## **Learning Outcomes**

This article explains how discounting links cash-flow timing to value, and how NPV, IRR, annuities, and perpetuities are applied in valuation and risk measurement. After reading, you will be able to:

- Compute present and future values using discount factors under different compounding conventions
- Calculate and interpret NPV and IRR, including common situations where IRR can mislead
- Value level-payment annuities and perpetuities and connect these building blocks to bonds and derivatives pricing

## **GARP FRM Part 1 Syllabus**

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

- Define discount factors and use a discount function to compute present and future values
- Apply compounding conventions and interpret the impact of compounding frequency on value
- Use discounting to value streams of cash flows (including bond-like annuities and perpetuities), and connect valuation to arbitrage-free pricing logic

## **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. A 2-year spot rate is 3% with annual compounding. What is the 2-year discount factor  
   a) 0.9709  
   b) 0.9426  
   c) 0.9151  
   d) 0.8850

2. A project costs \$10,000,000 today and pays \$3,000,000 at the end of each year for 4 years. If the discount rate is 8% (annual compounding), what is the NPV (approximately)?

3. In what type of cash-flow pattern can IRR be non-unique (more than one IRR)? State the condition in words.

4. True or False: The present value of an annuity due equals the present value of the corresponding ordinary annuity multiplied by $(1+r)$. Briefly explain.

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

- Can you convert a quoted nominal rate with $m$ compounding periods into an effective annual rate?
- When discounting a multi-year cash-flow stream, do you instinctively match the discount rate’s compounding convention to the cash-flow timing?
- Do you know when to prefer NPV over IRR for ranking projects or trades?

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

## **Introduction**

Most valuation problems in financial risk management reduce to a single idea: cash flows that arrive at different times are not directly comparable until they are converted to a common point in time (usually “today”). This conversion is done using discount factors or interest rates with clearly stated compounding conventions.

For FRM candidates, time value of money is not only “corporate finance math.” It sits inside core market models. Bond prices are the present value of coupons and principal discounted at appropriate rates. Derivatives valuation under risk-neutral pricing discounts an expected payoff at the risk-free rate (often using $e^{-rT}$ in continuous time). Risk measures such as DV01 and duration are sensitivities of these present values to yield changes.

> **Key Term: Time value of money**  
> The principle that a unit of currency received sooner is worth more than the same unit received later because it can earn a return (or avoid funding cost) over time.

## **Discounting and discount factors**

### Discount factors as the “price today” of future money

A **discount factor** $d(t)$ converts a cash flow at time $t$ into today’s value (time 0). If a cash flow of $CF_t$ is received at time $t$, its present value is:

$$PV(CF_t)=CF_t \times d(t)$$

For a stream of cash flows, present value is the sum of discounted cash flows:

$$PV=\sum_{t} CF_t \, d(t)$$

> **Key Term: Discount factor**  
> $d(t)$ is the present value of 1 unit of currency received at time $t$. It is a function of the interest-rate convention and the term structure used for discounting.

In an arbitrage-free setting (the “law of one price”), two instruments with identical future cash flows must have the same price. Discount factors are the tool that makes this precise: once $d(t)$ is known for all relevant maturities, any risk-free cash-flow stream has a unique price.

### Interest rates, compounding, and discount factors

To apply discounting correctly, you must match the quoted interest rate’s compounding convention to the discounting formula.

**Annual compounding:** with annual rate $r$ and time $t$ in years,

$$d(t)=\frac{1}{(1+r)^t}$$

**$m$-times-per-year compounding:** with nominal annual rate $R$ compounded $m$ times per year,

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

**Continuous compounding:** with continuously compounded rate $r_c$,

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

Continuous compounding is common in derivatives models. For example, risk-neutral option valuation discounts expected payoffs as $e^{-rT}$.

#### Converting between compounding conventions

If $R_1$ is quoted with compounding $m_1$ times per year and you want the equivalent nominal rate $R_2$ with compounding $m_2$ times per year, equate the one-year growth factors:

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

A useful intermediate is the **effective annual rate (EAR)**:

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

Then discounting with annual compounding can use $EAR$.

> **Key Term: Effective annual rate (EAR)**  
> The annualized growth rate that reflects intra-year compounding:
>
> $$
> EAR=\left(1+\frac{R}{m}\right)^m-1
> $$

### Worked Example 1.1

Compute the present value of \$100 received in 5 years when the interest rate is 8% per year compounded quarterly.

