Valuation-Rate and Compounding Consistency

Why Valuation Consistency Matters

A Redington comparison is only meaningful when assets and liabilities are valued at the same rate on the same time basis. A rate quote may be annual effective, nominal convertible several times per year, or expressed as a force of interest. Those are different ways to describe accumulation. They can represent the same economic rate, but only after conversion. Likewise, a cash flow due in years must not be discounted as if its date were a number of quarters or months.

A mismatch can change the apparent present-value surplus and can also distort the duration and convexity comparisons used for local immunization. The essential discipline is to identify the valuation-rate convention, convert equivalent quotes where needed, express each payment date in the corresponding periods, and use one rate variable consistently when taking derivatives.

Definition: A consistent valuation basis specifies (1) the interest-rate convention and its equivalent accumulation, (2) the time unit for payment dates, and (3) the rate variable with respect to which any sensitivity or derivative is calculated. Assets and liabilities must use the same basis for a Redington comparison.

Converting Rate Quotes to One Valuation Basis

Let $i$ be the effective annual rate and $v=1/(1+i)$. Under annual effective interest, a cash flow $C_t$ paid $t$ years from now has present value $C_t v^t$. If the rate is instead quoted as a nominal annual rate $i^{(m)}$, convertible $m$ times per year, the accumulation over $t$ years is $(1+i^{(m)}/m)^{mt}$, provided fractional periods are handled consistently with this convention.

1+i=\left(1+\frac{i^{(m)}}{m}\right)^m, \qquad i=\left(1+\frac{i^{(m)}}{m}\right)^m-1

The equivalent force of interest $\delta$ and discount rate $d$ satisfy $1+i=e^\delta=1/(1-d)$. These conversions preserve the accumulation over a year; they do not permit using the unconverted quoted number as if it were an annual effective rate.

i=e^\delta-1, \qquad \delta=\ln(1+i), \qquad d=\frac{i}{1+i}, \qquad v=1-d

For example, a nominal rate convertible quarterly uses a quarterly rate $i^{(4)}/4$. If a payment is due after $t$ years, the number of quarters is $4t$, not $t$. The same payment can instead be valued using the equivalent annual effective rate and the exponent $t$. Mixing the quarterly rate with a year-based exponent is the underlying error in many apparently plausible calculations.

Worked Example: A Quarterly Quote and a Payment in Eighteen Months

A hypothetical liability of $9,000 is due in 1.5 years. The valuation rate is 3% per quarter. Find its present value using both the quarterly rate and the equivalent annual effective rate. State the common period-count mistake.

Method. Use the stated quarterly accumulation directly, then convert to an annual effective rate as a check. There are $4(1.5)=6$ quarters to the payment date.

Quarterly calculation. The accumulation factor is $1.03^6=1.194052296$. Thus $9000/1.194052296=\$7,537.36$, rounded to the nearest cent.

Annual-effective check. Four quarters accumulate by $1.03^4=1.12550881$, so the equivalent annual effective rate is $1.12550881-1=0.12550881$, or 12.5509%. Over 1.5 years the accumulation factor is $(1.12550881)^{1.5}=1.03^6=1.194052296$. The present value is again $7,537.36.

Common mistake and correction. Using $9000/(1.03)^3$ counts only three quarters, although 1.5 years contains six quarters. That incorrect calculation gives approximately $8,236.27, overstating the present value. The payment date, not the number of years alone, determines the number of quarterly periods.

Check. Both valid methods use the same accumulation factor, $1.194052296$, and therefore produce the same present value.

Worked Example: Converting a Nominal Semiannual Rate

A hypothetical payment of $4,000 is due in three years. The quoted valuation rate is 8% nominal, convertible semiannually. Find the present value using the nominal quote and then using its equivalent annual effective rate.

Method. Convert the quote to a half-year rate, count the half-year periods, and verify by converting to an annual effective rate.

The rate per half-year is $0.08/2=0.04$. There are $2(3)=6$ half-year periods, so the accumulation factor is $1.04^6=1.265319018496$. The present value is $4000/1.265319018496=\$3,161.26$, rounded to the nearest cent.

