Rebalancing A Portfolio To A Target Duration

Why Rebalance to a Target Duration?

A bond portfolio’s duration depends on how its market value is distributed across bonds with different durations. When a portfolio must maintain a specified total market value, rebalancing means choosing how much to invest in each bond so their combined duration reaches a desired target. The central calculation is an allocation problem: duration is a market-value-weighted average, so the target determines the required weights.

This tutorial uses Macaulay duration, measured in years. The portfolio formula is straightforward, but it applies only when the durations and market values are measured on a consistent valuation basis. In particular, do not substitute face values for market values or mix Macaulay duration with modified duration.

Definition: For bonds with positive market values $P_1$ and $P_2$, and Macaulay durations $D_1$ and $D_2$, the market-value-weighted portfolio duration is $$ D_P=\frac{P_1D_1+P_2D_2}{P_1+P_2}. $$ If the total market value is fixed at $V=P_1+P_2$, the amounts invested must satisfy $P_1+P_2=V$ as well as the target-duration equation.

Deriving the Two-Bond Allocation

Suppose bond 1 has the lower duration, $D_1<D_2$. Let $x$ be the market value invested in bond 1. The remaining market value, $V-x$, is invested in bond 2. For a target portfolio duration $D^*$, the portfolio equation is

D^*=\frac{xD_1+(V-x)D_2}{V}.

Multiplying by $V$, expanding, and collecting the terms involving $x$ gives

VD^*=xD_1+VD_2-xD_2 =VD_2-x(D_2-D_1), $$ $$ x=V\frac{D_2-D^*}{D_2-D_1}.

The amount invested in bond 2 follows either by subtracting $x$ from $V$, or by solving for its weight directly. Thus the allocation weights are

\frac{P_1}{V}=\frac{D_2-D^*}{D_2-D_1}, \qquad \frac{P_2}{V}=\frac{D^*-D_1}{D_2-D_1}.

The weights sum to one: their numerators add to $D_2-D_1$, the denominator. This is also a useful check on a calculation. If short positions are prohibited, both weights must be nonnegative. That occurs exactly when $D_1\leq D^*\leq D_2$. A target outside that interval cannot be reached with a long-only allocation between these two bonds.

Key result: With a fixed total market value and two distinct bond durations, the target determines a unique allocation. For a long-only portfolio, the target must lie between the two durations, inclusive. If the durations are equal, changing the allocation cannot change the portfolio duration.

Worked Allocations

Worked Example: Allocate a Fixed Value to Reach the Target

A portfolio must have a market value of $100,000 and a Macaulay duration of 5 years. It can invest in two hypothetical zero-coupon bonds: Bond A matures in 2 years and has an annual effective yield of 5%; Bond B matures in 8 years and has an annual effective yield of 4%. Determine the market value invested in each bond. Assume annual effective yields and no transaction costs or taxes.

Method. A zero-coupon bond has a single payment at maturity, so its Macaulay duration equals its maturity. Therefore, $D_A=2$ years and $D_B=8$ years. The yields determine the bonds’ prices but do not change these single-payment durations.

Let $x$ be the amount invested in Bond A. Then $100{,}000-x$ is invested in Bond B. The duration equation is

5=\frac{2x+8(100{,}000-x)}{100{,}000}.

Multiply both sides by 100,000 and solve:

500{,}000=2x+800{,}000-8x, $$ $$ 6x=300{,}000, \qquad x=50{,}000.

The Bond B investment is $100{,}000-50{,}000=50{,}000$. The portfolio has market value $50{,}000+50{,}000=100{,}000$, and its duration is

\frac{50{,}000(2)+50{,}000(8)}{100{,}000} =\frac{100{,}000+400{,}000}{100{,}000} =5\text{ years}.

The result checks both the fixed-value condition and the target. No rounding is needed. A common mistake is to use the bonds’ yields as their durations; for zero-coupon bonds, duration is the time to the single payment, not the yield rate.

Worked Example: A Target at the Midpoint of Two Durations

An investor has $250,000 to allocate between two hypothetical zero-coupon bonds. Bond C matures in 3 years and has an annual effective yield of 4%; Bond D matures in 9 years and has an annual effective yield of 6%. The investor wants a portfolio Macaulay duration of 6 years. Find the amounts invested, assuming annual effective yields and no transaction costs or taxes.

The zero-coupon durations are $D_C=3$ years and $D_D=9$ years. Since the target 6 is between them, a long-only allocation is feasible. Let $x$ be the amount invested in Bond C:

6=\frac{3x+9(250{,}000-x)}{250{,}000}.

Multiplying by 250,000 and simplifying:

1{,}500{,}000=3x+2{,}250{,}000-9x, $$ $$ 6x=750{,}000, \qquad x=125{,}000.

The amount invested in Bond D is $250{,}000-125{,}000=125{,}000$. To check, the total market value is $250,000, and

D_P=\frac{125{,}000(3)+125{,}000(9)}{250{,}000} =\frac{375{,}000+1{,}125{,}000}{250{,}000} =6\text{ years}.

The equal allocation follows because the target is exactly halfway between 3 and 9 years. The yields are stated to establish the valuation assumptions, but the market-value allocation—not the yields—determines the portfolio duration.

