as the last one which is $-0.565$, but we can still calculate using the normal distribition with same mean and variance:
>
>$$\Phi^{-1}(0.05) = - 1.64485, \text{VAR} = 30.7\% - 1.64485 \times 36.7\% = -29.67\%$$
>
>$(\text{b})$ Suppose that the worst three rates of return in a sample of 36 monthly observations are $-17\%$, $–5\%$, and $-2\%$. Estimate the $5\text{% VaR}$.
>
>$36 \times 5\% = 1.8$, so that $-17\% + 0.8 \times (-5\% - (-17\%)) = -7.4\%$

### Normality over Time
### Deviation from Normality and Value at Risk
When returns are normal, knowing just the mean and standard deviation allows us to fully describe the entire distribution. In that case, we do not need to estimate VaR explicitly—we can calculate it exactly from the properties of the normal distribution. But when returns are not normal, the VaR conveys important additional information beyond mean and standard deviation, specifically, the probability of severe losses arising from deviations from normality.

**Kurtosis**: Measure of the fatness of the tails of a probability distribution relative to that of a normal distribution. Indicates likelihood of extreme outcomes. $\odot$ Positive values indicate higher frequency of extreme values than this benchmark.$\square$

**Skew**: Measure of the asymmetry of a probability distribution.$\odot$ Negative skew suggests that extreme negative values are more frequent than extreme positive ones.$\square$

When the two additional statistics are nonzero, be careful using the $\text{VaR}$

### Using Time Series of Returns
### Risk Premiums and Risk Aversion
**risk-free rate**: The rate of return that can be earned with certainty. (like on treasury bills)

**risk premium**: An expected return in excess of that on risk-free securities. (expected HPR - risk-free rate)

**excess return**: Rate of return in excess of the risk-free rate.

**risk aversion**: Reluctance to accept risk.

Investors taking on risk to earn a risk premium are speculating; gambling is the assumption of risk for no purpose beyond the enjoyment of the risk itself.

To determine an investor’s optimal portfolio strategy, now we only look at the complete portfolio and quantify his degree of risk aversion. Following the *principle of revealed preference: We infer preferences from the choices individuals are willing to make*, for portfolio $Q$ with a variance $\sigma_{Q}^2$, the degree of risk aversion is 

$$\boxed{A = \frac{E(r_Q) - r_f} {\sigma_{Q}^2}}$$

**price of risk**: $A$, the ratio of portfolio risk premium to variance.

And there's an estimate of average price of risk for all investors in a market. Suppose average net borrowing and lending are zero, in which case the average investor's complete portfolio can be represented by a stock-market index $M$. We use a small sample, such as the S&P 500 to do the calculation.

$$\bar{A} = \frac{\bar{r}_M - r_f} {\sigma_M^2}$$

This $\bar{A}$ is also called the **market price of risk**.

### The Sharpe Ratio
**Sharpe ratio**: Ratio of portfolio risk premium to standard deviation.

$$S = \frac{\text{Portfolio risk premium}} {\text{Standard deviation of portfolio excess return}} = \boxed{ \frac{E(\sigma_P) - r_f} {\sigma_P}}$$

$\odot$
A higher Sharpe ratio indicates a better reward per unit of SD, a stock with higher efficiency$\square$

**mean-variance analysis**: Ranking portfolios by their Sharpe ratios.

## The Historical Record
### History of U.S. Interest Rates, Inflation, and Real Interest Rates
### World and U.S. Risky Stock and Bond Portfolios
## Asset Allocation Across Risky and Risk-Free Portfolios
A simple strategy to control portfolio risk is to specify the fraction of the portfolio invested in broad asset classes such as stocks, bonds, and safe assets.

**capital allocation to risky assets**: The choice between risky and risk-free assets.

**complete portfolio**: The entire portfolio including risky and risk-free assets.

### The Risk-Free Asset
$\odot$
1. Commonly Treasury bills are viewed as the risk-free asset.
2. All the money market instruments are virtually immune to interest rate risk.
3. The risk-free asset has a standard deviation of zero but unknown return rate.
$\square$

### Portfolio Expected Return and Risk
Assume $y$ fraction will be allocated to the risky portfolio, and the left $(1-y)$ for the risk free asset. We use $r_C$ for the return of complete portfolio and $r_P$ for the risky asset return. Then we have

$$E[r_C] = y(E[r_P]) + (1-y)r_f \Rightarrow E[r_C] - r_f = y(E[r_P] - r_f)$$

And besides, since the standard deviation of risk-free asset is $0$, we have $\sigma_C = y \cdot \sigma_P$.