The equivalent annual effective rate is $(1.04)^2-1=0.0816$, or 8.1600%. Valuing three years at that rate gives the same factor: $(1.0816)^3=(1.04)^6=1.265319018496$. The resulting present value is again $3,161.26.

Common mistake and correction. Applying the annual effective rate as 8% would use a different accumulation factor, because 8% nominal convertible semiannually is not 8% effective annually. The conversion shows the effective annual rate is 8.16%.

Check. Both methods represent six half-year accumulations of 4%, so agreement is required.

Consistency in Redington Derivatives

Suppose a portfolio has deterministic cash flows $C_t$ at times $t$ years, and all cash flows are valued at one flat annual effective rate $i$. Its present value is $P(i)=\sum_t C_t(1+i)^{-t}$. For the conventional duration and convexity interpretation, take the cash flows in the portfolio to be nonnegative and the present value to be positive. Assets and liabilities may each be valued this way; the surplus is $S(i)=A(i)-L(i)$.

P(i)=\sum_t C_t(1+i)^{-t}, \qquad P'(i)=-\sum_t tC_t(1+i)^{-t-1}, \qquad P''(i)=\sum_t t(t+1)C_t(1+i)^{-t-2}

The Macaulay duration at $i$ is the present-value-weighted mean time, $D(i)=\sum_t tC_t(1+i)^{-t}/P(i)$. The derivative formulas imply $P'(i)=-P(i)D(i)/(1+i)$. Hence, when asset and liability present values match at the valuation rate, equality of their Macaulay durations is equivalent to equality of their first derivatives with respect to $i$. The second-derivative comparison must also use that same rate variable.

Redington comparison on a consistent basis: At the valuation rate $i_0$, compare asset and liability present values, first derivatives, and second derivatives as functions of the same annual effective rate. The usual local conditions are $A(i_0)=L(i_0)$, $A'(i_0)=L'(i_0)$, and $A''(i_0)>L''(i_0)$. For nonnegative cash flows, the first two conditions express present-value and duration matching, while the strict second-derivative inequality provides the local convexity margin.

Why Equivalent Rate Variables Preserve a Valid Comparison

A consistent change of rate variable does not itself reverse a valid Redington convexity comparison. The important qualifications are that the same smooth conversion is applied to assets and liabilities and that present value and first derivative are matched at the valuation point.

To see why, write the surplus in a new rate variable $q$, with $i=g(q)$. The chain rule gives the first derivative and second derivative below. At a matched point $S(i_0)=0$ and $S'(i_0)=0$; in particular, the second-derivative comparison uses the first-derivative match. The term involving $g''(q)$ therefore vanishes, leaving a positive square multiplying the original second derivative.

\frac{dS}{dq}=S'(i)g'(q), \qquad \frac{d^2S}{dq^2}=S''(i)\bigl(g'(q)\bigr)^2+S'(i)g''(q)

Thus, at a matched point with $g'(q)\ne 0$, the sign of the surplus curvature is preserved. This does not justify mixing rate bases: if assets are differentiated with respect to one variable and liabilities with respect to another, or if the conversion is applied to only one side, the quantities being compared are not the derivatives of a single consistently defined surplus.

Worked Example: Verifying Redington Conditions Under an Equivalent Quote

A hypothetical asset portfolio pays $55 at year 1 and $66.55 at year 3. A liability pays $121 at year 2. The valuation rate is 10% effective annually. Verify the present-value and duration matches and the strict convexity condition. Then state the equivalent nominal semiannual quote and explain what happens to the curvature comparison under that rate variable.

Method. Value each dated cash flow using the annual effective rate, then compare present-value-weighted times and second derivatives with respect to $i$.

The year-1 asset present value is $55/1.1=50$. The year-3 asset present value is $66.55/1.1^3=66.55/1.331=50$. Thus asset present value is $50+50=100$. The liability present value is $121/1.1^2=121/1.21=100$.

