Adding Cash To A Bond Portfolio’s Duration

Why Adding Cash Changes Duration

A fixed cash balance does not change in market value when bond yields move, under the usual assumption that the cash remains a fixed dollar amount over the measurement interval. Bonds in the same portfolio can gain or lose value as yields change. Adding cash therefore changes the proportion of the portfolio exposed to bond-price movements, even though it does not change the bonds themselves.

The calculation is a market-value-weighted average: each component’s duration is multiplied by its share of the total portfolio value. A cash holding has duration zero in this model, so it contributes to the denominator but not to the duration-weighted numerator. The result is a lower duration than that of the bond holdings alone, provided the added cash has positive value and the bonds have positive duration.

Definition: Suppose a portfolio consists of bonds with market values $V_1,\ldots,V_n$ and Macaulay durations $D_1,\ldots,D_n$, plus a fixed cash amount $C$. Its market value is $V_P=\sum_{j=1}^{n}V_j+C$. If the cash has zero duration and all component durations use compatible time units, portfolio Macaulay duration is $$ D_P=\frac{\sum_{j=1}^{n}V_jD_j}{V_P}. $$

The denominator includes the cash. Omitting it would calculate the duration of the bond holdings alone, not the duration of the full portfolio. The cash is treated as a value at time zero: it is already held, and its value is assumed not to change in response to the yield shift being analyzed.

Why the Market-Value Weighting Works

Macaulay duration is a present-value-weighted average of the times at which a security’s cash flows are received. For a bond $j$, let $CF_{jt}$ be its cash flow at time $t$, and let $v^t$ be the discount factor under the applicable yield convention. Its price is $V_j=\sum_t CF_{jt}v^t$, and its Macaulay duration is $\sum_t tCF_{jt}v^t/V_j$. Multiplying that duration by the bond’s market value cancels the denominator $V_j$, leaving the bond’s time-weighted present-value cash flows.

Adding those numerators across bonds and dividing by total portfolio value gives the formula above. Cash contributes its value to the total but has no future payment time to add to the numerator. Equivalently, regarding the cash as a value available at time zero gives it duration zero.

For a small yield change, modified duration gives the first-order estimate of percentage price change. For an individual bond, modified duration $D_{\text{mod},j}$ is defined locally by the negative percentage price derivative with respect to its yield. The derivative of total portfolio value is the sum of the component derivatives. Since fixed cash has zero derivative with respect to yield, the portfolio’s aggregate modified duration is

D_{\text{mod},P} =\frac{\sum_{j=1}^{n}V_jD_{\text{mod},j}}{V_P}.

This formula is valid when the components’ modified durations describe the same yield shock and compatible price sensitivities. If the bond holdings all use the same yield and compounding convention, then $D_{\text{mod},j}=D_j/(1+i)$ for annual effective yield $i$, so the portfolio modified duration is $D_P/(1+i)$. Do not divide a portfolio Macaulay duration by one yield when the component yields differ.

Assumption: “Cash” here means a fixed dollar amount whose market value is unchanged for the yield change under study. A deposit, money-market instrument, or short-term security with interest, a maturity value, or yield-sensitive market value may have nonzero duration. Treating such an asset as zero-duration would require a separate justification.

Worked Example: Adding Cash to Bonds With a Common Yield

A hypothetical investor holds two bond positions. Their market values are $600,000 and $400,000, and their Macaulay durations are 5.0 years and 8.0 years. Both bonds use a common annual effective yield of 5%. The investor adds $250,000 in fixed cash. Find the portfolio Macaulay duration, its modified duration, and the estimated dollar change in portfolio value for a 25-basis-point yield increase.

Method. First find the market-value-weighted duration of the bonds, then include cash in the total portfolio value. Because both bonds share the annual effective yield, convert the resulting Macaulay duration to modified duration by dividing by $1.05$.

