Present-Value Matching at the Valuation Rate

Why Present-Value Matching Comes First

An immunization calculation compares the value of assets with the value of liabilities on a common basis. Before considering how either value responds when yields change, the first question is whether the two values are equal at the stated valuation rate. If they are not equal initially, a later analysis of interest-rate sensitivity cannot make the starting balance disappear.

This tutorial focuses on that initial comparison. It covers how to discount each future cash flow at a specified base yield, how to interpret a surplus or shortfall, and how to adjust an unknown cash flow to obtain equality. Present-value equality is a starting condition for Redington immunization, not a sufficient condition for it. A portfolio may match at the base rate and still have a surplus that moves unfavorably when the rate changes.

Definition: Choose a valuation date and a base effective annual yield $i_0$. For a future cash flow $C_t$ payable $t$ years after the valuation date, its present value at the base yield is $C_t(1+i_0)^{-t}$. The present value of a stream is the sum of the present values of its cash flows. Write $A(i_0)$ for the asset present value, $L(i_0)$ for the liability present value, and $S(i_0)=A(i_0)-L(i_0)$ for the surplus.

Set the Valuation Basis Before Calculating

The base yield must be applied consistently to both sides of the comparison. In the examples below, $i_0$ is an effective annual yield, cash flows occur at annual times, and the valuation date is immediately before the first listed future payment. The discount factor for one year is $v_0=(1+i_0)^{-1}$; for $t$ years it is $v_0^t$. Thus a cash flow at time $t$ has present value $C_t v_0^t$.

The calculation assumes that the listed cash flows are fixed amounts in the same currency, with no default, reinvestment uncertainty, transaction costs, taxes, or omitted payments. It also assumes that the stated yield is the valuation basis for both assets and liabilities. If a problem specifies a different compounding convention, payment timing, or valuation date, use that basis consistently instead; the annual formula above should not be applied unchanged.

Asset cash flows are treated as positive inflows and liability cash flows as positive obligations. Their present values are calculated separately and then compared. This makes the sign of the surplus explicit: $S(i_0)>0$ is a surplus, $S(i_0)<0$ is a shortfall, and $S(i_0)=0$ is present-value matching at the base yield.

A(i_0)=\sum_{k=1}^{m} C_k v_0^{t_k}, \qquad L(i_0)=\sum_{j=1}^{n} B_j v_0^{u_j}, \qquad S(i_0)=A(i_0)-L(i_0)

Here $C_k$ is an asset payment at time $t_k$, and $B_j$ is a liability payment at time $u_j$. The asset and liability payment dates do not have to coincide. Each stream is discounted from its own payment date to the same valuation date.

Why Equality Is the Starting Condition

Present value translates future dated amounts into values at one common date. If assets are intended to fund liabilities, equality at that date means the assets and liabilities have the same value under the chosen base yield. If their values differ, the difference is already present before any yield shock is considered.

The justification follows directly from the definition of surplus. At the base yield, the surplus is the asset present value less the liability present value. Therefore a zero surplus occurs exactly when those two present values are equal. This is an accounting identity on the chosen valuation basis; it does not depend on a claim about how values behave away from the base yield.

S(i_0)=0 \quad\Longleftrightarrow\quad A(i_0)-L(i_0)=0 \quad\Longleftrightarrow\quad A(i_0)=L(i_0)

For Redington immunization, equality is necessary as the initial balance condition. It is not, by itself, a guarantee that the surplus remains nonnegative after yields move. The later analysis of interest-rate sensitivity imposes further conditions. At this stage, the practical task is to calculate the two present values correctly and establish whether the base-rate balance is zero.

Worked Example: Diagnose a Base-Rate Shortfall

At an effective annual base yield of 5%, an asset portfolio will pay $105 at the end of year 1 and $210 at the end of year 2. A liability of $300 is due at the end of year 2. Calculate each present value and the surplus. State whether the streams match at the base yield.

Method. Discount every payment to time 0 at 5%, then subtract the liability present value from the asset present value. The cash flows are assumed certain and annual, with no other payments.

Asset present value. The one-year discount factor is $1/1.05$, and the two-year discount factor is $1/(1.05)^2=1/1.1025$. Therefore,

A(0.05)=\frac{105}{1.05}+\frac{210}{1.1025} =100+190.476190\ldots =290.476190\ldots

Liability present value. The $300 liability is due in two years, so

L(0.05)=\frac{300}{1.1025} =272.108843\ldots

Surplus and conclusion.

S(0.05)=290.476190\ldots-272.108843\ldots =18.367346\ldots

Rounded to the nearest cent at the final step, the asset present value is $290.48, the liability present value is $272.11, and the surplus is $18.37. The streams do not match at the base yield; they have a surplus. The common mistake this method avoids is discounting the year-2 asset payment by the one-year factor. Each payment must be discounted for its own number of years.

Worked Comparisons on a Common Basis

A comparison is meaningful only if both sides use the same valuation date, yield, and timing convention. A liability payment may occur on a date when no asset payment occurs; that does not prevent a present-value comparison. Each stream is valued separately, and the resulting totals are compared.

