# LO 4.4: Summarize how to map a fixed income portfolio into positions o f standard

LO 4.4: Summarize how to map a fixed income portfolio into positions o f standard instruments.
To illustrate principal, duration, and cash flow mapping, we will use a two position fixed- income portfolio consisting of a one-year bond and a five-year bond. You will notice in the following examples that the primary difference between these mapping techniques is the consideration of the timing and amount of cash flows.
Suppose a portfolio consists of two par value bonds. One bond is a one-year \$100 million bond with a coupon rate of 3.5%. The second bond is a five-year \$100 million bond with a coupon rate of 5%. In this example, we will differentiate between the timing and cash flows used to map the VaR for this portfolio using principal mapping, duration mapping, and cash flow mapping. The risk percentages (or VaR percentages) for zero-coupon bonds with maturities ranging from one to five years (at the 95% confidence level) are as follows:
Maturity
1 2 3 4 5
VAR % 0.4696 0.9868 1.4841 1.9714 2.4261
2018 Kaplan, Inc.
Page 41
Topic 4 Cross Reference to GARP Assigned Reading – Jorion, Chapter 11
Principal mapping is the simplest of the three techniques as it only considers the timing of the redemption or maturity payments of the bonds. While this simplifies the process, it ignores all coupon payments for the bonds. The weights in this example are both 50% (i.e., \$100 million / \$200 million). Thus, the weighted average life of this portfolio for the two bonds is three years [0.50(1) + 0.50(5) = 3].
As Figure 3 illustrates, the principal mapping technique assumes that the total portfolio value of \$200 million occurs at the average life of the portfolio, which is three years. Note that the VaR percentage at the 95% confidence level is 1.4841 for a three-year zero-coupon bond. We compute the VaR under the principal method by multiplying the VaR percentage times the market value of the average life of the bond, as follows:
Principal mapping VaR = \$200 million x 1.4841% = \$2,968 million
Figure 3: Fixed-Income Mapping Techniques
Year 1 2
2.768
3 4 5
CFs fo r
5- Year Bond
\$5 \$5
\$5 \$5 \$105
CFs fo r
1-Year Bond
\$103.5
\$0
\$0 \$0 \$0
Spot Rates 3.50% 3.90%
4.19% 4.21% 5.10%
M apping Technique
Principal
D uration
\$200
\$200
\$200
\$200
PV(CF) \$104.83 \$4.63
\$4.42 \$4.24 \$81.88 \$200.00
In the last three columns of Figure 3, you can see the differences in the amounts and timing of cash flows for all three methods. To calculate the VaR of this fixed-income portfolio using duration mapping, we simply replace the portfolio with a zero-coupon bond that has the same maturity as the duration of the portfolio. Figure 4 demonstrates the calculation of Macaulay duration for this portfolio. The numerator of the duration calculation is the sum of time, t, multiplied by the present value of cash flows, and the denominator is simply the present value of all cash flows. Duration is then computed as \$553.69 million / \$200 million = 2.768.
Figure 4: Duration Calculation
Year 1 2 3 4 5
C F for
5- Year Bond
CF fo r
1-Year Bond
\$5 \$5 \$5 \$5 \$105
\$103.5
\$0 \$0 \$0 \$0
Spot Rate PV(CF) 3.50% \$104.83 3.90% \$4.63 \$4.42 4.19% 4.21% \$4.24 \$81.88 5.10% \$200.00
t X PV(CF) \$104.83 \$9.26 \$13.26 \$16.96 \$409.38 \$553.69
Page 42
2018 Kaplan, Inc.
Topic 4 Cross Reference to GARP Assigned Reading – Jorion, Chapter 11
The next step is to interpolate the VaR for a zero-coupon bond with a maturity of 2.768 years. Recall that the VaR percentages for two-year and three-year zero-coupon bonds were 0.9868 and 1.4841, respectively.
The VaR of a 2.768 year maturity zero-coupon bond is interpolated as follows:
0.9868 + (1.4841 – 0.9868) x (2.768 – 2 ) = 0.9868 + (0.4973 x 0.768) = 1.3687
We now have the information needed to calculate the VaR for this portfolio using the interpolated VaR percentage for a zero-coupon bond with a 2.768 year maturity:
Duration mapping VaR = \$200 million x 1.3687% = \$2,737 million
In order to calculate the VaR for this fixed-income portfolio using cash flow mapping, we need to map the present value of the cash flows (i.e., face amount discounted at the spot rate for a given maturity) onto the risk factors for zeros of the same maturities and include the inter-maturity correlations. Figure 3 summarizes the required calculations. The second column of Figure 3 provides the present value of cash flows that were computed in Figure 3. The third column of Figure 5 multiplies the present value of cash flows times the zero- coupon VaR percentages.
Figure 5: Cash Flow Mapping
Year 1 2 3 4 5
X
104.83 4.63 4.42 4.24 81.88
Undiversified VaR
x x V 0.4923 0.0457 0.0656 0.0836 1.9864 2.674
1Y 1
0.894 0.887 0.871 0.861
Correlation Matrix (R) 2F 4Y 0.894 0.871 0.964 0.992
3Y 0.887 0.99 1 1 1 0.99 0.964 0.954
0.992 0.987
1
0.996
5T 0.861 0.954 0.987 0.996
1
xAVaR 1.17 0.115 0.170 0.217 5.168 6.840
If the five zero-coupon bonds were all perfectly correlated, then the undiversified VaR could be calculated as follows:
Diversified VaR
Undiversified VaR = ^^|xj x Vj
N
i= l
In this example, the undiversified VaR is computed as the sum of the third column: 2.674.
2018 Kaplan, Inc.
Page 43
Topic 4 Cross Reference to GARP Assigned Reading – Jorion, Chapter 11
The correlation matrix provided in the fourth through eighth columns of Figure 5 provides the inter-maturity correlations for the zero-coupon bonds for all five maturities. The diversified VaR can be computed using matrix algebra as follows:
Diversified VaR = ol^Jx ‘^ ^ x = ^ (x x V )rR(x x V)
Where x is the present value of cash flows vector, Vis the vector of VaR for zero-coupon bond returns and R is the correlation matrix. The last column of Figure 5 summarizes the computations for the matrix algebra. The square root of the sum of this column (6.840) is the diversified VaR using cash flow mapping and is calculated as 2.615.
Notice that in order to calculate portfolio diversified VaR using the cash flow mapping method, we need to incorporate the correlations between the zero-coupon bonds. As you can see, cash flow mapping is the most precise method, but it is also the most complex.
Professors Note: The complex calculations required fo r cash flow mapping would be very time consuming to perform using a fin an cial calculator. Therefore, this calculation it is highly unlikely to show up on the exam.
S t r e s s T e s t i n g