Stress-Testing Nonparallel Yield Changes

Why Nonparallel Yield Stresses Matter

An asset and a liability portfolio can have similar values and still respond differently when yields change by maturity. A short-term yield increase combined with a long-term yield decrease, for example, is not a parallel shift. The change in surplus depends on how much asset and liability value is exposed in each maturity region, not just on their total values or average durations.

A useful stress test therefore has two parts. First, estimate the change using maturity-specific key-rate sensitivities. Then, when cash flows and stressed yields are specified, revalue each maturity bucket under the stressed yields. The estimate shows which exposures matter; exact repricing captures the nonlinear effect of a finite yield change.

Definition: A nonparallel yield stress assigns different yield changes to different maturities. A twist typically moves shorter and longer yields in opposite directions. A bend moves an intermediate maturity region differently from the shorter and longer regions. These labels describe the pattern of shocks; the actual stressed yields must be specified to value the portfolios.

For the calculations below, each cash-flow maturity $t$ has a stated annual effective base yield $y_t$, and the cash flows in that bucket are discounted at that yield. A yield shock $\Delta y_t$ changes that bucket's yield to $y_t+\Delta y_t$. This is a stated scenario convention, not a claim that every market curve is represented by a single yield for each cash-flow date. If a scenario instead defines interpolated curve shifts, the exposures must reflect that interpolation.

Key-Rate Estimates of Surplus Change

Let $P_t^A$ and $P_t^L$ be the base present values of asset and liability cash flows at maturity $t$, respectively. Define the net present value in that bucket as $N_t=P_t^A-P_t^L$. A positive $N_t$ means assets exceed liabilities in that bucket; a negative $N_t$ means liabilities exceed assets.

With annual effective yield $y_t$, the first-order dollar sensitivity of a bucket's present value to its yield is $-tP_t/(1+y_t)$. Thus, for a small change $\Delta y_t$, the first-order change in surplus is:

\Delta S \approx -\sum_t \frac{tN_t}{1+y_t}\Delta y_t

The factor $tN_t/(1+y_t)$ is a signed dollar exposure. It is not a percentage duration: it already includes the bucket's dollar size and the asset-minus-liability sign. Use decimal rate changes in the formula: a rise of 0.20 percentage point is $\Delta y=0.002$, not $0.20$.

Why the First-Order Formula Holds

For a fixed cash flow at time $t$, discounted at an annual effective yield $y$, its present value is $P(y)=C(1+y)^{-t}$, where $C$ is the payment. Differentiating with respect to the yield gives:

\frac{dP}{dy}=-tC(1+y)^{-t-1}=-\frac{tP(y)}{1+y}

For a small yield change, the first-order change is the derivative times the change, so $\Delta P\approx -tP\Delta y/(1+y)$. Apply this to asset and liability cash flows in each bucket and subtract the liability change from the asset change. Since $N_t=P_t^A-P_t^L$, the result is the surplus formula above. The approximation omits curvature; it is most reliable for small shocks, while exact repricing is preferable for larger shocks or when precision matters.

Worked Example: Combining Mixed Bucket Exposures

At a flat base yield of 4% effective annually, the net asset-minus-liability present values are $40 at year 1, $-$30$ at year 2, and $10 at year 4. The yield changes in those buckets are, respectively, +0.20, −0.30, and +0.10 percentage point. Estimate the first-order surplus change and identify the largest contribution.

Method. Apply the signed key-rate formula bucket by bucket. Convert the shocks to decimals: $0.002$, $-0.003$, and $0.001$. The denominator is $1.04$ in every bucket.

