## **Learning Outcomes**

This article explains how continuous compounding fits into time value of money, how discount factors are defined and used, and how to convert rates between compounding conventions. After reading, you will be able to:

- Compute present values and future values using continuously compounded rates and discount factors
- Convert interest rates between discrete compounding frequencies and continuous compounding
- Derive discount factors from spot rates and connect discount factors to forward rates and derivative pricing

## **GARP FRM Part 1 Syllabus**

For the GARP FRM Part 1 exam, you are expected to understand time value of money and discounting with a focus on the following syllabus points:

- Define discount factor and use a discount function to compute present and future values
- Calculate and interpret the impact of different compounding frequencies on a bond’s value
- Define spot rate and compute discount factors given spot rates
- Interpret forward rates and compute forward rates given spot rates (including continuously compounded formulations)

## **Test Your Knowledge**

_Attempt these questions before reading this article. If you find some difficult or cannot remember the answers, remember to look more closely at that area during your revision._

1. The discount factor for \$1 received in 3 years when the continuously compounded spot rate is 6.25% is closest to:  
   a) 0.9420  
   b) 0.8869  
   c) 0.8290  
   d) 0.7500

2. Convert a 5.00% continuously compounded annual rate into an equivalent nominal rate with semiannual compounding.

3. In one or two sentences, explain why option pricing models often use continuously compounded interest rates even when government bonds are quoted with semiannual compounding.

4. True or False (brief explanation required): If the term structure is flat at a continuously compounded rate $r$, then the discount function satisfies $d(t+s)=d(t)d(s)$.

## **What Do You Know Already?**

- Can you write the present value formula under $m$-times-per-year compounding and under continuous compounding?
- Can you move between a quoted “nominal” rate (e.g., semiannual compounding) and a continuously compounded rate?
- If you are given a spot rate curve $r(t)$, can you produce the corresponding discount factor curve $d(t)$?

Use these prompts to gauge your starting point; then focus your revision where your answers were least confident.

## **Introduction**

Discounting is the mechanism that turns future cash flows into today’s values. It is used everywhere in risk management: pricing bonds, valuing swaps, computing option values under risk-neutral pricing, and converting future exposures or losses into present-value terms for reporting and limit systems. For FRM candidates, discounting is not only a “time value of money” topic; it is the language through which term structures, arbitrage arguments, and valuation models connect.

Compounding frequency matters because quoted interest rates are not complete descriptions unless the compounding convention is stated. A rate quoted with annual compounding, semiannual compounding, or continuous compounding will produce different future values and therefore different present values. Continuous compounding is especially common in derivatives pricing because it produces compact formulas and aligns naturally with exponential growth and log returns.

> **Key Term: Discount factor**  
> A discount factor $d(t)$ is the present value today of \$1 received at time $t$. Equivalently, it is the multiplier that converts a cash flow at time $t$ into its present value: $PV = CF_t \times d(t)$.

## **Continuous compounding as a time value model**

Interest can be compounded at different frequencies. If a nominal annual rate $R$ is compounded $m$ times per year, then the per-period rate is $R/m$, and there are $mt$ compounding periods in $t$ years. The future value (FV) of a present value (PV) is:

$$FV = PV\left(1+\frac{R}{m}\right)^{mt}$$

Discounting is the inverse operation:

$$PV = \frac{FV}{\left(1+\frac{R}{m}\right)^{mt}} = FV\left(1+\frac{R}{m}\right)^{-mt}$$

As the compounding frequency increases (annual $\rightarrow$ semiannual $\rightarrow$ quarterly $\rightarrow$ monthly), the FV increases slightly for a given quoted nominal rate, and the PV decreases slightly for a given future cash flow. This is why lenders prefer more frequent compounding, while borrowers prefer less frequent compounding.

### Continuous compounding

Continuous compounding is the limiting case as $m \rightarrow \infty$. If $r$ is the continuously compounded annual rate, then:

$$FV = PV\,e^{rt}$$
$$PV = FV\,e^{-rt}$$

This matches the intuition that growth is exponential in time when the growth rate is constant.

> **Key Term: Continuously compounded rate**  
> A continuously compounded rate $r$ is defined so that an amount grows as $e^{rt}$. It is also the continuously compounded spot rate when it applies to a single maturity $t$.

A useful bridge to remember is the one-year relationship between a continuously compounded rate and an effective annual growth factor:

- If the continuously compounded rate is $r$, then the effective annual return is $e^{r}-1$.
- If the effective annual return is $R_{\text{eff}}$, then the equivalent continuous rate is $\ln(1+R_{\text{eff}})$.

