Redington Surplus and Parallel Yield Shifts

Surplus at a Common Yield

An asset portfolio and a liability stream can each be valued by discounting their cash flows at a yield. To compare the two values meaningfully, use the same yield and valuation date for both. Their difference is the surplus. When the yield changes, each present value changes according to its own cash-flow amounts and timing; the surplus changes by the difference between those two value changes.

This tutorial uses annual effective yields and annual cash-flow dates. A parallel yield shift means adding the same amount to the yield at every maturity. For example, shifting the common yield from 8% to 10% is a parallel increase of 2 percentage points. It does not mean increasing each yield by 2% of its original value.

Definition: At a common annual effective yield $y$, the asset value $A(y)$ is the present value of the asset cash flows, and the liability value $L(y)$ is the present value of the liability cash flows. The surplus is $$ S(y)=A(y)-L(y). $$ If asset cash flow $C_j$ occurs at time $t_j$ and liability cash flow $B_k$ occurs at time $u_k$, then $$ A(y)=\sum_j C_j(1+y)^{-t_j}, \qquad L(y)=\sum_k B_k(1+y)^{-u_k}. $$ These expressions assume that all cash flows are valued at the same date, that the annual effective yield is used consistently, and that $y>-1$. Here, cash flows are positive amounts paid at their stated dates.

For a parallel shift of $h$, the new common yield is $y+h$, which must also exceed $-1$. Reprice both sides at that yield:

$$ S(y+h)=A(y+h)-L(y+h). $$

The change in surplus is the new surplus minus the original surplus:

\Delta S=S(y+h)-S(y) =\bigl[A(y+h)-A(y)\bigr]-\bigl[L(y+h)-L(y)\bigr].

This identity keeps the signs straight. If both values fall after a yield increase, their changes are both negative; subtracting the liability change means the change in surplus is not simply the asset-value change. The relative size of the two changes matters.

Why the Parallel-Shift Formula Holds

Under annual compounding, one dollar payable at time $t$ has present value $(1+y)^{-t}$. After a parallel shift, the yield used for that same cash flow is $y+h$, so its new present value is $(1+y+h)^{-t}$. The change for that cash flow is therefore the new discounted amount minus the old discounted amount. Adding those changes across asset cash flows gives $A(y+h)-A(y)$; adding them across liability cash flows gives $L(y+h)-L(y)$. Since surplus is assets minus liabilities, its change is the asset change minus the liability change.

\Delta S = \sum_j C_j\left[(1+y+h)^{-t_j}-(1+y)^{-t_j}\right] - \sum_k B_k\left[(1+y+h)^{-u_k}-(1+y)^{-u_k}\right].

This is an exact repricing relationship, not a small-change approximation. It applies when the same annual effective yield shift is applied to every cash-flow date. It does not describe a nonparallel change in which different maturities receive different yield changes. Nor does the formula itself show that a portfolio is immunized: it calculates the surplus at the two yields being compared.

Worked Examples: Repricing Both Sides

Worked Example: A Ten-Percentage-Point Parallel Increase

At a valuation yield of 0%, a portfolio has asset cash flows of $100 at the end of year 1 and $100 at the end of year 3. It has a liability payment of $180 at the end of year 2. Find the original surplus, the surplus after a parallel increase to 10%, and the change in surplus. Report final dollar amounts to the nearest cent.

Method. Value all cash flows at the original common yield, then reprice both the assets and the liability at 10%. The change in surplus is new surplus minus original surplus.

At 0%, the discount factor for each year is 1. Thus,

A(0)=100+100=200,\qquad L(0)=180, \qquad S(0)=200-180=20.

At 10%, the annual accumulation factor is $1.10$. The asset value is

A(0.10)=\frac{100}{1.10}+\frac{100}{1.10^3} =\frac{1000}{11}+\frac{100000}{1331} =\frac{221000}{1331} \approx 166.040571.

The liability value is

L(0.10)=\frac{180}{1.10^2} =\frac{18000}{121} =\frac{198000}{1331} \approx 148.760331.

Therefore, the new surplus is

S(0.10)=\frac{221000-198000}{1331} =\frac{23000}{1331} \approx 17.280240421 \approx \$17.28.

The change, calculated from the unrounded values, is

\Delta S=\frac{23000}{1331}-20 =-\frac{3620}{1331} \approx -2.719759579 \approx -\$2.72.

The surplus falls by $2.72. Common mistake avoided: rounding the individual present values before subtracting can alter the displayed intermediate digits. Keep the exact fractions, or full-precision values, until the final rounding.

Worked Example: An Exact Shift with Convenient Discount Factors

At a common yield of 0%, assets pay $250 at the end of year 1 and $125 at the end of year 2. Liabilities require $100 at the end of year 1 and $200 at the end of year 2. Find the surplus at 0% and after a parallel increase to 25%, and state how much it changes. Report final amounts to the nearest cent.

At 0%, the asset value is $250+125=375$, and the liability value is $100+200=300$. Thus the original surplus is $375-300=\$75$.

At 25%, one-year and two-year discount factors are

v=\frac{1}{1.25}=0.8,\qquad v^2=0.64.