Worked Example: Rebalance Existing Holdings at Fixed Value

A portfolio holds $150,000 of a hypothetical 3-year zero-coupon bond and $100,000 of a hypothetical 9-year zero-coupon bond. Their annual effective yields are 4% and 5%, respectively. The portfolio must keep its $250,000 market value but increase its Macaulay duration to 6 years. How much market value should be transferred from the shorter-duration bond to the longer-duration bond? Assume annual effective yields and no transaction costs or taxes.

The bond durations equal their maturities: 3 and 9 years. Let $x$ be the amount transferred from the 3-year bond to the 9-year bond. The new holdings are $150{,}000-x$ and $100{,}000+x$; their sum remains $250,000.

The target-duration equation is

6=\frac{3(150{,}000-x)+9(100{,}000+x)}{250{,}000}.

The target requires duration-dollar exposure of $6(250{,}000)=1{,}500{,}000$ dollar-years. Before the transfer, the duration-dollar exposure is

$$ 3(150{,}000)+9(100{,}000) =450{,}000+900{,}000 =1{,}350{,}000. $$

Each dollar transferred from the 3-year bond to the 9-year bond increases duration-dollar exposure by $9-3=6$ dollar-years. The required increase is $1{,}500{,}000-1{,}350{,}000=150{,}000$, so

6x=150{,}000, \qquad x=25{,}000.

Transfer $25,000. The new holdings are $125,000 in the 3-year bond and $125,000 in the 9-year bond. Their market values still sum to $250,000, and their duration is

\frac{3(125{,}000)+9(125{,}000)}{250{,}000} =\frac{375{,}000+1{,}125{,}000}{250{,}000} =6\text{ years}.

The common mistake is to calculate the needed duration change but overlook that a transfer changes both holdings. Accounting for both changes shows that each dollar transferred changes duration-dollar exposure by the difference $9-3$, not by 9.

Feasibility and Common Errors

For a long-only portfolio, the portfolio duration is a weighted average of the component durations, with nonnegative weights that sum to one. It therefore cannot be less than the shorter duration or greater than the longer duration. A target at an endpoint requires placing all the market value in the bond at that endpoint. A target strictly between the durations requires positive investments in both bonds.

If short positions are allowed, the algebraic allocation can still be calculated for a target outside the interval, but one holding will be negative and the other will exceed the total market value. That is a leveraged position, not a long-only allocation. Before solving, identify whether the problem permits short sales; do not treat a negative result as an acceptable long investment.

The market-value weighting formula follows from the definition of Macaulay duration. For a bond with cash flows $C_t$ at times $t$, its price is the sum of the present values of those cash flows, and its duration is the sum of time multiplied by present value, divided by price. For two bonds on the same valuation basis, adding their present-value time totals and dividing by their combined price gives the market-value-weighted average of their individual durations. This explains why the weights are prices, or market values, rather than face amounts.

For example, suppose two bonds have market values $P_1,P_2$ and durations $D_1,D_2$. Their combined duration numerator is $P_1D_1+P_2D_2$, and their combined market value is $P_1+P_2$. Dividing the first quantity by the second yields the portfolio formula. This aggregation is the conventional Macaulay-duration measure for the portfolio. Keep the valuation date, duration units, and duration definition consistent across the component bonds.

Wrong: take the simple average. That gives the portfolio duration only when the two market values are equal. In the first worked example, equal holdings happen to make the simple average $(2+8)/2=5$, but unequal holdings require unequal weights.
Wrong: weight by face value. Face value is not the market value invested. Use each bond’s current market value in the duration average.
Wrong: mix duration measures. Macaulay duration is expressed in time units; modified duration is a different measure. Do not insert one measure into an equation specified for the other.
Wrong: ignore feasibility. A target outside the range of component durations cannot be reached using only positive holdings in those bonds.

Practical check: After solving, verify that the two market values sum to the required total and that their market-value-weighted duration equals the target. For a long-only allocation, also check that neither holding is negative.

Check Your Understanding

For each problem, use Macaulay duration and market-value weights. Unless a question says otherwise, assume no transaction costs or taxes.

A portfolio must invest $180,000 between two hypothetical zero-coupon bonds maturing in 2 and 8 years. Both have annual effective yields of 5%. Find the amounts needed for a 3.5-year duration target. (Answer: $135,000 in the 2-year bond and $45,000 in the 8-year bond.)
A $420,000 portfolio is allocated between two hypothetical zero-coupon bonds with maturities 4 and 10 years and annual effective yields of 3% and 6%. Find the long-only allocation for a target duration of 7 years. (Answer: $210,000 in each bond.)
A portfolio holds $150,000 in a hypothetical 3-year zero-coupon bond and $100,000 in a hypothetical 9-year zero-coupon bond. Their annual effective yields are 4% and 5%. With total market value fixed at $250,000, how much should be transferred from the shorter-duration bond to reach a 6-year duration? (Answer: Transfer $25,000 from the 3-year bond to the 9-year bond.)
A $100,000 portfolio may hold only long positions in two hypothetical zero-coupon bonds with maturities 3 and 7 years. Their annual effective yields are 4% and 5%. Can the portfolio reach a 9-year Macaulay duration? Explain. (Answer: No. A long-only weighted average of 3 and 7 years cannot exceed 7 years.)
A $100,000 portfolio is to be allocated between hypothetical zero-coupon bonds with maturities 2 and 8 years and annual effective yields of 3% and 6%. If short positions are permitted, determine the algebraic holdings for a 9-year target. (Answer: $-$16,666.67$ in the 2-year bond and $116,666.67 in the 8-year bond.)

What You'll Learn