Because exponential functions and logarithms appear repeatedly in derivatives pricing (and in lognormal asset price models), continuous compounding becomes the default convention in many valuation formulas.

## **Discount factors and the discount function**

A discount factor curve (also called a discount function) is a mapping from time to maturity $t$ into a present-value multiplier $d(t)$. If you know $d(t)$ for all relevant maturities, you can value any deterministic set of cash flows:

$$PV = \sum_{i=1}^{N} CF(t_i)\,d(t_i)$$

This is the backbone of fixed-income valuation: price equals the sum of discounted cash flows.

> **Key Term: Discount function**  
> The discount function $d(t)$ is the term structure of discount factors. It summarizes the time value of money across maturities and is sufficient to price risk-free cash flows.

### Discount factors from spot rates

Spot rates (zero-coupon rates) provide the same information as discount factors, but expressed as an interest rate rather than a present-value multiplier.

If the $t$-year spot rate is expressed with semiannual compounding (common in U.S. Treasury conventions), then:

$$d(t)=\left(1+\frac{r(t)}{2}\right)^{-2t}$$

If the $t$-year spot rate is expressed with continuous compounding, then:

$$d(t)=e^{-r(t)t}$$

Both formulas say the same thing: discounting is just the reciprocal of compounding, with the convention determining the exact functional form.

### Spot rates from discount factors

In practice you often move in both directions: instruments imply discount factors, and discount factors imply spot rates.

For continuous compounding:

$$r(t) = -\frac{1}{t}\ln(d(t))$$

For semiannual compounding:

$$r(t) = 2\left(d(t)^{-1/(2t)} - 1\right)$$

These conversions are mechanical but exam-relevant because problems may supply either spot rates or discount factors.

### Worked Example 1.1

Compute the present value of \$1,000 to be received in 3 years when the continuously compounded annual rate is 5%.

> **Answer:**  
> Use $PV = FV\,e^{-rt}$.
>
> 1. Identify inputs: $FV=1000$, $r=0.05$, $t=3$.
> 2. Discount:
>    $$
>    PV = 1000\,e^{-0.05\times 3} = 1000\,e^{-0.15}
>    $$
> 3. Compute:
>    $$
>    e^{-0.15}\approx 0.86071 \;\Rightarrow\; PV \approx 1000\times 0.86071 = 860.71
>    $$
>
> The present value is approximately \$860.71.

## **Rate conversion to and from continuous compounding**

Risk managers constantly translate between rate conventions:

- Treasury cash bonds often use semiannual compounding.
- Money market instruments may use Actual/360 or Actual/365 conventions and quote simple or periodically compounded rates.
- Derivatives valuation commonly uses continuous compounding in formulas.

Equivalent rates are defined to produce the same future value over the same horizon.

### Converting between two discrete compounding frequencies

If $R_1$ is compounded $m_1$ times per year and we want the equivalent rate $R_2$ compounded $m_2$ times per year, set the one-year accumulation factors equal:

$$(1 + R_1/m_1)^{m_1} = (1 + R_2/m_2)^{m_2}$$

Solving for $R_2$ yields the exam-usable conversion formula:

$$R_2 = \left[\left(1+\frac{R_1}{m_1}\right)^{m_1/m_2}-1\right]m_2$$

### Converting discrete compounding to continuous compounding

Let $R_m$ be the nominal annual rate compounded $m$ times per year, and let $R_c$ be the equivalent continuously compounded annual rate. Equate one-year growth:

$$e^{R_c} = \left(1+\frac{R_m}{m}\right)^m$$

So:

$$R_c = m\ln\left(1+\frac{R_m}{m}\right)$$

and the reverse conversion is:

$$R_m = m\left(e^{R_c/m}-1\right)$$

> **Key Term: Equivalent rates**  
> Two quoted interest rates (possibly with different compounding conventions) are equivalent if they generate the same accumulation factor over the same horizon.

### Worked Example 2.1

A rate is 6% per year compounded semiannually. Find the equivalent continuously compounded annual rate.

> **Answer:**  
> Here $R_m=0.06$ and $m=2$.
>
> 1. Apply the conversion:
>    $$
>    R_c = 2\ln\left(1+\frac{0.06}{2}\right)=2\ln(1.03)
>    $$
> 2. Compute:
>    $$
>    \ln(1.03)\approx 0.0295588 \;\Rightarrow\; R_c \approx 2\times 0.0295588 = 0.0591176
>    $$
>
> The equivalent continuously compounded rate is approximately 5.91% per year.