Asset duration is $(50(1)+50(3))/100=200/100=2$ years. Liability duration is $100(2)/100=2$ years. Present value and duration therefore match.

For the second derivatives, the asset amount is $[50(1)(2)+50(3)(4)]/1.1^2=(100+600)/1.21=578.5124$. The liability amount is $100(2)(3)/1.21=600/1.21=495.8678$. Therefore $A''(i_0)-L''(i_0)=82.6446>0$, so the strict convexity condition holds. Values here are rounded to four decimal places.

The equivalent nominal rate convertible semiannually is $i^{(2)}=2(\sqrt{1.10}-1)$, approximately 9.7618%. Its half-year accumulation factor is $\sqrt{1.10}$, so each dated cash flow has the same present value under either convention. If derivatives are taken with respect to $i^{(2)}$, then $i=(1+i^{(2)}/2)^2-1$, and $di/di^{(2)}=\sqrt{1.10}$ at this point. Since the surplus first derivative is zero, its second derivative in the new variable is $1.10$ times the second derivative in $i$; the positive curvature difference remains positive.

Common mistake and correction. Comparing an asset duration computed from annual-effective present values with a liability duration computed using an unconverted nominal quote would mix valuation bases. Convert the quote first, or value both sides directly using the same nominal convention and dates.

Check. Both sides have present value 100 and duration 2 years, and the asset second derivative exceeds the liability second derivative by 82.6446 in the annual-effective rate variable.

Practical Checks Before Comparing Portfolios

Before performing any Redington test, write down the rate convention and the payment-date unit explicitly. A compact audit prevents most basis errors:

Identify the quote.
Record whether the rate is effective annually, nominal convertible $m$ times per year, or a force of interest. Do not silently treat the quoted number as an annual effective rate.

Convert or define the accumulation.
Use the equivalent annual effective rate, or retain the quoted convention and write its accumulation factor explicitly.

Translate every date.
For a nominal $m$-thly quote, a payment $t$ years away corresponds to $mt$ conversion periods under the stated convention.

Use one derivative variable.
Calculate both sides’ sensitivities with respect to the same rate variable. If changing variables, apply the chain rule to the surplus.

Check equivalence.
Revalue at least one cash flow using the converted rate. The two accumulation factors should agree before proceeding with the portfolio comparison.

A useful diagnostic is to compare accumulation factors before comparing present values. If the same dated payment produces different factors under two supposedly equivalent quotes, the conversion or period count is wrong. Once values agree, confirm that duration and convexity use consistent present-value weights and the same derivative variable. These checks make the resulting Redington conclusion interpretable rather than an artifact of notation.

Key takeaway: Equivalent rate quotes produce equivalent valuations only when their compounding and payment periods are aligned. Redington conditions compare derivatives of one consistently defined surplus; matching the rate basis is part of the mathematics, not merely a presentation choice.

Check Your Understanding

For each problem, use one consistent valuation basis and check the payment-period count.

A $6,000 payment is due in two years at 5.4% nominal interest convertible monthly. State the monthly rate, number of months, and present-value expression. (Answer: monthly rate $0.054/12=0.0045$; 24 months; $6000/(1.0045)^{24}$.)
A hypothetical payment of $5,000 is due in two years, and the force of interest is 6% per year. Find the equivalent annual effective rate and present value to the nearest cent. (Answer: $i=e^{0.06}-1\approx 6.1837\%$; $5000e^{-0.12}=\$4,434.60$.)
A $10,000 payment is due in 2.5 years under a rate of 2% per quarter. Give the correct number of quarters and present-value expression. (Answer: 10 quarters; $10000/(1.02)^{10}$.)
A nominal rate convertible semiannually is 7.2%. Find its equivalent annual effective rate. (Answer: $(1+0.072/2)^2-1=7.3296\%$.)
At a matched valuation point, suppose the surplus has $S'(i_0)=0$, $S''(i_0)=40$, and $i=(1+q/2)^2-1$ with $di/dq=1.05$ at the point. Find $d^2S/dq^2$ there. (Answer: $40(1.05)^2=44.1$, which is positive.)

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