Equal NPV, Different Cash-Flow Risk

Equal Present Values Do Not Mean Equal Risk

Two projects can have the same present value at a stated yield and still respond differently when that yield changes. The reason is timing: an amount received sooner is discounted for fewer periods than an amount received later. Equal present values summarize value at one rate, but they do not by themselves describe sensitivity to other rates.

This comparison is useful when evaluating projects, assets, or other promised cash-flow streams under a common valuation yield. The analysis below assumes annual cash flows at year-end and a flat effective annual yield. It focuses on parallel changes in that yield; if rates vary by maturity, each cash flow must instead be valued using the relevant maturity-specific rate.

Definition: For cash flows $C_t$ payable at the end of year $t$, valued at an annual effective yield $y>-1$, the present value is $V(y)=\sum_{t=1}^{n} C_t(1+y)^{-t}$. Two streams have equal present value at a base yield $y_0$ if their values $V(y_0)$ are equal. Equal value at $y_0$ does not imply equal value at any other yield.

When all cash flows are nonnegative and the present value is positive, each discounted cash flow can be expressed as a proportion of total present value. These proportions show how the value is distributed across payment dates. They provide a direct way to calculate duration and convexity.

Definition: At yield $y$, the present-value weight of cash flow $C_t$ is $w_t=C_t(1+y)^{-t}/V(y)$. The Macaulay duration is $D_M=\sum_t t w_t$, measured in years. For annual effective compounding, modified duration is $D_{\mathrm{mod}}=D_M/(1+y)$. The modified convexity is $C_{\mathrm{mod}}=\sum_t t(t+1)w_t/(1+y)^2$.

The weights sum to one. Macaulay duration is therefore a weighted average payment time when all cash flows are nonnegative. Modified duration measures the approximate proportional price response to a small change in yield, while modified convexity describes the curvature that a duration-only estimate omits.

Why Duration and Convexity Describe Sensitivity

For a fixed set of cash flows, differentiating the present-value function with respect to its annual effective yield gives its first and second derivatives. The first derivative is negative for positive cash flows: increasing the yield decreases the present value. The second derivative is positive, so the value function is convex.

V(y)=\sum_{t=1}^{n}C_t(1+y)^{-t}

V'(y)=-\sum_{t=1}^{n}tC_t(1+y)^{-t-1} \qquad V''(y)=\sum_{t=1}^{n}t(t+1)C_t(1+y)^{-t-2}

Dividing the negative first derivative by present value gives modified duration; dividing the second derivative by present value gives modified convexity. A second-order Taylor approximation around a base yield $y_0$ then gives the following relationship. It is an approximation for a sufficiently small yield change, not an alternative to exact repricing.

\frac{\Delta V}{V} \approx -D_{\mathrm{mod}}\Delta y +\frac{1}{2}C_{\mathrm{mod}}(\Delta y)^2

The linear duration term predicts a larger price decline for a stream with greater modified duration when yields rise by the same small amount. Convexity adjusts that estimate, and can affect comparisons when durations are similar. For a specified yield change, exact repricing from the cash flows is definitive; duration and convexity help explain the result and estimate it.

Worked Examples

Worked Example: Equal Value, Different Durations

At a base yield of 5% effective annually, two positive-cash-flow projects each have a present value of $2,000. Project X has 80% of its base present value at year 1 and 20% at year 5. Project Y has 50% at year 2 and 50% at year 4. Compare their durations and their exact values if the yield rises to 6%.

Method and setup. The given proportions are present-value weights at 5%, not proportions of the undiscounted cash-flow amounts. For an exact repricing, each base-yield present-value contribution at time $t$ is multiplied by $(1.05/1.06)^t$. Set $q=105/106$, so $q\approx0.990566038$.

Duration comparison. For X, $D_{M,X}=0.8(1)+0.2(5)=1.8$ years. For Y, $D_{M,Y}=0.5(2)+0.5(4)=3.0$ years. Their modified durations at the base yield are $1.8/1.05\approx1.7143$ and $3/1.05\approx2.8571$, respectively. Y has the greater duration and should be more sensitive to a small yield increase.

Exact repricing of X. The needed powers are $q^5\approx0.953711829$. Therefore,

V_X(6\%) =2{,}000\left[0.8q+0.2q^5\right] \approx2{,}000\left[0.792452830+0.190742366\right] \approx \$1{,}966.39.

Exact repricing of Y. Here $q^2\approx0.981221075$ and $q^4\approx0.962794798$. Thus,

V_Y(6\%) =2{,}000\left[0.5q^2+0.5q^4\right] \approx2{,}000\left[0.490610538+0.481397399\right] \approx \$1{,}944.02.

The exact declines from the common $2,000 base value are $33.61 for X and $55.98 for Y. Y loses more, consistent with its greater duration. Common mistake avoided: using the correct powers matters: rounding or miscalculating $q^2$, $q^4$, or $q^5$ before weighting can change the reported cents.

