Summarise the way a bond portfolio gets expressed through a set of standard benchmark instruments.

FRM® Part II · Market Risk Measurement and Management · VaR Mapping · FRM2.MR.5.d

The idea

Instead of tracking every bond, a fixed-income portfolio is expressed through a set of benchmark instruments, typically zero-coupon bonds at standard maturities called vertices, for example 3 months, 1, 2, 3, 5, 7, 10 and 30 years. Each position's cash flows are discounted to present value and allocated to the nearest vertices, keeping value and risk unchanged. The present values are then summed vertex by vertex across all bonds, with short positions entered as negatives, giving one net exposure vector. VaR follows from that vector and the vertex volatilities and correlations. Let z be the normal quantile, xᵢ the exposure at vertex i, σᵢ its volatility and ρᵢⱼ the correlation of vertices i and j. Summing over every pair of vertices, the portfolio risk is VaR = z × √(Σ xᵢ × xⱼ × σᵢ × σⱼ × ρᵢⱼ). Offsetting positions at the same vertex net out before risk is computed, and imperfectly correlated vertices diversify. The same vector shows where risk is concentrated and which benchmark instruments can hedge it. Exam point: net the signed exposures first, then apply correlations.

Formulas

VaR from vertex exposures

VaR = z × √(Σ xᵢ × xⱼ × σᵢ × σⱼ × ρᵢⱼ)

z: the standard normal quantile
xᵢ, xⱼ: the net present-value exposures at vertices i and j, shorts negative
σᵢ, σⱼ: the volatilities of vertices i and j
ρᵢⱼ: the correlation between vertices i and j, 1 when i = j

Sum over every pair of vertices. Net the signed exposures first; entering a short as long reverses the sign of its cross terms.

Minimum-variance hedge at a second vertex

hedge = x × ρ × σx ÷ σh

x: the exposure to be hedged
ρ: the correlation between the two vertices
σx: the volatility of the exposed vertex
σh: the volatility of the hedging vertex

The hedge is the opposite position. A less volatile hedging vertex needs more than the exposure; a correlation below 1 trims the amount.

A practice question

From the question bank, with the reason each wrong option is wrong.

After a bond portfolio has been mapped to benchmark vertices, the exposure assigned to a vertex is:

the sum of the face values of all bonds maturing at that vertex.Why not: Uses face values; exposures are present values after discounting and allocation.
the average yield of the bonds near the vertex, weighted by duration.Why not: Confuses an exposure in currency units with a yield; the vector holds present values.
the net present value of the cash flows allocated to it, with short positions entered as negatives. - correct
the vertex's volatility multiplied by the total portfolio value.Why not: Mixes the exposure with the volatility, which is a separate input to the VaR calculation.

The portfolio is summarised by a vector of net present values, one per vertex. Volatilities and correlations are then applied to this vector to obtain VaR.

A harder one

Bond A is long and maps to USD 5 million at the 1-year vertex and USD 8 million at the 3-year vertex. Bond B is a short position that maps to USD 7 million at the 1-year vertex and USD 2 million at the 3-year vertex. Daily volatilities are 0.50 percent (1-year) and 1.20 percent (3-year), with a correlation of 0.6. The one-day 95 percent VaR of the combined portfolio is closest to:

USD 129,000Why not: Treats the net 1-year exposure as long USD 2 million instead of short, losing the offset with the long 3-year exposure.
USD 120,000Why not: Sets the vertex correlation to zero.
USD 135,000Why not: Adds the two stand-alone VaRs, assuming perfect correlation and ignoring the sign of the short.
USD 109,000 - correct

Net exposures: 1-year = 5 - 7 = -2m, 3-year = 8 - 2 = +6m. Stand-alone amounts: -0.010m and +0.072m. Variance = 0.010² + 0.072² + 2 x 0.6 x (-0.010)(0.072) = 0.004420, so sigma = 0.0665m and VaR = 1.645 x 0.0665m = 0.109m.

The hardest tier

A mapped portfolio has exposures of USD 100 million at the 1-year vertex, minus USD 60 million at the 3-year vertex and USD 40 million at the 5-year vertex. Daily volatilities are 0.40, 0.70 and 1.20 percent. Correlations are 0.90 (1-year and 3-year), 0.70 (1-year and 5-year) and 0.85 (3-year and 5-year). The one-day 95 percent VaR is closest to:

USD 2,138,000Why not: Adds the three stand-alone VaRs, which assumes perfect correlation and all positions long.
USD 2,002,000Why not: Enters the 3-year short as a long exposure while still applying the correlations.
USD 718,000 - correct
USD 1,238,000Why not: Treats the three vertices as uncorrelated.

Stand-alone amounts: 0.400m, -0.420m and 0.480m. With the short sign and the correlations, the variance is 0.4² + 0.42² + 0.48² - 2(0.9)(0.4)(0.42) + 2(0.7)(0.4)(0.48) - 2(0.85)(0.42)(0.48) = 0.1905. Sigma = 0.4365m and VaR = 1.645 x 0.4365 = 0.718m.

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