Worked Example: Verify Matching With Different Cash-Flow Patterns

At a 5% effective annual base yield, assets pay $105 at the end of year 1 and $110.25 at the end of year 2. Liabilities are $52.50 at the end of year 1 and $165.375 at the end of year 2. Calculate both present values and determine whether they match.

Method. Use the same discount factor at each time for both streams. All payments are fixed annual cash flows, and no payments other than those listed are included.

At year 1, the discount factor is $1/1.05$. At year 2, it is $1/1.1025$. The asset present value is

A(0.05)=\frac{105}{1.05}+\frac{110.25}{1.1025} =100+100 =200

The liability present value is

L(0.05)=\frac{52.50}{1.05}+\frac{165.375}{1.1025} =50+150 =200

The surplus is $S(0.05)=200-200=0$. The streams therefore match in present value at the 5% base yield. The common mistake this calculation avoids is comparing the undiscounted totals: the asset payments sum to $215.25 and the liability payments sum to $217.875, but those totals are not the values at time 0. Discounting each dated payment gives the appropriate comparison.

This example establishes only equality at the specified base yield. The different cash-flow patterns mean that equality alone does not establish what the surplus will be at another yield.

Worked Example: Find the Payment Required for Present-Value Equality

At an effective annual base yield of 5%, liabilities are $105 at the end of year 1, $110.25 at the end of year 2, and $115.7625 at the end of year 3. Assets pay $52.50 at the end of year 1, $110.25 at the end of year 2, and an unknown amount $X$ at the end of year 3. Find $X$ so that asset and liability present values are equal.

Method. First value the liability stream and the known asset payments. Then set the present value of the remaining asset payment equal to the difference. The yield is effective annually, and all payments occur at the stated year-ends.

The liability present value is

L(0.05) =\frac{105}{1.05} +\frac{110.25}{1.1025} +\frac{115.7625}{1.157625} =100+100+100 =300

The known asset payments have present value

\frac{52.50}{1.05}+\frac{110.25}{1.1025} =50+100 =150

To match the liability present value, the year-3 asset payment must contribute the remaining $150 at time 0. Thus

\frac{X}{1.157625}=150 \qquad\Longrightarrow\qquad X=150(1.157625)=173.64375

Rounded to the nearest cent, the required payment is $173.64. Check: the asset present value is $50+100+150=300$, equal to the liability present value of $300. The common mistake avoided here is treating the residual $150 present value as the required year-3 payment. It is the amount needed at time 0; the payment at year 3 must be accumulated by the factor $1.05^3=1.157625$.

Common Calculation Traps

Present-value matching is conceptually simple, but errors in timing, signs, or valuation basis can reverse the conclusion. Use the following checks before declaring that a portfolio matches.

Use each payment’s own time. A year-2 amount is discounted by $(1+i_0)^{-2}$, not by the one-year factor. For a payment at time $t$, use the factor corresponding to $t$.
Discount both sides. Comparing undiscounted totals does not test present-value equality when payments occur at different times.
Keep the base yield fixed throughout the comparison. Do not value assets at one rate and liabilities at another unless the problem explicitly defines distinct bases and asks for that comparison.
Use consistent signs. When $A$ and $L$ are recorded as positive values, surplus is $A-L$. A negative result is a shortfall, not a surplus.
Do not infer full immunization from a zero surplus. $A(i_0)=L(i_0)$ establishes equality at the base yield only. It does not establish how the surplus behaves under a yield change.

For instance, if an asset stream pays $105 at year 1 and $210 at year 2, its present value at 5% is

\frac{105}{1.05}+\frac{210}{1.1025} =100+190.476190\ldots =290.476190\ldots

If the liability is $300 at year 2, its present value is $300/1.1025=272.108843\ldots$, not $300. The base-rate surplus is therefore $290.476190\ldots-272.108843\ldots=18.367346\ldots$, or $18.37 to the nearest cent. Keeping the year-2 discounting on both sides is essential to the comparison.

Key takeaway: Present-value matching means that the asset and liability cash flows, discounted to the same valuation date at the same specified base yield, have equal totals. This zero-surplus condition is the starting balance for Redington immunization; it is not a complete immunization test.

Check Your Understanding

For each problem, use the stated effective annual base yield, show the discounting at each payment date, and compare asset and liability present values.

At 5%, assets pay $105 at year 1 and $220.50 at year 2. Liabilities pay $105 at year 1 and $220.50 at year 2. Find both present values and the surplus. (Answer: Asset PV = $300.00; liability PV = $300.00; surplus = $0.00.)
At 5%, assets pay $105 at year 1 and $210 at year 2. A liability of $300 is due at year 2. Find the two present values and the surplus. (Answer: Asset PV = $290.48; liability PV = $272.11; surplus = $18.37.)
At 4%, liabilities are $104 at year 1 and $216.32 at year 2. Assets pay $52 at year 1 and $X$ at year 2. Find $X$ for present-value equality. (Answer: $X=$$270.40.)
At 10%, assets pay $110 at year 1 and $121 at year 2. Liabilities pay $55 at year 1 and $181.50 at year 2. Calculate both present values and state what equality at this base yield does not, by itself, establish. (Answer: Asset PV = $200.00; liability PV = $200.00; equality at the base yield does not by itself establish immunization for yield changes.)

What You'll Learn