### Worked Example 2.2

A continuously compounded annual rate is 5%. Find the equivalent nominal annual rate compounded semiannually.

> **Answer:**  
> Here $R_c=0.05$ and $m=2$.
>
> 1. Use the reverse conversion:
>    $$
>    R_m = 2\left(e^{0.05/2}-1\right)
>    $$
> 2. Compute:
>    $$
>    e^{0.025}\approx 1.025315 \;\Rightarrow\; R_m \approx 2(0.025315)=0.05063
>    $$
>
> The equivalent semiannually compounded nominal rate is approximately 5.06% per year.

### Worked Example 2.3

A continuously compounded 3-year spot rate is 6.25%. Compute the 3-year discount factor.

> **Answer:**  
> For continuous compounding, $d(t)=e^{-rt}$.
>
> 1. Inputs: $r=0.0625$, $t=3$.
> 2. Discount:
>    $$
>    d(3)=e^{-0.0625\times 3}=e^{-0.1875}
>    $$
> 3. Compute:
>    $$
>    e^{-0.1875}\approx 0.8290
>    $$
>
> The 3-year discount factor is approximately 0.8290.

## **Forward rates expressed with continuous compounding**

A forward rate is a future interest rate implied by today’s term structure. The core idea is an arbitrage-based indifference: investing to $T_2$ directly should produce the same return as investing to $T_1$ and then reinvesting from $T_1$ to $T_2$, otherwise a riskless profit would be available (the “law of one price” logic applied to time).

For continuously compounded spot rates $R_1$ for maturity $T_1$ and $R_2$ for maturity $T_2$, the continuously compounded forward rate between $T_1$ and $T_2$ is:

$$F(T_1,T_2)=\frac{R_2T_2 - R_1T_1}{T_2-T_1}$$

This formula is heavily used because it avoids compounding-within-compounding algebra.

> **Key Term: Forward rate**  
> The forward rate $F(T_1,T_2)$ is the interest rate for borrowing/lending over a future interval $[T_1,T_2]$ that is implied by today’s spot rates (or discount factors) under no-arbitrage conditions.

### Forward rates from discount factors

Using $d(t)=e^{-R(t)t}$, we can write the forward rate directly in terms of discount factors:

$$\frac{d(T_2)}{d(T_1)} = e^{-F(T_1,T_2)(T_2-T_1)}$$

So:

$$F(T_1,T_2) = -\frac{1}{T_2-T_1}\ln\left(\frac{d(T_2)}{d(T_1)}\right)$$

This relationship is conceptually important: discount factors contain the entire term structure, so forward rates are just a re-expression of that same information.

### Worked Example 3.1

Continuously compounded spot rates are 4.25% for 3 years and 4.40% for 3.5 years. Compute the 6-month forward rate from year 3 to year 3.5.

> **Answer:**  
> Use $F=\frac{R_2T_2-R_1T_1}{T_2-T_1}$ with $R_1=0.0425$, $T_1=3.0$, $R_2=0.0440$, $T_2=3.5$.
>
> 1. Compute the numerator:
>    $$
>    0.0440\times 3.5 - 0.0425\times 3.0 = 0.1540 - 0.1275 = 0.0265
>    $$
> 2. Divide by $0.5$:
>    $$
>    F = \frac{0.0265}{0.5}=0.0530
>    $$
>
> The 6-month forward rate is 5.30% (continuously compounded, annualized).

## **Applications of continuous discounting in valuation and risk**

Continuous compounding is not just a mathematical convenience; it shows up explicitly in FRM valuation models.

### Risk-neutral valuation in option pricing

In risk-neutral pricing, the value of a derivative is the discounted expected payoff under the risk-neutral probability measure:

$$V_0 = e^{-rT}\,E^{\mathbb{Q}}[\text{Payoff at }T]$$

Here $e^{-rT}$ is the discount factor when the risk-free rate $r$ is continuously compounded.

In the binomial model, this same exponential structure appears in the risk-neutral probabilities. For a one-step tree with step length $t$ (in years), up/down factors $U$ and $D$, and continuously compounded risk-free rate $r$, the risk-neutral up probability is:

$$\pi_u = \frac{e^{rt}-D}{U-D}$$

and $\pi_d = 1-\pi_u$. The presence of $e^{rt}$ is not accidental: it encodes the idea that under the risk-neutral measure, the expected growth rate of the asset (net of payouts such as a continuous dividend yield) matches the risk-free rate, and discounting is done at that same risk-free rate.

