Interest-Rate Shocks and Discounting Effects

Why Separate the Two Interest-Rate Effects?

An interest-rate shock can affect a long-term contract through more than one channel. A change in the return earned on invested assets changes projected investment income. A change in the rate used to discount future benefit payments changes their present value. Those changes can move surplus in the same direction, but they are not the same calculation.

For example, if an insurer expects to earn more on assets, projected assets at a future date may increase. If the rate used to value fixed future benefits also rises, the present value of those benefits generally decreases. Combining the two before calculating their separate effects makes it harder to explain the result and can cause an error such as applying an asset return directly to a liability cash flow.

The calculations below use simplified deterministic scenarios to isolate the channels. In practice, asset returns, market values, liability cash flows, and valuation discount rates may have different timing and may respond differently to a market shock. The assumptions for each calculation must therefore be stated explicitly.

Definition: The investment-income effect is the change in projected asset accumulation or investment income when the investment return assumption changes, holding other projection inputs fixed. The discounting effect is the change in the present value of liability cash flows when the discount rate changes, holding those cash flows fixed. For surplus, defined here as assets minus liabilities, an increase in assets or a decrease in liabilities increases surplus.

Discounting Fixed Liability Cash Flows

Suppose fixed payments $C_t$ are due at the end of years $t=1,\ldots,n$, and the same effective annual discount rate $i$ applies at every duration. Their value at time 0 is

L(i)=\sum_{t=1}^{n} C_t v^t =\sum_{t=1}^{n}\frac{C_t}{(1+i)^t}, \qquad v=\frac{1}{1+i}.

If every $C_t$ is nonnegative, increasing $i$ decreases each discounted payment, and therefore decreases $L(i)$. This direction is a useful check, not a substitute for the calculation. A liability with cash flows that depend on interest rates, or with discount rates that vary by duration, requires the relevant rate-sensitive cash flows or duration-specific discount factors; the fixed-cash-flow formula alone does not capture those features.

Worked Example: Isolating Two Effects on One-Year Surplus

A hypothetical contract has assets of $500 at time 0. During the next year, those assets earn an effective annual investment return, and a fixed expense of $12 is paid at year-end. A fixed liability payment of $300 is also due at year-end. In the base scenario, the investment return and the liability discount rate are both 4%. In a joint stress, the investment return is 5% and the liability discount rate is 6%. Calculate base and stressed surplus, then attribute the change to the two channels. Assume no other cash flows, no default, and no tax.

Method. First calculate projected year-end assets from the investment return and expense. Separately calculate the time-0 present value of the fixed liability payment using its stated discount rate. Define surplus for this comparison as projected assets less the present value of the liability.

Base scenario. Projected assets are

$$ 500(1.04)-12=520-12=508. $$

The liability value is

\frac{300}{1.04}=288.461538\ldots

Thus base surplus is $508-288.461538\ldots=219.538462\ldots$, or $219.54.

Investment-return effect alone. Change the investment return to 5%, but keep the liability discount rate at 4%. Assets become

$$ 500(1.05)-12=525-12=513. $$

Surplus at this intermediate step is $513-288.461538\ldots=224.538462\ldots$. The investment effect is therefore $224.538462\ldots-219.538462\ldots=$ +$5.00.

Discounting effect alone, after the investment change. Now change the liability discount rate to 6%, leaving projected assets at $513. The stressed liability value is

\frac{300}{1.06}=283.018868\ldots

Joint-stress surplus is $513-283.018868\ldots=229.981132\ldots$, or $229.98. The discounting effect is $229.981132\ldots-224.538462\ldots=5.442670\ldots$, or +$5.44. The total change is $5.00 + $5.44 = +$10.44, which agrees with $229.98 − $219.54 after rounding.

Check and common mistake. The asset calculation uses the investment return, while the liability calculation uses the discount rate. Substituting the 5% investment return into the stressed liability value would give $300/1.05$, which is not the stated 6% liability valuation. The correct joint-stress liability is $300/1.06$. Round only the final dollar results to the nearest cent.