> **Answer:**  
> The quarterly rate is $0.08/4=0.02$ and there are $5 \times 4=20$ quarters.
>
> $$PV=\frac{100}{(1.02)^{20}} \approx 67.30$$
>
> The present value is approximately \$67.30.

### Term structure discounting (multiple discount rates)

In practice, especially for fixed income, discounting often uses a **term structure**: different maturities have different rates and therefore different discount factors. If spot rates $s(t)$ are given for each maturity, you compute $d(t)$ maturity-by-maturity and value cash flows by matching each cash flow to its maturity-specific discount factor.

This is consistent with Treasury pricing logic: if you know discount factors for all maturities, you can value any Treasury cash-flow stream; conversely, observing prices of a set of Treasury instruments can allow you to imply discount factors.

## **Net present value in valuation and risk**

### Definition and decision rule

**Net present value (NPV)** is present value of benefits minus present value of costs, all measured at time 0 using an appropriate discount rate or curve:

$$NPV=\sum_{t=0}^{T} CF_t \, d(t)$$

Often $CF_0$ is the initial investment and is negative, so:

$$NPV=-C_0+\sum_{t=1}^{T}\frac{CF_t}{(1+r)^t}$$

> **Key Term: Net present value (NPV)**  
> The value added today by entering a set of cash flows, defined as the sum of discounted cash flows (including the initial cost). A positive NPV means value is created relative to the discount rate used.

Decision rule (single independent project):

- Accept if $NPV>0$
- Reject if $NPV<0$

Ranking rule (mutually exclusive choices):

- Prefer the highest NPV at the relevant discount curve, subject to constraints (capital, liquidity, risk limits)

### Choosing the discount rate is a risk decision

For Treasury cash flows, discounting uses risk-free discount factors. For risky cash flows (corporate projects, risky bonds, loans), discounting usually adds compensation for risk, often expressed as a **spread** over the risk-free term structure. Conceptually:

- Build a base curve (risk-free discount factors)
- Add a risk premium (credit, liquidity, optionality, funding) to obtain a risky discount curve
- Discount expected cash flows using the risky curve (or discount expected cash flows and model losses explicitly, depending on context)

A risk manager should recognize that NPV is highly sensitive to:

- interest-rate level and curve shape
- spreads (credit and liquidity conditions)
- cash-flow forecasting error

This is why scenario analysis and stress testing often include rate shocks and spread shocks to see how NPV changes under adverse conditions.

### Worked Example 2.1

A project costs \$10,000,000 today and pays \$3,000,000 at the end of each year for 4 years. The discount rate is 8% with annual compounding. Compute NPV.

> **Answer:**  
> Use the present value of a 4-year ordinary annuity:
>
> $$PV = 3{,}000{,}000 \times \frac{1-(1.08)^{-4}}{0.08}$$
>
> Compute the annuity factor:
>
> $$(1.08)^{-4}\approx 0.7350$$
>
> $$\frac{1-0.7350}{0.08}=\frac{0.2650}{0.08}\approx 3.3125$$
>
> So:
>
> $$PV \approx 3{,}000{,}000 \times 3.3125 = 9{,}937{,}500$$
>
> $$NPV \approx -10{,}000{,}000 + 9{,}937{,}500 = -62{,}500$$
>
> The NPV is approximately \-\$62,500, so at 8% the project is slightly value reducing.

## **Internal rate of return and its pitfalls**

### Definition and interpretation

The **internal rate of return (IRR)** is the discount rate that sets NPV to zero:

$$0=-C_0+\sum_{t=1}^{T}\frac{CF_t}{(1+IRR)^t}$$

> **Key Term: Internal rate of return (IRR)**  
> The discount rate that makes the present value of inflows equal to the present value of outflows (NPV equals zero). It is a yield-like summary of a cash-flow stream.

Decision rule (single project):

- Accept if $IRR$ exceeds the hurdle rate (required return)

In fixed income, **yield to maturity (YTM)** is effectively an IRR: it is the single yield that equates the bond’s price to the discounted value of its promised cash flows under the yield’s compounding convention.

### Worked Example 3.1

A project costs \$100 today and pays \$60 at the end of Year 1 and \$60 at the end of Year 2. Compute IRR.