The bond market value is $$ \$600{,}000+\$400{,}000=\$1{,}000{,}000. $$ The duration-weighted numerator for the bonds is $$ \$600{,}000(5.0)+\$400{,}000(8.0) =\$3{,}000{,}000\text{ years}+\$3{,}200{,}000\text{ years} =\$6{,}200{,}000\text{ years}. $$ The full portfolio value, including cash, is $$ \$1{,}000{,}000+\$250{,}000=\$1{,}250{,}000. $$ Thus its Macaulay duration is $$ D_P=\frac{\$6{,}200{,}000\text{ years}}{\$1{,}250{,}000} =4.96\text{ years}. $$

The portfolio modified duration is $$ D_{\text{mod},P}=\frac{4.96}{1.05}=4.7238095\text{ years}. $$ A 25-basis-point increase is a yield change of $0.0025$. The estimated percentage price change is $$ -4.7238095(0.0025)=-0.0118095, $$ or approximately $-1.18095\%$. Applying that estimate to the full portfolio value gives $$ -\$1{,}250{,}000(0.0118095)=-\$14{,}761.90 $$ to the nearest cent. The estimated portfolio value after the change is therefore $\$1{,}250{,}000-\$14{,}761.90=\$1{,}235{,}238.10$.

Check. The bonds alone have Macaulay duration $6.2$ years. Adding positive zero-duration cash must lower it; $4.96<6.2$, as expected. The cash itself has no estimated price loss, so the dollar estimate also equals the bonds’ estimated loss: $-\$1{,}000{,}000(6.2/1.05)(0.0025)=-\$14{,}761.90$.

Common mistake avoided: Dividing the bond-only duration by $1.05$ and using that as the portfolio modified duration would ignore the cash dilution. That would estimate a larger portfolio percentage loss than the bond exposure warrants.

Tracking the Effect of an Additional Cash Deposit

When cash is added to a portfolio that already contains cash, use the new cash balance in the total value. The bond duration-weighted numerator stays unchanged if the bond positions and their durations do not change. Consequently, more fixed cash lowers portfolio duration, while the estimated dollar sensitivity of the bonds themselves remains the same for the same yield shock.

Worked Example: An Additional Deposit and a Yield Shock

A hypothetical portfolio has $900,000 of bonds with a Macaulay duration of 3.6 years and $100,000 of fixed cash. The bonds have a common annual effective yield of 4%. The investor adds another $200,000 of fixed cash. Find the old and new portfolio Macaulay durations, then estimate the dollar change in value for a 40-basis-point increase in yield.

Method. Calculate the bond duration-weighted value once. Divide it first by the old total value and then by the new total value. Use the new portfolio duration for the price-change estimate.

The bond duration-weighted numerator is $$ \$900{,}000(3.6)=\$3{,}240{,}000\text{ years}. $$ Before the deposit, total value is $\$900{,}000+\$100{,}000=\$1{,}000{,}000$, giving $$ D_{\text{old}}=\frac{\$3{,}240{,}000}{\$1{,}000{,}000}=3.24\text{ years}. $$ After the $\$200{,}000$ deposit, cash totals $\$300{,}000$, and portfolio value is $\$1{,}200{,}000$. Therefore, $$ D_{\text{new}}=\frac{\$3{,}240{,}000}{\$1{,}200{,}000}=2.70\text{ years}. $$ With a common annual effective yield of 4%, new portfolio modified duration is $$ D_{\text{mod,new}}=\frac{2.70}{1.04}=2.5961538\text{ years}. $$ The 40-basis-point increase is $0.0040$, so the estimated percentage change is $$ -2.5961538(0.0040)=-0.0103846, $$ or approximately $-1.03846\%$. The estimated dollar change is $$ -\$1{,}200{,}000(0.0103846)=-\$12{,}461.54 $$ to the nearest cent.

Check. The bond-only modified duration is $3.6/1.04=3.4615385$ years. Its estimated dollar loss for the shock is $-\$900{,}000(3.4615385)(0.0040)=-\$12{,}461.54$, the same as the portfolio estimate because the added cash is assumed unchanged.

Common mistake avoided: Applying the bond-only percentage loss to the larger total portfolio value overstates the dollar loss. The correct calculation either applies the new portfolio modified duration to total value or the bond-only modified duration to bond value.

When Component Yields Differ

The market-value-weighted formula for modified duration is especially useful when bond holdings have different yields. In that case, each component’s modified duration is calculated using its own yield and convention, and the portfolio modified duration is the weighted average of those component sensitivities, with cash still contributing zero. It is generally not correct to calculate a weighted Macaulay duration and divide it by one arbitrarily chosen yield.