Why the Discounting Formula Works

The present value formula follows from the definition of an effective annual rate. One unit invested at time 0 accumulates to $1+i$ after one year, to $(1+i)^2$ after two years, and to $(1+i)^t$ after $t$ years, assuming the same annual effective rate applies and interest is reinvested. Therefore, the amount that accumulates to $C_t$ at time $t$ is $C_t/(1+i)^t$ at time 0. Adding the time-0 values of all the fixed payments gives $L(i)$.

State the timing and rate.
Identify when each liability payment is due and which effective annual discount rate applies to that duration.

Discount each payment.
For a payment $C_t$ due at the end of year $t$, calculate $C_t/(1+i)^t$.

Sum the present values.
Add the discounted payments to obtain the liability value, then compare the base and stressed values.

Multiple Payments and Rate Sensitivity

The size of a liability’s response to a rate change depends on both the amount and timing of its payments. Payments further in the future are discounted for more periods and generally contribute more to sensitivity to a small change in a common annual rate. A useful first-order measure is modified duration.

Definition: For fixed cash flows discounted at a common effective annual rate $i$, the Macaulay duration is $D_{\mathrm{Mac}}=\sum_{t=1}^{n}tC_t v^t/L(i)$. The modified duration is $D_{\mathrm{mod}}=D_{\mathrm{Mac}}/(1+i)$. For a small change $\Delta i$, the first-order approximation is $\Delta L\approx-D_{\mathrm{mod}}L(i)\Delta i$. This approximation is not an exact revaluation and is less reliable for large changes or cash flows that change with rates.

To justify the approximation, differentiate the fixed-cash-flow value term by term:

\frac{dL}{di} =\sum_{t=1}^{n}\frac{-tC_t}{(1+i)^{t+1}} =-\frac{1}{1+i}\sum_{t=1}^{n}tC_tv^t =-D_{\mathrm{mod}}L(i).

For a small rate change, the first-order change is approximately the derivative times the change in rate, giving $\Delta L\approx(dL/di)\Delta i$. The negative sign agrees with the direction check: for nonnegative fixed payments, a rate increase reduces the liability value.

Worked Example: Revaluing a Two-Payment Liability

A hypothetical benefit fund has fixed payments of $100 at the end of year 1 and $100 at the end of year 2. The same effective annual discount rate applies at both durations. The base rate is 5%, and the stressed rate is 6%. Find both values and the actual change, then compare it with the modified-duration estimate. Assume the payments do not change with interest rates.

Base value. At 5%, the year-1 discount factor is $1/1.05=0.95238095\ldots$, and the year-2 factor is $1/(1.05)^2=1/1.1025=0.90702948\ldots$. Thus

L(0.05)=\frac{100}{1.05}+\frac{100}{1.1025} =95.238095\ldots+90.702948\ldots =185.941043\ldots

The base value is $185.94.

Stressed value. At 6%, the year-1 factor is $1/1.06=0.94339623\ldots$, and the year-2 factor is $1/(1.06)^2=1/1.1236=0.88999644\ldots$. Therefore

L(0.06)=\frac{100}{1.06}+\frac{100}{1.1236} =94.339623\ldots+88.999644\ldots =183.339267\ldots

The stressed value is $183.34, so the exact revaluation change is $183.339267\ldots-185.941043\ldots=-2.601776\ldots$, or −$2.60.

Duration estimate. The Macaulay duration at 5% is

D_{\mathrm{Mac}} =\frac{1(95.238095\ldots)+2(90.702948\ldots)}{185.941043\ldots} =\frac{276.6440\ldots}{185.941043\ldots} =1.4880\ldots\text{ years}.

Modified duration is $1.4880\ldots/1.05=1.4171\ldots$. With $\Delta i=0.01$, the estimated change is

\Delta L\approx-(1.4171\ldots)(185.941043\ldots)(0.01) =-2.634\ldots

The estimate is approximately −$2.63, compared with the exact decrease of $2.60. The small difference is expected because duration uses a first-order approximation. A common mistake is to treat the duration estimate as the exact stressed value; exact revaluation discounts each cash flow at the new rate. Dollar results are rounded to the nearest cent, and duration is shown to four decimal places.