At year 1, the quantity inside the sum is:

\frac{1(40)}{1.04}(0.002)=0.076923

At year 2, the net present value is negative and the yield change is also negative, so their product is positive:

\frac{2(-30)}{1.04}(-0.003)=0.173077

At year 4:

\frac{4(10)}{1.04}(0.001)=0.038462

The sum inside the formula is $0.076923+0.173077+0.038462=0.288462$. Therefore:

\Delta S\approx -0.288462\text{ dollars}\approx -\$0.29

The year-2 bucket has the largest first-order contribution to the sum, $0.173077$, and therefore contributes about $-$0.17$ to the surplus change. The year-1 and year-4 contributions to surplus are about $-$0.08$ and $-$0.04$, respectively. The contributions sum to approximately $-$0.29$, as required.

Common mistake avoided: A negative net bucket does not automatically mean a negative surplus change. At year 2, both the net exposure and the yield shock are negative, making their product positive; the leading minus sign then makes that bucket's surplus contribution negative.

Exact Repricing Under a Specified Stress

When each bucket has a single maturity and a single stated yield, exact repricing follows directly from the present-value definition. If the base present value in bucket $t$ is $P_t$, then its underlying cash flow is $P_t(1+y_t)^t$. Discounting that same cash flow at $y_t+\Delta y_t$ gives:

P_t^{\text{stress}} =P_t\left(\frac{1+y_t}{1+y_t+\Delta y_t}\right)^t

The cash flow is unchanged; only the yield used to discount it changes. For a portfolio, apply this ratio to every asset and liability bucket separately, then calculate stressed surplus as stressed assets minus stressed liabilities. Do not apply the ratio to total portfolio value unless all included cash flows have the same maturity and yield assumption.

Worked Example: Repricing a Two-Maturity Twist

A hypothetical asset portfolio has base present values of $80 at year 1 and $120 at year 3. A liability portfolio has base present values of $60 at year 1 and $140 at year 3. The annual effective base yield is 4% at both maturities. Under a twist, the year-1 yield rises by 0.50 percentage point and the year-3 yield falls by 0.25 percentage point. Find the first-order estimate and the exact change in surplus.

Method. First form the net base values: $N_1=80-60=20$ and $N_3=120-140=-20$. The shocks are $0.005$ and $-0.0025$.

The first-order estimate is:

\Delta S\approx -\left[ \frac{1(20)}{1.04}(0.005) +\frac{3(-20)}{1.04}(-0.0025) \right]

The two terms inside the brackets are $0.096154$ and $0.144231$, so:

\Delta S\approx -(0.096154+0.144231) =-0.240385\text{ dollars}

For exact repricing, the year-1 net bucket becomes $20(1.04/1.045)$. Its change is:

20\left(\frac{1.04}{1.045}-1\right) =20\left(\frac{208}{209}-1\right) =-0.095694

The year-3 net bucket becomes $-20(1.04/1.0375)^3$. Its change from the base value $-20$ is:

-20\left[\left(\frac{1.04}{1.0375}\right)^3-1\right] =-0.144927

Thus, exact surplus change is $-0.095694-0.144927=-0.240621$, or −$0.24 to the nearest cent. The exact result is close to the first-order estimate, $-$0.240385$, but is not identical because the finite shocks introduce curvature. The year-3 bucket supplies the larger adverse contribution.

Common mistake avoided: The liability excess at year 3 is a negative net bucket, and its yield falls. That combination increases the present value of the net liability excess and reduces surplus. Omitting either the negative bucket sign or the negative yield shock gives the wrong contribution.

Separating Maturity-Region Effects

A total surplus change can conceal offsetting movements. It is useful to report each bucket's contribution as well as the total. Under a twist, one region can improve surplus while another worsens it. A bend can also produce a large change in an intermediate region even when the short and long ends move less.

The sign should be interpreted economically, not inferred from the word “up” or “down” alone. A yield increase lowers the value of a positive asset cash flow, but it also lowers the value of a liability. Whether that helps surplus depends on which side has the larger exposure at that maturity.