Worked Example: Equal Duration, Different Convexities

At a 5% effective annual base yield, projects A and B each have a present value of $10,000. A has half of its base present value at year 2 and half at year 4. B has half at year 1 and half at year 5. Compare their durations and convexities, then reprice each at 6%.

Method and duration. Calculate the present-value-weighted time for each project. Both have $D_M=0.5(2)+0.5(4)=3$ years for A and $D_M=0.5(1)+0.5(5)=3$ years for B. Their modified durations are both $3/1.05\approx2.8571$.

Convexity comparison. At the base yield, A's weighted average of $t(t+1)$ is $0.5(6)+0.5(20)=13$. For B it is $0.5(2)+0.5(30)=16$. Dividing each by $1.05^2=1.1025$, the modified convexities are approximately $11.7914$ for A and $14.5125$ for B. B has greater convexity despite the equal durations.

For exact repricing, again let $q=105/106$, with $q\approx0.990566038$, $q^2\approx0.981221075$, $q^4\approx0.962794798$, and $q^5\approx0.953711829$. Then

V_A(6\%)=10{,}000\left[0.5q^2+0.5q^4\right]\approx \$9{,}720.08

V_B(6\%)=10{,}000\left[0.5q+0.5q^5\right]\approx \$9{,}721.39

B is worth about $1.31 more than A after the yield increase. Its greater convexity accounts for the difference when the first-order duration effects are the same. Common mistake avoided: equal duration does not mean identical repricing; later and more dispersed cash flows can have different convexity.

Worked Example: Exact Repricing Versus a Duration Estimate

A project is worth $3,000 at a 5% effective annual yield. Half of its base present value comes from a year-1 payment and half from a year-3 payment. Find its exact value at a 6% yield, and compare the exact decline with the first-order duration estimate.

Method and duration. The base-yield Macaulay duration is $D_M=0.5(1)+0.5(3)=2$ years. Its modified duration is $2/1.05\approx1.9048$. The yield change is $\Delta y=0.01$, so the first-order proportional change is approximately $-1.9048(0.01)=-0.019048$. The estimated decline is $3{,}000(0.019048)\approx \$57.14$, giving an estimated value of about $2,942.86.

Exact repricing. With $q=105/106$, $q\approx0.990566038$ and $q^3\approx0.971964273$. The exact value is

V(6\%) =3{,}000\left[0.5q+0.5q^3\right] \approx3{,}000\left[0.495283019+0.485982137\right] \approx \$2{,}943.80.

The exact decline is approximately $56.20, which is smaller than the duration-only estimate of $57.14. The difference is consistent with positive convexity: the second-order term offsets part of the duration-predicted decline. Common mistake avoided: treating a duration estimate as an exact new value; use the cash-flow formula when an exact value is requested.

Interpreting Comparisons and Avoiding Traps

For positive cash flows valued under one flat annual yield, the comparison can be organized as follows:

Equal present values: establish a common base for comparing changes, but do not establish equal sensitivity.
Different durations: for a small common yield change, greater modified duration generally means greater proportional sensitivity in the opposite direction from the yield change.
Equal durations: do not conclude that the streams will reprice identically. Convexity and higher-order effects can still differ.
Specified yield change: exactly reprice each stream if possible. Use duration and convexity as explanatory measures or approximations.

The sensitivity statements require care. Duration is a local, first-order measure at a particular yield. For a large yield movement, an approximation can be noticeably inaccurate. Also, the weighted-average interpretation of duration depends on nonnegative cash flows. If a stream contains both positive and negative amounts, its present-value weights need not be positive, so duration may no longer behave like an average payment time and simple rankings can be misleading.

Key takeaway: Equal value at one yield is only one point of agreement between two cash-flow streams. Compare payment timing through duration, curvature through convexity, and exact sensitivity through repricing at the specified yield.

Check Your Understanding

Assume all cash flows are positive, paid at year-end, and valued using annual effective yields. Treat the listed proportions as present-value weights at the stated base yield.

A $4,000 stream has 25% of its base value at year 1 and 75% at year 3. The base yield is 5%. Find its exact value after the yield rises to 6%. (Answer: $4{,}000[0.25q+0.75q^3]$, where $q=105/106$, or approximately $3,906.46.)
At a 5% base yield, two $8,000 streams have weights (0.5 at year 1, 0.5 at year 5) and (0.5 at year 2, 0.5 at year 4), respectively. Reprice both at 6% and identify the larger value. (Answer: approximately $7,777.11 and $7,776.06, respectively; the first is larger by about $1.05.)
A stream has base-yield weights 0.4 at year 1 and 0.6 at year 4, with a 5% base yield. Find its Macaulay duration and modified convexity. (Answer: $D_M=2.8$ years and $C_{\mathrm{mod}}=12.8/1.05^2\approx11.61$.)
A project worth $6,000 at a 5% yield has modified duration 2.5. Estimate its value after a 0.75 percentage-point yield increase using duration only. (Answer: $6{,}000[1-(2.5)(0.0075)]\approx \$5{,}887.50$.)

What You'll Learn