>Every possible portfolio can be found on the line from point $F$ to $P$, with corresponding $y$ from $0$ to $1$.
>
>
>Notice that for this figure the intercept is the $r_f$ and the slope is Sharpe Ratio $S$. And some are risk lover who would take a leverage position investing in the risky portfolio.
>
>**e.g.** for $y=0.75$, do calculation:
>
>expected return: $E(r_C) = 7\% + 0.75(15\% - 7\%) = 13\%$
>
>risk premium: $E(r_C) - r_f = 13\% - 7\% = 6\%$
>
>standard deviation: $\sigma_C = 0.75 \times \sigma_P = 0.75 \times 22\% = 16.5\%$
>
>Shape Ratio: $S = \displaystyle \frac{13\% - 7\%} {16.5\%} = 0.36$
>
>$\odot$ And no matter how much is your $y$, the Shape Ratio never change.$\square$

### The Capital Allocation Line
**capital allocation line (CAL)**: Plot of risk-return combinations available by varying portfolio allocation between a risk-free asset and a risky portfolio.

Its slope is the Sharpe Ratio, or sometimes called the *reward-to-volatility* ratio.

### Risk Aversion and Capital Allocation
More risk-averse, less of the risky asset, more of the risk-free asset, nearer to point $F$ in the CAL. Just don't like taking any possible risk. (The opposite is called the risk-tolerant investors)

We let $A$ be the degree of risk aversion.

$$y = \frac{\text{Available risk premium to variance ratio}} {\text{Required risk premium to variance ratio}} = \frac{[E(r_P) - r_f]/\sigma_P^2} {A} = \frac{E(r_P) - r_f} {A\sigma_P^2}$$

Here the *available risk premium* can be just as the *market price of risk*, $\bar{A}$.

>**e.g.** Consider the following, find the deviation and mean of the complete portfolio
>
>$\text{Complete Portfolio } (C)\begin{cases}
\text{ Risk-Free Asset } (F): 1-y = 30\% \\[1em]
\text{ Risky Portfolio } (P): y = 70\% \begin{cases}
\text{ Risky asset } w_1 = 40\% \\ 
\text{ Risky asset } w_2 = 60\% 
\end{cases}
\end{cases}$
>
>We can choose money market fund as the risk-free asset, with $r_f = 2\%$, $\sigma_f \equiv 0$; and for the risky asset, we have $E(r_1) = 5\%$, $\sigma_1 = 8\%$, $E(r_2) = 10\%$, $\sigma_2 = 12\%$, and $\sigma_{1,2} = -10\%$.
>
>Now
>
>$$\begin{align}E(r_P) &= E(w_1r_1 + w_2r_2)\\
&= w_1E(r_1) + w_2E(r_2) \\
&= 0.4 \times 5\% + 0.6 \times 10\% \\
&= 8\%\end{align}$$
>
>and for the deviation we have 
>
>$$\begin{align} \sigma_P^2 &= \text{Var}(w_1r_1 + w_2r_2) \\
&= w_1^2\text{Var}(r_1) + w_2^2\text{Var}(r_2) + 2w_1w_2\text{Cov}(r_1,r_2) \\
&= 0.4^2 \times 8\%^2 + 0.6^2 \times 12\%^2 + 2 \times 0.4 \times 0.6 \times (-10\%\%) \\
&= 57.28\%\% \\
\Rightarrow \sigma_P &= \sqrt{57.28\%\%} = 7.57\% \end{align}$$
>
>Up to now,
>
>$$\text{Complete Portfolio } 100\% (C)\begin{cases}
1-y &= 30\%(F) &r_f = 2\%,\sigma_f \equiv 0 \\[1em]
y &= 70\%(P) &E(r_p) = 8\%, \sigma_p = 7.57\%
\end{cases}$$
>
>$$\begin{align}E(r_C) &= E(yr_P + (1-y)r_f)\\
&= yE(r_P) + (1-y)E(r_f) \\
&= 0.7 \times 8\% + 0.3 \times 2\% \\
&= 6.2\%\end{align}$$
>
>$$\begin{align} \sigma_C^2 &= \text{Var}(yr_P + (1-y)r_f) \\
&= y^2\text{Var}(r_P) + (1-y)^2\text{Var}(r_f) + 2y(1-y)\text{Cov}(r_P,r_f) \\
&= 0.7^2 \times 7.57\%^2 + 0.3^2 \times 0\%^2 + 2 \times 0.4 \times 0.6 \times 0 \\
&= 28.08\%\% \\
\Rightarrow \sigma_C &= \sqrt{28.08\%\%} = 5.3\% \end{align}$$