Attribution, Assumptions, and Common Traps

For a surplus measure $S=A-L$, where $A$ is projected assets and $L$ is the liability value, changing one input at a time makes the effects interpretable. If the investment assumption changes from $r_0$ to $r_1$ and the discount rate changes from $i_0$ to $i_1$, one possible sequential attribution is

\Delta S =\bigl[A(r_1)-A(r_0)\bigr] +\bigl[L(i_0)-L(i_1)\bigr],

provided the asset and liability calculations are separable and the specified cash flows are held fixed within each channel. If the scenario includes rate-dependent benefits, expenses, behavior, or other interactions, the total change may not be explained by these two isolated effects alone. In that case, state which inputs are held fixed and calculate the interaction or residual separately rather than assigning it silently to one channel.

The order of sequential attribution can also matter when the channels interact. For example, if the investment return affects future benefit amounts, then changing the return may alter the cash flows subsequently discounted. The first calculation path may assign the interaction differently from a path that changes the discount rate first. Report the chosen sequence or use a clearly specified attribution convention.

Do not assume the two rates are interchangeable. The investment return describes asset accumulation; the discount rate values liability cash flows. They may be equal in a simplified example, but that equality must be an assumption.
Do not discount a payment due at time 1 for two periods. Match each exponent to the payment date, using the stated time unit.
Do not infer the total financial effect from liability discounting alone. A rate shock may also affect investment income and, where relevant, market values of assets. Include only the channels specified in the scenario.
Check the direction before rounding. With positive fixed future payments, a higher discount rate should reduce their present value. If the calculation shows otherwise, check the rate, exponent, or subtraction order.

Separating these effects matters because a surplus movement is useful only when its drivers are clear. A higher projected investment return can improve asset accumulation, while a higher discount rate can reduce the measured value of fixed liabilities. Neither observation alone establishes the full outcome for a product or portfolio: the result depends on the scenario definition, timing, and whether the cash flows themselves change.

Key takeaway: Recalculate assets using the investment-return assumption and liabilities using the discount-rate assumption. Attribute each change against a stated base, hold other inputs fixed when isolating a channel, and verify that the combined effects reconcile to the total change.

Check Your Understanding

For each question, keep the investment and discounting calculations separate. Round dollar answers to the nearest cent.

A hypothetical liability pays $120 at the end of year 1 and $240 at the end of year 2. Both payments are fixed. Calculate its value at a 4% effective annual discount rate and at a 5% rate, and state the change. (Answer: $337.28 at 4%; $331.97 at 5%; decrease of $5.31.)
A fund has assets of $800 at the start of a year. They earn an effective annual investment return of 3.5% in the base scenario and 4.5% in a stress. A fixed $18 expense is paid at year-end. Find projected year-end assets in each scenario and the investment effect. (Answer: $810.00 base; $818.00 stressed; investment effect +$8.00.)
A fixed payment of $500 is due at the end of one year. Calculate its present value at 5% and 7%, and state the discounting effect on the liability value. (Answer: $476.19 at 5%; $467.29 at 7%; liability decrease of $8.90.)
A hypothetical portfolio has $600 of assets, earns 4% in the base scenario and 5% in the stress, and pays a fixed $10 expense at year-end. A separate fixed liability payment of $250 is due at year-end; its discount rate is 4% in the base scenario and 6% in the stress. Calculate base and stressed surplus and attribute the change. (Answer: base surplus $373.62; stressed surplus $384.15; investment effect +$6.00; discounting effect +$4.54; total increase $10.54.)
A liability consists of positive, fixed payments at future dates and is valued using one common effective annual rate. State the direction of its value change when that rate rises, and give one reason that a duration estimate may differ from an exact revaluation. (Answer: its value decreases; duration is a first-order approximation, so the difference reflects higher-order effects for a finite rate change.)

This treatment is aligned with the SOA Exam ALTAM (Advanced Long-Term Actuarial Mathematics) syllabus.

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