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金融代写|金融数学代写Financial Mathematics代考|Interest Rate Risk Management

Fixed-income portfolio management is usually focused on managing the interest rate risk of a portfolio relative to a benchmark or a related set of liabilities. Benchmarking is the process of identifying an index or other source of observable returns and using those returns as a point of reference for evaluating the success of an investment strategy. For example, a fixedincome manager for a mutual fund might desire that the effect of interest rate shifts on the manager’s portfolio be very close to the effects of the same interest rate shifts on the value of the manager’s benchmark such as a published index of bond returns. A fixed-income manager of a pension fund may be more concerned about making sure that the bond portfolio’s ability to provide funds to meet anticipated liabilities (pension benefits) is protected from interest rate shifts.

The process of ensuring that the goals for a fixed-income portfolio (benchmarking or liability funding) will be met despite interest rate changes is known as immunization. One of the premier methods of obtaining immunization is duration matching. In the case of a mutual fund manager attempting to track a benchmark such as a major bond index, the manager will tend to make sure that the duration of her portfolio is very close to the duration of the benchmark. In the case of a pension fund manager attempting to ensure that the fund’s investments will be able to meet the pension obligations that manager will tend to make sure that the duration of her portfolio is very close to the duration of the plan’s anticipated liabilities (assuming the plan is fully funded with the value of the investments equaling the present value of the liabilities).

Another method of interest rate risk management is cash flow matching (i.e., matching the various cash flow longevities of the portfolio to the cash flow longevities of the benchmark or liability stream). The problem with cash flow matching is that it can be very inflexible with regard to the investment opportunities that the manager can select.
Duration provides a first-order solution that permits flexibility. The inclusion of convexity matching tends to improve the accuracy of the immunization but also tends to restrict the flexibility of the manager to include diverse investment opportunities.
In Chapter 5, we will discuss hedging strategies to manage interest rate risk using duration in detail. In the next subsection, we will discuss other risks associated with fixed-income security: credit risk and default risk. The management of credit risk and default risk will be discussed in detail in Chapter 8 .

金融代写|金融数学代写Financial Mathematics代考|Introduction to Credit Risk and Default Risk

Credit risk refers to uncertainty regarding a borrower’s ability to make timely interest and principal payments. Credit risk includes: (1) volatility of market prices caused by changes in a borrower’s perceived ability to meet debt obligations, (2) fluctuations in the risk premiums (credit spreads) throughout the credit markets, and (3) losses due to defaults. Default risk is the risk that the borrower will make only partial payments, no payments, and/or delayed payments.
In the case of fixed income assets that have no risk of default, the present value or market value of the asset can be found by discounting the promised cash flows at the riskless interest rates corresponding to the longevity of each cash flow. In the case of fixed-income investments (not derivatives) with default risk, the present value or market value of the asset is found by discounting the promised cash flows with a risky discount rate that is equal to the sum of a riskless interest rate and a credit spread, s. Equation $2.22$ modifies the general asset valuation formula (Equation 2.10) by including a discount factor that captures the effect of potential default through the inclusion of a credit spread
$$
\tilde{B}(0, T)=\sum_{i=1}^n C_{t_i} e^{-\left(r_i+s\right) t_i}
$$
where $\tilde{B}(0, T)$ is the $T$ year risky bond price today.
Equation $2.22$ is based on a term-structure approach in which cash flows of different longevity are discounted at interest rates corresponding to the cash flow’s longevity. Equation $2.23$ illustrates the use of a credit spread in a yield-based valuation model in which all cash flows are discounted at the same rate $(y)$ in which $y$ contains the credit spread
$$
\tilde{B}(0, T)=\sum_{i=1}^n C_{t_i} e^{-y t_i}
$$
where $y$ is the risky bond yield.
If the term structure of risk-free interest rates is flat, then Equation $2.22$ becomes:
$$
\tilde{B}(0, T)=\sum_{i=1}^n C_{t_i} e^{-(r+s) t_i}
$$
Comparing Equations (2.23) and (2.24), note that we use $y=r+s$ in Equation $2.24$ in which $s$ is the yield spread or credit spread.

