Negative Forward Rates And Accumulation

Introduction

A forward rate can be negative without implying that an investment balance becomes negative. Under the usual positive-accumulation convention, a negative effective forward rate means that a dollar invested over the specified interval grows to less than one dollar. The amount remains positive; it is the change in value that is negative.

The interpretation depends on the rate convention and the length of the interval. A quoted nominal rate must be compounded at its stated frequency, while an annual effective rate must be annualized if the forward interval is not one year. This tutorial focuses on checking the accumulation factor first, then interpreting the rate consistently. It is aligned with the SOA Exam FM (Financial Mathematics) syllabus.

Forward Accumulation and a Negative Effective Rate

Let $D(0,t)$ denote the time-0 price of a zero-coupon payment of one unit at time $t$. Assume the discount factors are positive and that the rates over the interval are expressed using conventional positive accumulation factors. For an interval from time $s$ to time $t$, its length is $h=t-s>0$.

Definition: The forward accumulation factor from $s$ to $t$ is $A_{s,t}=D(0,s)/D(0,t)$. The annual effective forward rate $i_{s,t}$ over that interval is defined by $A_{s,t}=(1+i_{s,t})^h$, so $i_{s,t}=A_{s,t}^{1/h}-1$. For a one-year interval, the forward accumulation factor is $1+i_{s,t}$.

The distinction between the interval’s total factor and its annual effective rate matters whenever $h\ne1$. The factor $A_{s,t}$ describes the growth from the start to the end of the whole interval. The annual effective rate is the constant annual rate that would produce that same factor over $h$ years.

Key result: If $0<A_{s,t}<1$, then $i_{s,t}<0$, but $i_{s,t}>-1$. Conversely, a conventional negative annual effective forward rate satisfies $-1<i_{s,t}<0$ and produces a positive accumulation factor below one. A negative accumulation factor is not consistent with this conventional effective-rate model.

The lower bound follows directly from the rate definition. Since $A_{s,t}>0$, its positive $h$-th root is also positive, so $1+i_{s,t}>0$ and $i_{s,t}>-1$. If $A_{s,t}<1$, then $A_{s,t}^{1/h}<1$, which gives $i_{s,t}<0$. A rate of $-100\%$ would produce a zero factor; a rate below $-100\%$ would produce a negative one-year factor, neither of which represents a conventional positive accumulation of invested funds.

Why the Discount-Factor Ratio Gives the Forward Factor

The ratio $D(0,s)/D(0,t)$ follows by comparing two ways to match the same payment at time $t$. An amount $C$ payable at time $s$ has present value $C D(0,s)$. If that amount is accumulated to time $t$ by the forward factor $A_{s,t}$, its time-$t$ value is $C A_{s,t}$, whose present value is $C A_{s,t}D(0,t)$. Equating the present values gives:

C D(0,s)=C A_{s,t}D(0,t) \quad\Longrightarrow\quad A_{s,t}=\frac{D(0,s)}{D(0,t)}.

This identity is useful even when the forward rate is negative: calculate the ratio, verify that it is positive, and then apply the stated rate convention. Do not infer the sign of the forward rate just by looking at one discount factor in isolation.

Worked Examples

Worked Example: A Negative One-Year Effective Forward Rate

Suppose $D(0,1)=0.9800$ and $D(0,2)=1.0000$. Find the one-year effective forward rate from year 1 to year 2. Interpret the factor for a hypothetical investment of $10,000 made at year 1.

Method. First form the forward accumulation factor from the discount-factor ratio. Since the interval is one year, the factor equals one plus the annual effective forward rate.

Step 1: Calculate the factor. $A_{1,2}=D(0,1)/D(0,2)=0.9800/1.0000=0.9800.$

Step 2: Calculate the rate. $i_{1,2}=A_{1,2}-1=0.9800-1=-0.0200=-2.00\%.$

Step 3: Interpret the accumulation. Under this hypothetical rate, $10,000 at year 1 accumulates to $10{,}000(0.9800)=\$9{,}800$ at year 2, a decrease of $200 over the year.

Check. The calculated rate is above $-100\%$, and $1+i_{1,2}=1-0.0200=0.9800$, the original positive factor. The method avoids treating a negative rate as though it were a negative balance.

Worked Example: Annualizing a Negative Forward Return over Three Years

Suppose the discount factors at years 2 and 5 are $D(0,2)=0.9000$ and $D(0,5)=1.0000$. Find the total forward accumulation factor from year 2 to year 5 and the equivalent annual effective forward rate. How much would a hypothetical $20,000 investment at year 2 accumulate to by year 5 under this factor?

Method. Use the discount-factor ratio for the full three-year factor, then take its three-year root to obtain the annual effective rate.

Step 1: Identify the interval. Its length is $h=5-2=3$ years.

Step 2: Calculate the full-interval factor. $A_{2,5}=0.9000/1.0000=0.9000.$

Step 3: Annualize the factor. $i_{2,5}=0.9000^{1/3}-1$. The cube root is approximately $0.9654894$, so the annual effective forward rate is approximately $-0.0345106$, or $-3.4511\%$, rounded to four decimal places as a percentage.

Step 4: Calculate the accumulated amount. $20{,}000(0.9000)=\$18{,}000.$ The decrease over the full three years is $2,000.

Check. Applying the annual effective factor for three years gives approximately $0.9654894^3=0.9000$, as required. The method avoids calling the three-year total decrease of 10% an annual effective rate of $-10\%$.