Worked Example: Exact Repricing at Opposite Ends of the Curve

At a 4% annual effective base yield, a hypothetical asset portfolio has a base present value of $100 at year 1, and a hypothetical liability portfolio has a base present value of $100 at year 3. The year-1 yield falls by 0.50 percentage point, while the year-3 yield rises by 0.50 percentage point. The base surplus is zero. Calculate the exact stressed surplus change, compare it with the first-order estimate, and identify the maturity-region effects.

Method. Reprice each cash flow at its own stressed yield. The asset yield becomes $3.5\%$, and the liability yield becomes $4.5\%$.

The stressed asset value is:

100\left(\frac{1.04}{1.035}\right) =100\left(\frac{208}{207}\right) =\$100.483092

The stressed liability value is:

100\left(\frac{1.04}{1.045}\right)^3 =100\left(\frac{208}{209}\right)^3 =\$98.571450

The base surplus is $100-100=0$, so the stressed surplus change is:

100.483092-98.571450 =\$1.911642 \approx +\$1.91

As a check, the first-order asset increase is $-100(1/1.04)(-0.005)=+$0.480769$, and the first-order liability change is $-100(3/1.04)(0.005)=-$1.442308$. Therefore the first-order surplus change is $0.480769-(-1.442308)=+$1.923077$, close to the exact increase of $1.91.

Both maturity regions improve surplus in this stress: the falling year-1 yield raises asset value, and the rising year-3 yield lowers liability value. Neither region drives an adverse effect here. The check is important because a mistaken sign can reverse the conclusion about which maturity region is problematic.

Common mistake avoided: Do not describe the year-3 liability exposure as adverse merely because it is a liability. The yield rises, so the liability's present value falls, improving surplus.

Practical Stress-Test Checks

Before interpreting a stress result, check the setup and the signs. For each maturity bucket, confirm the following:

Use decimal shocks. Convert percentage-point changes to decimal yield changes before substituting them.
Keep assets and liabilities distinct. Reprice them separately, or use $N_t=P_t^A-P_t^L$ consistently in the first-order formula.
Check the yield direction. For positive cash flows, a yield increase reduces present value and a yield decrease raises it.
Inspect each bucket's contribution. A small total can result from large gains and losses that offset one another.
Choose the right precision. Key-rate estimates are local approximations; use exact repricing when the scenario gives the cash flows or base bucket values and stressed yields.

These calculations are aligned with the SOA Exam FM (Financial Mathematics) syllabus. In an applied matching problem, they help diagnose whether a portfolio's surplus is exposed at the short end, the intermediate region, or the long end, and whether a proposed match remains robust under more than a parallel shift.

Check Your Understanding

For each problem, use the stated annual effective yields and the maturity-bucket convention defined above.

At a flat base yield of 3%, net asset-minus-liability present values are $30 at year 2 and $-$18$ at year 5. The corresponding yield changes are +0.20 and −0.10 percentage point. Find the first-order surplus change. (Answer: approximately −$0.20.)
An asset bucket has base present value $50 at year 2 and a 4% base yield. Its yield rises by 1.00 percentage point. Use exact repricing to find the change in asset value, rounded to the nearest cent. (Answer: −$0.95.)
At a 4% base yield, net present values are $-$25$ at year 1, $40 at year 3, and $-$10$ at year 5. Their yield changes are −0.30, +0.20, and +0.10 percentage point. Estimate the surplus change and name the largest adverse bucket contribution. (Answer: approximately −$0.25; the year-3 bucket.)
At a 5% base yield, assets have present values of $70 at year 1 and $50 at year 3; liabilities have present values of $40 at year 2 and $80 at year 4. The yield changes at years 1–4 are, respectively, +0.20, −0.40, +0.10, and +0.30 percentage point. Calculate the first-order surplus change and identify the largest adverse contribution. (Answer: approximately +$0.33; the year-2 bucket contributes approximately −$0.30.)
A portfolio has a positive net present value in a maturity bucket. If that bucket's yield falls and all other yields are unchanged, what is the direction of its first-order surplus contribution? (Answer: Positive; the value of the net asset exposure rises.)

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