金融代写|金融数学代写金融数学代考|利率风险管理

固定收益投资组合管理通常侧重于管理投资组合相对于基准或相关负债的利率风险。对标是确定一个指数或其他可观察收益来源的过程，并将这些收益作为评价投资策略成功与否的参照点。例如，共同基金的固定收益经理可能希望，利率变动对其投资组合的影响，与利率变动对其基准价值(如已公布的债券回报指数)的影响非常接近。养老基金的固定收益经理可能更关心的是，确保债券投资组合提供资金以满足预期负债(养老金福利)的能力不受利率变动的影响

在利率变化的情况下，确保固定收益投资组合的目标(基准或负债融资)将得到满足的过程被称为免疫。获得免疫的主要方法之一是持续时间匹配。在一个共同基金经理试图跟踪一个基准的情况下，例如一个主要的债券指数，经理将倾向于确保她的投资组合的持续时间非常接近基准的持续时间。如果一个养老基金经理试图确保基金的投资将能够满足养老金义务，该经理将倾向于确保其投资组合的持续时间非常接近计划的预期负债的持续时间(假设该计划的资金充足，投资价值等于负债的现值)

利率风险管理的另一种方法是现金流匹配(即，将投资组合的各种现金流寿命与基准或负债流的现金流寿命进行匹配)。现金流匹配的问题在于，在经理所能选择的投资机会方面，它可能非常缺乏灵活性。
Duration提供了一阶解决方案，允许灵活性。包含凸度匹配往往会提高免疫的准确性，但也往往会限制经理人包含不同投资机会的灵活性。在第五章中，我们将详细讨论利用期限管理利率风险的套期保值策略。在下一小节中，我们将讨论与固定收益证券相关的其他风险:信用风险和违约风险。第8章将详细讨论信用风险和违约风险的管理

金融代写|金融数学代写金融数学代考|信用风险与违约风险简介

.

信用风险是指借款人是否有能力及时支付利息和本金的不确定性。信用风险包括:(1)由于借款人偿债能力的变化而引起的市场价格波动，(2)整个信贷市场的风险溢价(信用价差)波动，以及(3)违约造成的损失。违约风险是借款人只支付部分付款、不付款和/或延迟付款的风险。在没有违约风险的固定收益资产中，资产的现值或市场价值可以通过按每个现金流的存续期所对应的无风险利率对承诺现金流进行折现得到。对于有违约风险的固定收益投资(不是衍生品)，资产的现值或市场价值是通过对承诺现金流进行折现，折现率等于无风险利率和信贷利差之和，s.公式$2.22$修改了一般资产估值公式(公式2.10)，通过包含信用价差
$$
\tilde{B}(0, T)=\sum_{i=1}^n C_{t_i} e^{-\left(r_i+s\right) t_i}
$$
，其中$\tilde{B}(0, T)$是当前$T$年风险债券价格，从而包含了一个捕捉潜在违约影响的贴现因子。公式$2.22$基于期限结构方法，其中不同寿命的现金流按现金流寿命对应的利率折现。公式$2.23$说明了在基于收益的估值模型中使用信用利差，其中所有现金流以相同的比率贴现$(y)$，其中$y$包含信用利差
$$
\tilde{B}(0, T)=\sum_{i=1}^n C_{t_i} e^{-y t_i}
$$
，其中$y$是风险债券收益率。如果无风险利率期限结构是平坦的，则公式$2.22$变成:
$$
\tilde{B}(0, T)=\sum_{i=1}^n C_{t_i} e^{-(r+s) t_i}
$$
比较式(2.23)和(2.24)，注意我们在公式$2.24$中使用$y=r+s$，其中$s$是收益率差或信用差

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