**Brief formula summary**

$$y = \frac{E(r_C) - r_f} {E(r_P) - r_f} = \frac{\sigma_C} {\sigma_P}\\
S = \frac{E(r_P) - r_f} {\sigma_P} = \frac{E(r_C) - r_f} {\sigma_C}$$

## Passive Strategies and the Capital Market Line
**passive strategy**: Investment policy that avoids security analysis. $\odot$ A passive strategy is based on the premise that securities are fairly priced, and it avoids the costs involved in undertaking security analysis.$\triangleleft$

To avoid the costs of acquiring information on any individual stock or group of stocks, we may follow a "neutral" **diversification approach**. Select a diversified portfolio of common stocks that mirrors the corporate sector. This results in a *value-weighted* portfolio, which, for example, invests a proportion in GE stock that is equal to the ratio of GE’s market value to the market value of all listed stocks. And this strategy is called **indexing** (The rate of return on the portfolio replicates the return on the index).

**capital market line (CML)**: The capital allocation line using the market index portfolio as the risky asset.

Normally we can use S&P 500 index as the market index. It can be also seen as the CAL connecting 1-month T-bills and a broad index of common stocks (e.g., S&P 500 Index).

For those who want more excess return but less risk, it means higher **Sharpe ratio**, which is $\displaystyle \frac{E(P) - r_f} {\sigma}$.

## Summary
- Investors face a trade-off between risk and expected return. Historical data confirm our intuition that assets with low degrees of risk should provide lower returns on average than do those of higher risk.
- Shifting funds from the risky portfolio to the risk-free asset is the simplest way to reduce risk. Another method involves diversification of the risky portfolio. We take up diversification in later chapters.
- U.S. T-bills provide a perfectly risk-free asset in nominal terms only. Nevertheless, the standard deviation of real rates on short-term T-bills is small compared to that of assets such as long-term bonds and common stocks, so for the purpose of our analysis, we consider T-bills the risk-free asset. Besides T-bills, money market funds hold short-term, safe obligations such as commercial paper and CDs. These entail some default risk but relatively little compared to most other risky assets. For convenience, we often refer to money market funds as risk-free assets.
- A risky investment portfolio (referred to here as the risky asset) can be characterized by its Sharpe, or reward-to-volatility ratio. This ratio is the slope of the capital allocation line (CAL), the line connecting the risk-free asset to the risky asset. All combinations of the risky and risk-free asset lie on this line. Investors would prefer a steeper-sloping CAL, because that means higher expected returns for any level of risk.
- An investor’s preferred choice among the portfolios on the capital allocation line will depend on risk aversion. Risk-averse investors will weight their complete portfolios more heavily toward Treasury bills. Risk-tolerant investors will hold higher proportions of their complete portfolios in the risky asset.
- The capital market line is the capital allocation line that results from using a passive investment strategy that treats a market index portfolio, such as the Standard & Poor’s 500, as the risky asset. Passive strategies are low-cost ways of obtaining well-diversified portfolios with performance that will reflect that of the broad stock market.

## Key Terms
- arithmetic average
- asset allocation
- capital allocation to risky assets
- capital allocation line (CAL)
- capital market line (CML)
- complete portfolio
- dollar-weighted average return
- excess return
- expected return
- geometric average
- holding-period return (HPR)
- inflation rate
- kurtosis
- mean-variance analysis
- nominal interest rate
- passive strategy
- price of risk
- probability distribution
- real interest rate
- risk aversion
- risk-free rate
- risk premium
- scenario analysis
- Sharpe (or reward-to- volatility) ratio
- skew
- standard deviation
- value at risk (VaR)
- variance