Worked Example: Checking a Nominal Quarterly Forward Quote

A one-year forward interval has accumulation factor $0.92236816$. A quoted forward rate is nominal $j^{(4)}$, convertible quarterly. Determine the quoted rate and the equivalent annual effective rate. Check whether a report that calls the effective annual rate $-8\%$ is consistent with the factor.

Method. Under a nominal rate convertible quarterly, the quarterly rate is $j^{(4)}/4$, and four quarterly factors make up the one-year factor.

Step 1: Apply the nominal-rate convention. $0.92236816=(1+j^{(4)}/4)^4.$

Step 2: Take the positive fourth root. $0.92236816=0.98^4$, so $1+j^{(4)}/4=0.98$. Therefore $j^{(4)}/4=-0.02$, and $j^{(4)}=-0.08=-8\%$ nominal convertible quarterly.

Step 3: Find the equivalent annual effective rate. For a one-year interval, the annual effective rate is the factor minus one: $i=0.92236816-1=-0.07763184=-7.763184\%$, or $-7.7632\%$ rounded to four decimal places as a percentage.

Step 4: Check the report. An effective annual rate of $-8\%$ would produce a factor of $1-0.08=0.92$, not $0.92236816$. Thus the report is incorrect if it labels $-8\%$ as the annual effective rate. It is correct only as a nominal quarterly rate.

Check. Quarterly accumulation at $-2\%$ per quarter gives $0.98^4=0.92236816$, the stated factor. The method avoids treating a nominal quoted rate as if it were an effective annual rate.

Other Rate Conventions and Consistency Checks

The same positive forward factor can be expressed under different conventions, but the quoted numbers are not interchangeable. For an interval of length $h$, a nominal rate $j^{(m)}$ convertible $m$ times per year satisfies $A_{s,t}=(1+j^{(m)}/m)^{mh}$, when the interval and convention use $mh$ compounding periods. The periodic factor must be positive: $1+j^{(m)}/m>0$, or $j^{(m)}>-m$. A negative nominal rate can therefore be valid while still producing a positive total accumulation factor.

For a continuously compounded forward rate $\delta_{s,t}$, the convention is $A_{s,t}=e^{\delta_{s,t}h}$. Therefore $\delta_{s,t}=\ln(A_{s,t})/h$. Because the exponential is positive, any finite continuous rate gives a positive accumulation factor; the rate is negative exactly when that factor is less than one.

Convention	Accumulation factor over $h$ years	Condition for a negative rate
Annual effective	$(1+i_{s,t})^h$	$0<A_{s,t}<1$, giving $-1<i_{s,t}<0$
Nominal, convertible $m$ times yearly	$(1+j^{(m)}/m)^{mh}$	The periodic rate is negative and greater than $-1$
Continuous	$e^{\delta_{s,t}h}$	$0<A_{s,t}<1$, giving $\delta_{s,t}<0$

A practical consistency check is to start with the factor implied by the discount factors, then reconstruct it using the quoted convention. The reconstructed factor must equal the original ratio. This check catches both a sign error and a compounding-convention error.

Common Mistakes and Exam Traps

Using the ratio in the wrong order. The forward factor is $D(0,s)/D(0,t)$, not $D(0,t)/D(0,s)$. Reversing the ratio reverses the factor and may reverse the inferred sign.
Equating a negative rate with a negative factor. For example, an effective rate of $-2\%$ gives factor $0.98$, not $-0.02$. The rate is the factor minus one for a one-year interval.
Ignoring the interval length. Over three years, a factor of $0.90$ is a total decrease of 10%, but the equivalent annual effective rate is $0.90^{1/3}-1$, not $-10\%$.
Mixing nominal and effective rates. A nominal quarterly rate of $-8\%$ has a quarterly rate of $-2\%$ and an annual factor of $0.98^4$; it is not the same as an effective annual rate of $-8\%$.
Accepting an impossible conventional factor. A negative factor cannot be represented by a real annual effective rate under the usual positive-accumulation convention. Recheck the ratio, the stated convention, and the data before interpreting such a result.

Key takeaway: A negative forward rate means a positive accumulation factor below one. Calculate the factor from the discount-factor ratio, annualize it for the actual interval length, and verify it under the stated compounding convention before interpreting the quoted rate.

Check Your Understanding

For each problem, first identify the interval and rate convention, then calculate or verify the implied accumulation factor.

A one-year forward interval has $D(0,1)=0.9700$ and $D(0,2)=1.0000$. Find the annual effective forward rate. (Answer: $-3.00\%$)
The factor over a two-year forward interval is $0.9604$. Find the equivalent annual effective forward rate. (Answer: $-2.00\%$)
A forward accumulation factor is $0.9400$ over two years. Find the equivalent annual effective rate, rounded to four decimal places as a percentage. (Answer: approximately $-3.0464\%$)
A four-month forward interval has factor $0.9920$. Find its equivalent annual effective rate, rounded to four decimal places as a percentage. (Answer: approximately $-2.3809\%$)
A one-year forward quote is nominal $-6\%$, convertible semiannually. Find its accumulation factor and equivalent annual effective rate. (Answer: factor $0.9409$; annual effective rate $-5.91\%$)

Convention	Accumulation factor over \(h\) years	Condition for a negative rate
Annual effective	\((1+i_{s,t})^h\)	\(0<A_{s,t}<1\), giving \(-1<i_{s,t}<0\)
Nominal, convertible \(m\) times yearly	\((1+j^{(m)}/m)^{mh}\)	The periodic rate is negative and greater than \(-1\)
Continuous	\(e^{\delta_{s,t}h}\)	\(0<A_{s,t}<1\), giving \(\delta_{s,t}<0\)

What You'll Learn