Immunizing One Fixed Payment
Suppose an investor must make one known payment on a future date and uses coupon-bearing bonds to fund it. Matching the portfolio’s present value is necessary, but it is not enough: a change in the valuation yield can change asset and liability values at different rates. Redington immunization matches present value and first-order interest-rate sensitivity, then requires the assets to have strictly greater second-order sensitivity.
The focus here is one deterministic liability paid at time \(T\), and a portfolio of bonds whose cash flows occur at specified times. Assume annual effective yield \(i>-1\), annual cash-flow timing measured in years, no default, no taxes or transaction costs, and a parallel change in the same yield used to value every cash flow. The local Redington conditions do not promise protection against every large rate change or a nonparallel change in yields.
Write \(A(i)\) for the present value of the asset cash flows and \(L(i)\) for the present value of the liability. The surplus is \(S(i)=A(i)-L(i)\). For a single fixed payment, the Redington requirements at the valuation yield are \(S(i)=0\), \(S'(i)=0\), and \(S''(i)>0\). The first two conditions match present value and duration; the strict inequality makes the matched point a local minimum of the surplus.
Solving for Two Bond Weights
Suppose two bonds have present values \(V_1,V_2\) and durations \(D_1,D_2\), with positive market values. If the liability present value is \(P_L\) and its payment date is \(T\), the matching equations are \(V_1+V_2=P_L\) and \(V_1D_1+V_2D_2=P_LT\). If \(D_1<T<D_2\), solving gives positive weights:
These are market values, not automatically face amounts. To obtain the number of bonds or face amount to purchase, divide each required market value by that bond’s price per unit of face value, using consistent units. The duration formula uses all coupon and redemption cash flows, not just the bond’s maturity date.
Discount every cash flow at the specified valuation yield. Record bond prices as present values per stated unit of face value.
Weight each payment date by that payment’s share of the bond’s present value.
Use the two equations to find market-value allocations, then convert them into holdings if needed.
Compare the portfolio’s present-value-weighted second timing moment with the liability’s. It must be strictly greater.
Why the Convexity Check Works
The second derivative gives a direct way to check the final condition. For deterministic cash flows \(C_t\), differentiating each discounted amount twice gives \(t(t+1)C_t(1+i)^{-t-2}\). This identity applies to the assumed fixed cash flows and common yield; a yield-dependent or contingent cash flow would require a different model.
For a positive-present-value stream, let \(p_t=C_t(1+i)^{-t}/P\). These weights sum to 1, so they define a probability distribution over payment times. Its mean is the Macaulay duration \(D\). Dividing the second derivative by present value gives \(P''(i)/P=E[t(t+1)]/(1+i)^2\). Thus, when asset and liability present values and durations match, their second derivatives are ordered by their second moments of payment time.
For a single liability at \(T\), the payment-time variance is zero and its second moment is \(T^2\). For the asset portfolio, matching duration gives \(E[t]=T\), and therefore:
This is the key justification: with positive asset cash flows, matched present value and duration, and cash flows spread across more than one date, the asset timing variance is positive. The strict second-derivative condition then holds. The explicit numerical check remains useful: it verifies the cash-flow data, the weights, and the assumptions actually used.
Worked Example: Constructing the Portfolio at a 10% Yield
A hypothetical fund must pay \(121\) at the end of year 2. At an annual effective valuation yield of \(10\%\), it can purchase two annual-coupon bonds, each with face value \(100\) and annual coupon rate \(10\%\): a 2-year bond and a 3-year bond. Find the face amounts to hold and check the Redington conditions.
Method and present values. The 2-year bond pays \(10\) at year 1 and \(110\) at year 2. Its price is \(10/1.1+110/1.1^2=9.090909+90.909091=100\). The 3-year bond pays \(10,10,110\). Its price is \(10/1.1+10/1.1^2+110/1.1^3=9.090909+8.264463+82.644628=100\). The liability present value is \(121/1.1^2=100\). Thus the asset and liability present values can match with a total asset market value of \(100\).
Durations. For the 2-year bond, \(D_1=(1(9.090909)+2(90.909091))/100=190.909091/100=1.909091\). For the 3-year bond, \(D_2=(1(9.090909)+2(8.264463)+3(82.644628))/100=273.553719/100=2.735537\). The liability duration is \(T=2\), which lies between the bond durations.
Market-value weights. The 2-year bond allocation is \(100(2.735537-2)/(2.735537-1.909091)=89\). The 3-year bond allocation is the remaining \(100-89=11\). Equivalently, the exact duration-weighted average is \(0.89(1.909091)+0.11(2.735537)=2.000000\). Because each bond costs \(100\) per \(100\) of face value at this yield, the fund holds face amounts \(89\) and \(11\), respectively.