A common exam pattern is: compute an expected payoff at maturity, then discount using $e^{-rT}$.

### Bond pricing with continuously compounded yields

If a bond’s yield is expressed with continuous compounding and its cash flows $c_t$ arrive at times $t$, then the price is:

$$P = \sum_{t} c_t e^{-yt}$$

This form is compact and leads directly to a clean expression for yield-based duration:

$$D = \frac{1}{P}\sum_t t\,c_t e^{-yt}$$

Even if the market convention for quoting a bond yield is semiannual compounding, continuous compounding often appears in analytics (and in many term-structure and derivatives models). The exam expects you to be comfortable moving between conventions so that discounting is consistent.

### Discount factors, curve construction, and arbitrage consistency

Discount factors are also the building blocks for curve construction. For example, if you know the price of a risk-free zero-coupon instrument paying a single cash flow at maturity, the discount factor is simply:

$$d(T)=\frac{\text{Price}}{\text{Cash flow at }T}$$

For coupon instruments, the “law of one price” implies that the bond price must equal the sum of its cash flows discounted at maturity-specific discount factors. This is why, as you move out along the maturity spectrum, you can solve sequentially for discount factors using earlier ones (bootstrapping). In practice, market participants often obtain discount factors from Treasury STRIPS, OIS curves, or swap curves, depending on the collateral and credit assumptions relevant to the valuation.

### **Exam Warning**

> The exam frequently tests convention consistency. Common errors include:
>
> - Using $d(t)=e^{-rt}$ with a rate that is not continuously compounded (or using $(1+r/m)^{-mt}$ with a continuously compounded rate).
> - Forgetting that $t$ must be in years (or year fractions) that match the quoted convention.
> - Forgetting to multiply by time in the exponent (writing $e^{-r}$ instead of $e^{-rt}$).
> - Converting rates incorrectly by dividing or multiplying by $m$ without using $\ln(\cdot)$ or $e^{(\cdot)}$.
> - Mixing spot rates and forward rates in a valuation without checking the implied no-arbitrage relationship.

## **Summary**

Continuous compounding treats interest accrual as occurring at every instant, producing exponential accumulation and discounting: $FV=PV\,e^{rt}$ and $PV=FV\,e^{-rt}$. This convention is widely used in derivatives pricing and appears throughout FRM valuation models, including binomial option pricing and risk-neutral discounting.

Discount factors $d(t)$ are the central objects of discounting: they convert each future cash flow into a present value, and a full discount function is sufficient to price any deterministic set of risk-free cash flows. Spot rates and discount factors contain the same information; the difference is whether the time value of money is expressed as a rate or as a present-value multiplier.

Finally, FRM candidates must be able to translate between compounding conventions. Equivalent-rate formulas connect discrete compounding, continuous compounding, and discount factors, and they are essential for avoiding valuation and hedging errors caused by inconsistent rate inputs.

## **Key Point Checklist**

_This article has covered the following key knowledge points:_

- Continuous compounding present and future value formulas: $FV=PV\,e^{rt}$ and $PV=FV\,e^{-rt}$
- Discount factor definition and cash flow valuation: $PV=\sum_i CF(t_i)d(t_i)$
- Discount factors from spot rates:
  - Semiannual: $d(t)=\left(1+\frac{r(t)}{2}\right)^{-2t}$
  - Continuous: $d(t)=e^{-r(t)t}$
- Rate conversions:
  - Between discrete frequencies: $R_2=\left[\left(1+\frac{R_1}{m_1}\right)^{m_1/m_2}-1\right]m_2$
  - Discrete to continuous: $R_c=m\ln\left(1+\frac{R_m}{m}\right)$
  - Continuous to discrete: $R_m=m\left(e^{R_c/m}-1\right)$
- Continuous forward rate relationship: $F(T_1,T_2)=\frac{R_2T_2-R_1T_1}{T_2-T_1}$
- Where continuous discounting appears in FRM valuation models (risk-neutral option pricing, binomial trees, continuously compounded yield analytics)

## **Key Terms and Concepts**

- Time value of money
- Compounding frequency
- Continuously compounded rate
- Equivalent rates
- Effective annual rate (effective return)
- Discount factor
- Discount function (discount curve)
- Spot rate (zero-coupon rate)
- Forward rate
- Risk-neutral discounting
- Law of one price (no-arbitrage pricing)
- Bootstrapping (sequential curve construction)


Time value of money and discounting - Continuous compounding and discount factors