## Key Formulae
- Arithmetic average of $n$ returns, $\displaystyle (r_1 + r_2 + \cdots + r_n) \over n$
- Geometric average of $n$ returns, $(1 + r_1)(1 + r_2)\cdots (1 + r_n)^{1/n}-1$ or $\displaystyle \left( p_{\text{final}} \over p_{\text{start}}\right)^{1/n} - 1$
- Continuously compounded rate of return, $\text{APR} = r_{cc}= \ln(1 + \text{Effective annual rate})$
- Real rate of return: $\displaystyle \left( 1 + {\text{Nominal Return Rate}} \over 1 + {\text{Inflation Rate}} \right)^{1/n} - 1$
- Real rate of return (continuous compounding): $r_{\text{nominal}} - \text{Inflation Rate}$
- Expected return: $\sum [p(\text{Scenario}) \times \text{ Return in scenario}]$
- Variance: $\sum [p(\text{Scenario}) \times \text{ Deviation from mean in scenario}^2]$
- Standard Deviation: $\sqrt{\text{Variance}}$
- Sharpe Ratio: $\displaystyle {\text{Portfolio risk premium} \over \text{Standard Deviation of excess return}} = \frac{E(r_p) - r_f} {\sigma_p}$
- Optimal capital allocation to the risky asset: $y = \displaystyle \frac{E(r_p) - r_f} {A\sigma_p^2}$

# Assignment
1. You put up $\$50$ at the beginning of the year for an investment. The value of the investment grows $4\%$ and you earn a dividend of $\$3.50$. Your HPR was  
$$4\% + {\$3.50 \over \$50} = 11\%$$

2. The arithmetic average of $-11\%$, $15\%$, and $20\%$ is  
$${-11\% + 15\% + 20\% \over 3 }= 8\%$$

3. The geometric average of $-12\%$, $20\%$, and $25\%$ is
$$[(1-12\%)(1+20\%)(1+25\%)]^{1/3} - 1 = 9.7\%$$

4. You have an **APR** of $7.5\%$ with continuous compounding. The **EAR** is  
$$\textbf{EAR} = e^{\textbf{APR}} - 1 = 7.79\%$$

5. You have an **EAR** of $9\%$. The equivalent **APR** with continuous compounding is
$$\textbf{APR} = \ln(1 + \textbf{EAR}) = 8.62\%$$

6. During the 1926-2010 period which one of the following asset classes provided the lowest real return?
    - Long-term U.S. Treasury bonds
7. (***VIP***) If you are promised a nominal return of $12\%$ on a 1-year investment, and you expect the rate of inflation to be $3\%$, what real rate do you expect to earn
$$\left( 1.12 \over 1.03 \right) -1 = 8.74\%$$
8. You invest all of your money in 1-year T-bills. Which of the following statements is (are) correct?
    - Your **nominal return** on the T-bills is riskless.
    - Your **nominal Sharpe ratio** is zero.
    - WRONG: *Your real return on the T-bills is riskless.*
9. $$\begin{array}{c:c:c}
\text{Year} & \text{Beginning of Year Price} & \text{Number of Shares Bought or Sold} \\ \hline
2008 & 95 & 240 \text{ bought } \\ \hline
2009 & 100 & 190 \text{ bought } \\ \hline
2010 & 96 & 215 \text{ sold } \\ \hline
2011 & 99 & 215 \text{ sold } \\ \hline
\end{array}$$
What is the geometric average return for the period?
$$\left( 99 \over 95\right)^{1/\mathbf{3}} - 1 = 1.38\%$$

608e774f9bcb2d2d4546f3cb489051629dee9932
ipynb
Jupyter Notebook
FinMath/Intermediate Investment/Note_Chap05.ipynb
XavierOwen/Notes
d262a9103b29ee043aa198b475654aabd7a2818d
MIT
2018-11-27T10:31:08.000Z
2019-01-20T03:11:58.000Z
FinMath/Intermediate Investment/Note_Chap05.ipynb
XavierOwen/Notes
d262a9103b29ee043aa198b475654aabd7a2818d
MIT
FinMath/Intermediate Investment/Note_Chap05.ipynb
XavierOwen/Notes
d262a9103b29ee043aa198b475654aabd7a2818d
MIT
2020-07-14T19:57:23.000Z
2020-07-14T19:57:23.000Z
Qwen/Qwen-72B
1. YES
2. YES