Strict convexity check. For the 2-year bond, its present-value weights at years 1 and 2 are \(1/11\) and \(10/11\). Its timing second moment is therefore \(1/11+4(10/11)=41/11=3.727273\). For the 3-year bond, the present-value weights are \(121/1331,\ 110/1331,\ 1100/1331\), so its second moment is \((121+4(110)+9(1100))/1331=10461/1331=7.859504\). The portfolio second moment is \(0.89(3.727273)+0.11(7.859504)=4.181818\), strictly greater than the liability’s \(2^2=4\). The normalized \(t(t+1)\) moments are consequently \(6.181818\) for assets and \(6\) for the liability. Since the present values match, \(A''(0.10)>L''(0.10)\).
Check. Asset present value is \(89+11=100\), equal to the liability present value; duration is \(2\), equal to the liability duration; and the asset second moment is strictly greater. All three Redington conditions hold. The method avoids treating the 2-year bond’s maturity as its duration.
When a Maturity-Date Shortcut Fails
A coupon bond pays part of its value before maturity, so its duration is generally earlier than its maturity. A portfolio chosen by comparing maturity dates alone can therefore miss the first-order condition even if its present value is correct. The following diagnosis isolates that trap.
Worked Example: Diagnosing a Maturity-Based Match
Use the 10% yield and the 2-year, 10%-coupon bond from the previous calculation. A liability has present value \(100\) and is payable at the end of year 2. An investor buys \(100\) of the bond at market value and argues that the liability is matched because both have a 2-year horizon. Is the portfolio immunized?
Method. Compare the actual duration and the first derivatives, rather than comparing only the bond’s maturity and the liability date. The asset present value is \(100\), as required, but its duration is \(1.909091\), not \(2\).
For a stream with present value \(P\) and duration \(D\), the first derivative is \(-PD/(1+i)\). Thus the asset derivative is \(-100(1.909091)/1.1=-173.553719\). The liability derivative is \(-100(2)/1.1=-181.818182\). The surplus derivative is \(A'-L'=-173.553719-(-181.818182)=8.264463\), not zero.
Correction and check. The portfolio is not immunized: it matches present value but fails duration matching. Adding the 3-year bond and using the \(89/11\) market-value allocation calculated above brings duration to \(2\) and gives strictly greater asset convexity. The mistake is substituting a bond’s maturity for the present-value-weighted average date of all its payments.
Interpreting the Result and Its Limits
A positive-weight portfolio whose duration matches a single fixed payment must have cash flows with a positive timing variance to satisfy strict convexity. In practice, two bonds are useful when their durations bracket the liability date: the matching equations then produce positive market values. If both durations lie on the same side of \(T\), the two-equation solution cannot give positive weights for both bonds. That is a feasibility issue, not a reason to ignore duration.
The result is local. If the yield moves by a small amount \(h\), the surplus has the approximation \(S(i+h)\approx S(i)+hS'(i)+\tfrac12h^2S''(i)\). At an immunized point, the first two terms vanish and the positive second derivative makes the leading local surplus change positive. Higher-order effects can matter for larger changes, and a parallel-yield assumption does not cover a change in the shape of the yield curve. Revalue cash flows when the rate movement is not small or the model assumptions change.
Check Your Understanding
Use the stated durations and timing moments as given. Market-value allocations are measured at the valuation yield.
- A liability has present value \(200\) and is payable at time \(2\). Two assets have durations \(1.8\) and \(2.6\). Find the market-value allocation to each and check strict convexity if their timing second moments are \(3.4\) and \(7.0\). (Answer: \(150\) in the duration-\(1.8\) asset and \(50\) in the duration-\(2.6\) asset; portfolio second moment \(4.3>4\), so strict convexity holds.)
- A fixed liability has present value \(500\) and is payable at time \(3\). Assets have durations \(2\) and \(4\), with timing second moments \(5.2\) and \(17\). Find the weights and state whether strict convexity holds. (Answer: \(250\) in each asset; the portfolio second moment is \(11.1>9\), so strict convexity holds.)
- A single coupon bond has present value \(90\) and duration \(1.95\). A fixed liability has present value \(90\) and is payable at time \(2\). Does holding the bond satisfy the first-order Redington conditions? (Answer: No. Present value matches, but duration \(1.95\) does not equal \(2\).)
- A liability has present value \(120\) and is payable at time \(3\). Two assets have durations \(2.4\) and \(3.4\), and timing second moments \(6.2\) and \(12.4\). Find both allocations and check strict convexity. (Answer: \(48\) in the duration-\(2.4\) asset and \(72\) in the duration-\(3.4\) asset; portfolio second moment \(9.92>9\), so strict convexity holds.)