# 金融代写|利率理论代写portfolio theory代考|Biased Bootstrapping and Scenario Analysis

## 金融代写|利率理论代写portfolio theory代考|Time Series Dependence

The most important assumption of the bootstrap is the independence of returns through time. Unfortunately, there are many financial times series for which this is not a reasonable assumption. Treasury bill returns, for example, have a high degree of autocorrelation from one year to the next (the value next year is highly dependent on this year’s value). Inflation has this characteristic as well. It is possible to address this problem, however, it requires some additional statistical modeling and estimation.

Suppose, for example, we wished to create a pseudo-history of U.S. inflation. First, we must estimate a model of the inflation process $I$ :
$$I_t=a+b I_{t-1}+e_t$$
where $I_{t-1}$ is last year’s inflation rate and $t$ ranges from 1926 to 2007. For example, if $t=$ 1928 , then $I_{t-1}$ is inflation in 1927. The coefficients $a$ and $b$ can be estimated from a linear regression, and $e_t$ is the error term from the regression for year $t$. Let $e_t{ }^$ be an error term drawn randomly with replacement from the actual residuals for 1926 to 2007 (79 residuals). ${ }^{18}$ Let $I_t^$ be the variable we use to indicate the bootstrapped value of inflation at time $t$. To construct the bootstrapped series, we begin with an actual starting value, $\mathrm{I}_{1926}$, the inflation rate in $1926 . I^{ }{1927}$ is calculated as $a+b I{1926}+e_t^$, where $e_t^$ is drawn from the 79 residuals. The next bootstrapped inflation year in the sequence builds on the previous value: $I^{ }{1928}$ $=a+b I^{ }{1927}+e_{t+1}{ }^$, where this $e_{t+1} *$ is drawn with replacement for the period $t+1$ from the 76 regression residuals as before. This process continues until an entire 79-year pseudohistory of inflation is constructed. This method, based on bootstrapping the errors in the autocorrelation model, now preserves the time series dependency of annual inflation, as well as its approximate historical mean and standard deviation. The methodology can also be easily combined with a multiple-asset bootstrap to preserve the correlation between asset returns and autocorrelated series such as inflation or Treasury bills.

## 金融代写|利率理论代写portfolio theory代考|THE ECONOMIC PROPERTIES OF UTILITY FUNCTIONS

The first restriction placed on a utility function is that it be consistent with more being preferred to less. This attribute, known in the economic literature as nonsatiation, simply says that the utility of more $(X+1)$ dollars is always higher than the utility of less $(X)$ dollars. Thus, if we want to choose between two certain investments, we always take the one with the largest outcome. If we are concerned with end-of-period wealth, this property states that more wealth is always preferred to less wealth. If utility increases as wealth increases, then the first derivative of utility, with respect to wealth, is positive. Thus the first restriction placed on the utility function is a positive first derivative.

The second property of a utility function is an assumption about an investor’s taste for risk. Three assumptions are possible: the investor is averse to risk, the investor is neutral toward risk, or the investor seeks risk. Risk aversion, risk neutrality, and risk seeking can all be defined in terms of a fair gamble. Consider the gambles (options) shown in Table 11.11.
The option “invest” has an expected value of $(1 / 2)(2)+(1 / 2)(0)=\$ 1$. Assume that an investor would have to pay$\$1$ to undertake this investment and obtain these outcomes. Thus, if the investor chooses not to invest, the $\$ 1$is kept. This is the alternative: do not invest. The expected value of the gamble is exactly equal to the cost. The position of the investor may be improved or hurt by undertaking the investment, but the expectation is that there will be no change in position. Because the expected value of the gamble shown in Table 11.11 is equal to its cost, it is called a fair gamble. Risk aversion means that an investor will reject a fair gamble. In terms of Table 11.11 , it means$\$1$ for certain will be preferred to an equal chance of $\$ 2$or$\$0$. Risk aversion implies that the second derivative of utility, with respect to wealth, is negative. If $U(W)$ is the utility function and $U^{\prime \prime}(W)$ is the second derivative, then risk aversion is usually equated with an assumption that $U^{\prime \prime}(W)<0$. Let us examine why this is true.

If an investor prefers not to invest, then the expected utility of not investing must be higher than the expected utility of investing, or
$$U(1)>\frac{1}{2} U(2)+\frac{1}{2} U(0)$$
Multiplying both sides by 2 and rearranging, we have
$$U(1)-U(0)>U(2)-U(1)$$

# 利率理论代考

## 金融代写|利率理论代写portfolio theory代考|Time Series Dependence

bootstrap 最重要的假设是回报随时间的独立性。不幸的是，对于许多金融时间序列来 说，这不是一个合理的假设。例如，短期国库券收益从一年到下一年具有高度的自相关 性 (明年的价值高度依赖于今年的价值) 。通货膨胀也有这个特点。解决这个问题是可 能的，但是，它需要一些额外的统计建模和估计。

$$I_t=a+b I_{t-1}+e_t$$

$a+b$ 我 ${1926}+e_{-} t^{\wedge}$ ， 在哪里 $e_{-} t^{\wedge}$ 是从 79 个残差中提取的。序列中的下一个自举通胀 年建立在之前的值之上: $I 1928=\mathrm{a}+\mathrm{b} \mid \wedge\left{{1927}+e_{-}{t+1}\langle}^{\wedge}\right.$, 这个在哪里 $e_{t+1} *$ 抽取期 间更换 $t+1$ 和以前一样来自 76 个回归残差。这个过程一直持续到构建了整个 79 年的 通货膨胀假历史。这种方法基于引导自相关模型中的误差，现在保留了年度通货膨胀的 时间序列依赖性，以及它的近似历史均值和标准差。该方法还可以轻松地与多资产自举 相结合，以保持资产回报与自相关序列（例如通货膨胀或国库券) 之间的相关性。

## 金融代写|利率理论代写portfolio theory代考|THE ECONOMIC PROPERTIES OF UTILITY FUNCTIONS

“投资”选项的预期值为 $(1 / 2)(2)+(1 / 2)(0)=\$ 1$. 假设投资者必须支付$\$1$ 进行这项投 资并获得这些成果。因此，如果投资者选择不投资，则 $\$ 1$保持。这是另一种选择: 不 投资。赌博的期望值恰好等于成本。进行投资可能会改善或损害投资者的地位，但期望 地位不会发生变化。因为表 11.11 所示赌博的期望值等于其成本，所以称为公平奢博。 风险规避意味着投资者将拒绝公平的赌博。就表 11.11 而言，这意味着$\$1$ 因为某些人 会比平等的机会更受欢迎 $\$ 2$或者$\$0$. 风险规避意味着关于财富的效用的二阶导数是负 的。如果 $U(W)$ 是效用函数，并且 $U^{\prime \prime}(W)$ 是二阶导数，那么风险犬恶通常等同于一个 假设 $U^{\prime \prime}(W)<0$. 让我们来看看为什么这是真的。 如果投资者不愿意投资，那么不投资的预期效用一定高于投资的预期效用，或者 $$U(1)>\frac{1}{2} U(2)+\frac{1}{2} U(0)$$

$$U(1)-U(0)>U(2)-U(1)$$

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