Can you assist me with the following work. the second attachment is an example of how to get the answers

Overview The purpose of this assignment is to introduce you to the mapping of cash flows. Your task will be to allocate the cash flows between the standard RiskMetrics vertices. Data Suppose that on March 27, 2015, an investor owns 100,000 of the French OAT benchmark 7.5% maturing in April 2025. This bond pays coupon flows of 7,500 each over the next 10 years and returns the principal investment at maturity. One of these flows occurs in 6.08 years, between the standard vertices of 5 and 7 years (for which volatilities and correlations are available). RiskMetrics data for March 27, 2015 Risk Metrics Vertex 5yr 7yr Yield,% 7.628 7.794 % Price volatility (1.65 t), % 0.533 0.696 Correlation Matrix ij 5yr 1.000 0.963 7yr 0.963 1.000 To Do 1. Please follow the steps in the example in Lecture Notes #6. First, calculate the actual cash flow's interpolated yield. 2. Then determine the actual cash flow's present value. You may use PV=Cash Flow/(1+yield)^(time period). 3. Calculate the standard deviation of the price return on the actual cash flow. Please remember that the volatility represents 1.65xt. 4. Calculate both and (1-). Again you need to convert 1.65 t into t. 5. Then allocate the present value of cash flow into RiskMetrics vertex cash flows. That means cash flow for vertex 5 and cash flow for vertex 7. Week 6 - VaR Mapping and Bond VaR Welcome to week 6 Bond VaR. With the background knowledge on bond portfolio management last week, we will study VaR mapping and its application in bond VaR. 1. Mapping for risk measurement When we deal with a very large portfolio positions, it is practically impossible to model all positions individually as risk factors. Mapping is a method where many positions are aggregated into a small set of exposures without loss of risk information. Sometimes mapping is the only solution when the characteristics of the instrument change over time. For example, the risk profile of bonds changes over time. We cannot use the history of prices on a bond directly. The price each time represents at least different maturities. Instead, the bond must be mapped on yields that best represent its current profile. 2. Mapping fixed income portfolios There are three mapping systems for fixed income portfolios: principal, duration, and cash flows. Principal mapping: one risk factor is chosen that corresponds to the average portfolio maturity. Duration mapping: one risk factor is chosen that corresponds to the portfolio duration. Cash flow mapping: the portfolio cash flows are grouped into maturity buckets. Mapping should preserve the market value of the position. Ideally, it also should preserve its market risk. (Example) We have a two-bond portfolio consisting of a $100 million 5-year, 6 percent issue and a $100 million 1-year, 4 percent issue. Both issues are selling at par, implying a market value of $200 million. The portfolio has an average maturity of 3 years and a duration of 2.733 years. Term (year) 6% 5-year 4% 1-year Spot rate Cash flow Principal Duration 1 6 104 4,000 105.77 0 0 2 6 0 4,618 5.48 0 0 2.733 - - - 200 3 0 6 5.192 5.15 $200 0 4 6 0 5.716 4.80 0 0 5 106 0 6.112 78.79 0 0 (Principal mapping) It only considers the timing of redemption payments only. It has a 3year maturity and the risk for 3-year zeros is 1.484% percent (from the next table). Then the VaR is $200x1.484/100=$2.97 million. Although it is simple, it ignores intermittent coupon payments. (Duration mapping) 1 D P D* T t 1 Ct (1 y ) t t D (1 y ) dP D * dy P dP ( ) D * ( dy ) P In the next table, the risk for two and three year zeros are 0.9868 and 1.4841. Using linear interpolation, the risk will be 0.987+(1.484-0.987)x(2.733-2)=1.351 percent. With $200 million portfolio, the duration-based VaR is $200 X1.351/100=$2.70 million. (Cash