Worked Example: Combining Modified Durations at Different Yields

A hypothetical portfolio contains Bond A with market value $500,000 and modified duration 4.2 years, Bond B with market value $300,000 and modified duration 6.0 years, and $200,000 in fixed cash. The bond yields differ, and each stated modified duration is calculated using its own bond’s yield convention. Estimate the portfolio’s percentage and dollar value change for a parallel yield increase of 30 basis points.

Method. Weight each bond’s modified duration by its market value, include cash only in total portfolio value, and apply the resulting portfolio sensitivity to the parallel yield change.

Total portfolio value is $$ \$500{,}000+\$300{,}000+\$200{,}000=\$1{,}000{,}000. $$ The modified-duration numerator is $$ \$500{,}000(4.2)+\$300{,}000(6.0) =\$2{,}100{,}000\text{ years}+\$1{,}800{,}000\text{ years} =\$3{,}900{,}000\text{ years}. $$ Thus, $$ D_{\text{mod},P}=\frac{\$3{,}900{,}000}{\$1{,}000{,}000} =3.9\text{ years}. $$ A 30-basis-point increase is $0.0030$, so the estimated percentage change is $$ -3.9(0.0030)=-0.0117=-1.17\%. $$ The estimated dollar change is $$ -\$1{,}000{,}000(0.0117)=-\$11{,}700.00. $$

Check. The bonds’ estimated dollar changes sum to $-\$500{,}000(4.2)(0.0030)-\$300{,}000(6.0)(0.0030)=-\$6{,}300-\$5{,}400=-\$11{,}700$. Cash contributes no change under the stated assumption.

Common mistake avoided: Do not convert a combined Macaulay duration using one of the two different bond yields. The given component modified durations already measure each bond’s local sensitivity; weighting them directly is the appropriate calculation for this common yield shift.

Common Traps and Practical Interpretation

The most frequent error is to use the bond holdings’ value as the denominator after cash has been added. That produces the duration of the bond subportfolio, not the full portfolio. Another error is to apply a percentage change to bond value when the percentage was calculated using portfolio modified duration, or vice versa. Keep each duration and value base paired consistently.

Key takeaway: For fixed zero-duration cash, add its market value to the portfolio denominator and add zero to the duration-weighted numerator. For price sensitivity, either use portfolio modified duration with total portfolio value or use each bond’s modified duration with that bond’s market value.

The resulting estimate is a first-order approximation for a small yield change, not an exact repricing. It is aligned with the SOA Exam FM (Financial Mathematics) syllabus. If cash earns interest or is represented by a security whose market value responds to yields, first determine whether the zero-duration assumption is appropriate rather than assigning it automatically.

Check Your Understanding

Use the stated fixed-cash assumption and keep durations in years. Each problem is independent.

A bond holding worth $700,000 has Macaulay duration 5 years. An investor adds $300,000 in fixed cash. What is the new portfolio Macaulay duration? (Answer: 3.5 years)
A $1,200,000 bond portfolio has Macaulay duration 6.3 years and a common annual effective yield of 4%. It receives $300,000 in fixed cash. Find the new portfolio modified duration and estimated dollar change for a 20-basis-point yield increase. (Answer: modified duration $4.8462$ years; estimated change $-\$14{,}538.46$)
A portfolio contains bonds with market values $400,000 and $600,000 and modified durations 2.5 and 7.0 years, plus $500,000 in fixed cash. Find the portfolio modified duration and estimated dollar change for a 10-basis-point parallel yield increase. (Answer: modified duration $3.4667$ years; estimated change $-\$5{,}200.00$)
A portfolio has $2,000,000 of bonds with Macaulay duration 4.2 years and $500,000 of cash. It receives another $500,000 of fixed cash. Find the portfolio Macaulay duration before and after the deposit and the decrease in duration. (Answer: before, 3.36 years; after, 2.80 years; decrease, 0.56 years)
Two bond holdings have market values $250,000 and $450,000 and modified durations 3.2 and 5.6 years, respectively. Their yields differ. The portfolio also has $300,000 of fixed cash. Find the portfolio modified duration and estimate the dollar change for a 25-basis-point parallel yield increase. (Answer: modified duration $3.32$ years; estimated change $-\$8{,}300.00$)

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