The repriced asset value is

$$ A(0.25)=250(0.8)+125(0.64)=200+80=280. $$

The repriced liability value is

$$ L(0.25)=100(0.8)+200(0.64)=80+128=208. $$

So the new surplus is $280-208=\$72$, and its change is $72-75=-\$3$. The final surplus is $72.00 and the surplus has decreased by $3.00.

The asset and liability values both decrease, but by different amounts: assets fall by $95, while liabilities fall by $92. Subtracting the liability decrease from the asset decrease gives a net surplus decrease of $3. Common mistake avoided: a fall in both present values does not imply that the surplus is unchanged.

Worked Example: A Small Surplus Change from a Two-Point Shift

At an 8% common annual effective yield, a portfolio has asset payments of $108 at the end of year 1 and $116.64 at the end of year 2. The liability is $174.96 at the end of year 2. Revalue the portfolio after a parallel increase to 10%. Find the original and new surplus and their difference, to the nearest cent.

At 8%, the asset values are $108/1.08=100$ and $116.64/1.08^2=100$. The liability value is $174.96/1.08^2=150$. Thus

A(0.08)=200,\qquad L(0.08)=150,\qquad S(0.08)=50.

At 10%, the first-year asset payment has present value $108/1.10=1080/11$. The second-year asset payment and the liability have values

\frac{116.64}{1.10^2}=\frac{11664}{121}, \qquad \frac{174.96}{1.10^2}=\frac{17496}{121}.

Consequently,

A(0.10)=\frac{1080}{11}+\frac{11664}{121} =\frac{23544}{121} \approx 194.578512397,

and

L(0.10)=\frac{17496}{121} \approx 144.595041322.

The new surplus and its change are

S(0.10)=\frac{6048}{121} \approx 49.983471074, \qquad \Delta S=\frac{6048}{121}-50=-\frac{2}{121} \approx -0.016528926.

To the nearest cent, the new surplus is $49.98 and the surplus has decreased by $0.02. The near-offsetting changes in asset and liability values leave a small net change, but they do not make the change exactly zero. Common mistake avoided: do not infer an unchanged surplus just because the displayed values appear close; subtract using unrounded values first.

Interpreting and Checking the Result

For positive cash flows and yields above $-1$, increasing the yield reduces the present value of each payment. The surplus response depends on which side loses more value in dollars. A liability value that falls more than the asset value can cause surplus to rise; an asset value that falls more can cause surplus to fall. The direction cannot be determined from the yield shift alone.

For a reliable calculation, use this sequence:

Fix the valuation basis.
Identify the common yield, compounding convention, valuation date, and cash-flow dates.

Calculate the original values.
Discount asset and liability payments separately at the original yield, then subtract liability value from asset value.

Apply one shift to every maturity.
For a parallel shift $h$, replace $y$ by $y+h$ for every asset and liability cash flow.

Recompute surplus and change.
Find $S(y+h)=A(y+h)-L(y+h)$, then calculate $\Delta S=S(y+h)-S(y)$.

Round only at the end.
Retain full precision in the present values and in the subtraction used to find the change.

A useful independent check is to calculate the changes in the two values separately and verify that

\Delta S=\Delta A-\Delta L, \quad\text{where}\quad \Delta A=A(y+h)-A(y),\qquad \Delta L=L(y+h)-L(y).

For example, if both values decrease, $\Delta A$ and $\Delta L$ are negative. Subtracting the negative liability change adds its magnitude back. This sign convention is a frequent source of errors.

A parallel-shift surplus calculation is useful for describing how an asset-liability position responds to a specified change in yield. It is not, by itself, a complete immunization test. The surplus at one yield, or its change between two yields, does not establish how the portfolio behaves for all nearby yield changes. Those broader conditions require additional analysis.

Key takeaway: Value assets and liabilities at the same yield before comparing them. After a parallel shift, reprice every cash flow using the shifted common yield, subtract liability value from asset value, and compare the new surplus with the original one. Keep full precision until the final reported amounts.

Check Your Understanding

For each problem, use annual effective yields, calculate both values at each stated yield, and round final dollar answers to the nearest cent.

At 0%, assets pay $120 at year 1 and liabilities pay $100 at year 2. Find the original surplus and the surplus after a parallel increase to 10%. (Answer: Original surplus $20.00; new surplus $26.45; increase $6.45.)
At 0%, assets pay $200 at year 1 and $100 at year 2. Liabilities pay $100 at year 1 and $100 at year 3. Find the surplus at 0% and after a parallel increase to 25%, and the change. (Answer: Original surplus $100.00; new surplus $92.80; change $-\$7.20$.)
At 0%, assets pay $100 at year 2 and liabilities pay $90 at year 1. Find the surplus after a parallel increase to 10% and its change from the original surplus. (Answer: New surplus $0.83; change $-\$9.17$.)
At 0%, assets pay $400 at year 1 and $100 at year 2. Liabilities pay $250 at year 1 and $100 at year 3. Reprice after a parallel increase to 25%. (Answer: Original surplus $150.00; new surplus $132.80; change $-\$17.20$.)
At 8%, assets pay $108 at year 1 and $116.64 at year 2; liabilities pay $174.96 at year 2. Reprice after a parallel increase to 10%. (Answer: Original surplus $50.00; new surplus $49.98; change $-\$0.02$.)

What You'll Learn