flow mapping) Term (year) 1 2 3 4 5 6% 5-year 6 6 6 6 106 4% 1-year 104 0 0 0 0 Spot rate Cash flow 4,000 105.77 4,618 5.48 5.192 5.15 5.716 4.80 6.112 78.79 The cash-flow mapping consists of grouping all cash flows on term-structure \"vertices\" that correspond to maturities for which volatilities are provided. Each cash flow is represented by the present value of the cash payment, discounted at the appropriate zero-coupon rate. Risk(%) Term CF X*V Correlation Matrix (year) *($m) (%) 0.4696 1 105.77 49.66 1 0.9868 2 5.48 5.40 .897 1 1.4841 3 5.15 7.65 .886 .991 1.9714 4 4.80 9.47 .866 .976 .994 2.4261 5 78.79 191.15 .855 .966 .988 .988 Total 200.00 263.35 1Y 2Y 3Y VaR 4Y 5Y ($m) 1 1 1 VaR($m) Undiversified $ 2.63 Diversified $ 2.57 Please note that Risk(%) as well as V in the table represent . Now you can verify both undiversified and diversified VaR. Also note that R is the correlation matrix. The diversified VaR: = = ( )( ) Undiversified VaR: = | | =1 One observation from this example is that on a risk basis, there is no difference between holding the simple 5-year bond or the corresponding five zero coupon bonds. If someone out there already calculated the risk of the vertices like the example above, our life will be much easier. Actually that's what has happened in JP Morgan Riskmetrics. 3. Mapping cash flows onto RiskMetrics vertices Financial instruments, in general, can generate numerous cash flows, each one occurring at a unique time. This gives rise to an unwieldy number of combinations of cash flow dates when many instruments are considered. In order to get these intractable number of volatilities and correlations under control, we need to simplify the time structure of these cash flows. The RiskMetrics method is the answer to this task. The RiskMetrics method of simplifying time structure involves cash flow mapping, i.e., redistributing (mapping) the observed cash flows onto so-called RiskMetrics vertices, to produce RiskMetrics cash flow. 3.1. 1m 3m RiskMetrics vertices 6m 12m 2yr 3yr 4yr 5yr 7yr 9yr 10yr 15yr 20yr 30yr (two important properties) 1. These vertices are fixed. 2. RiskMetrics data set provides volatilities and correlations for each of these vertices. Three Conditions. Cash flow: Market value is preserved. Cash flow: Market risk is preserved. Sign is preserved. (Example) Allocating a cash flow in year 6 to the 5- and 7-year vertices (1) Calculating the actual Cash flow's interpolated yield ^ ^ y6 y5 (1 ) y7 , ^ 0 1 ( Equation 1) y6 interpolated 6 year zero yield , 0.5 ^ linearweig htingcoeff icient . y5 5 year zero yield y7 7 year zero yield (2) determine the actual Cash flow's present value From the 6-year zero yield, y6 , we determine the present value, P6 , of the cash flow occurring at the 6-year vertex. (3) Calculate the standard deviation of the price return on the actual cash flow: From 5and 7 , calculate 6 . ^ ^ ^ 6 5 (1 ) 7 0 1 ( Eq 2) ^ where linear wei ghting coefficien t from eq 1 5 standard deviation of the 5 year yield 7 standard deviation of the 7 year yield (4) Calculating both and (1-). Variance ( r6 yr ) Variance[r5 yr (1 )r7 yr ], or 2 2 2 6 2 5 2 (1 ) 5,7 5 7 (1 )2 7 ( Eq. 3) 2 2 2 2 2 ( 5 7 2 5,7 5 7 ) 2 ( 2 5,7 5 7 2 7 ) ( 7 6 ) 0 b b2 4ac 2a (Numerical Example) Daily RiskMetrics data set: y5 5 yr yield 6.605% y7 7 yr yield 6.745% 1.65 5 Volitility on the 5 yr bond price return 0.5770% 1.65 7 Volitility on the7 yr bond price return 0.8095% 5, 7 Correlatio n b / n the 5 yr and 7 yr bondreturn 0.9975 Calculation results: y6 6 yr yield 6.675% ^ ( From Eq 1, 0.5) 6 s tan dard deviation on the 6 yr bond price return 0.4202% 2 6 var iance on the 6 yr bond price return 1.765 10 3 % ( From eq 2) 2 5 var iance on the 5 yr bond price return 1.223 10 3 % 2 7 var iance on the 7 yr bond price return 2.406 10 3% a 2.14 10 6 b 1.39 10 5 c 6.41 10 6 .4966 Now you are ready to calculate the VaR