> **Answer:**  
> Solve:
>
> $$0=-100+\frac{60}{1+r}+\frac{60}{(1+r)^2}$$
>
> Let $x=1+r$. Multiply both sides by $x^2$:
>
> $$0=-100x^2+60x+60$$
>
> Rearrange:
>
> $$100x^2-60x-60=0$$
>
> Divide by 20:
>
> $$5x^2-3x-3=0$$
>
> Quadratic formula:
>
> $$x=\frac{3\pm\sqrt{9+60}}{10}=\frac{3\pm\sqrt{69}}{10}$$
>
> Take the positive root:
>
> $$x\approx \frac{3+8.306}{10}=1.1306$$
>
> Therefore:
>
> $$r=x-1\approx 0.1306=13.06\%$$
>
> The IRR is approximately 13.1% per year.

### Why IRR can mislead

IRR is popular because it produces one number, but risk managers should know when it fails.

1. **Non-unique IRR (multiple IRRs)**  
   If cash flows change sign more than once (for example: negative, then positive, then negative), the NPV equation can have multiple roots, meaning multiple IRRs or no real IRR.

2. **Reinvestment assumption**  
   Interpreting IRR as a “return” implicitly assumes interim cash flows are reinvested at the IRR. NPV analysis assumes reinvestment at the discount rate (often closer to a market opportunity cost). For very high IRRs, the reinvestment assumption can be unrealistic.

3. **Scale and timing problems (ranking conflicts)**  
   A smaller project can have a higher IRR but a lower NPV than a larger project. For mutually exclusive choices, NPV at the correct discount curve is usually the correct ranking tool because it measures value added in currency units.

4. **Term structure matters**  
   IRR collapses a term structure into one number. If discount rates vary by maturity (common in practice), NPV using maturity-matched discount factors is the cleaner approach.

### **Exam Warning**

> Common exam errors include (1) using IRR to rank mutually exclusive projects when NPV ranking differs, (2) forgetting that multiple sign changes can create multiple IRRs, and (3) mixing compounding conventions (for example, using a nominal semiannual yield as if it were an annual effective rate).

## **Annuities as building blocks for fixed income**

### Ordinary annuity and annuity due

An **ordinary annuity** pays a fixed amount $C$ at the **end** of each period for $N$ periods at a per-period rate $r$.

Present value:

$$PV_{\text{annuity}}=C\frac{1-(1+r)^{-N}}{r}$$

Future value:

$$FV_{\text{annuity}}=C\frac{(1+r)^N-1}{r}$$

An **annuity due** pays at the **beginning** of each period. It is worth one extra period of growth (or, equivalently, is discounted one period less):

$$PV_{\text{annuity due}}=PV_{\text{ordinary}}\times(1+r)$$

> **Key Term: Annuity**  
> A level-payment stream for a finite number of periods. Ordinary annuities pay at period-end; annuities due pay at period-beginning.

### Loan amortization as an annuity problem

If a loan balance today is $PV$ and the borrower makes level payments $PMT$ for $N$ periods at rate $r$ per period, then:

$$PV=PMT\frac{1-(1+r)^{-N}}{r}$$

Solve for the payment:

$$PMT=PV\cdot \frac{r}{1-(1+r)^{-N}}$$

### Worked Example 4.1

A borrower takes a \$250,000 mortgage for 30 years at 6% nominal interest compounded monthly. What is the monthly payment?

> **Answer:**  
> Monthly rate $r=0.06/12=0.005$. Number of payments $N=30\times 12=360$.
>
> $$PMT=250{,}000\cdot\frac{0.005}{1-(1.005)^{-360}}$$
>
> Approximate $(1.005)^{-360}\approx 0.166$.
>
> $$
> PMT\approx 250{,}000\cdot\frac{0.005}{1-0.166}
> =250{,}000\cdot\frac{0.005}{0.834}
> $$
>
> $$PMT\approx \frac{1{,}250}{0.834}\approx 1{,}498.8$$
>
> The monthly payment is approximately \$1,498.80.

### Where annuities appear in FRM valuation

- **Coupon bonds:** coupons form an annuity-like stream; principal is a final lump sum. A bond price is the sum of discounted coupons plus discounted principal.
- **Interest rate swaps:** the fixed leg is an annuity-like stream of fixed coupons; the floating leg resets and is valued differently, but discounting still uses term-structure discount factors.
- **Risk measures:** DV01 and duration are computed from discounted cash flows; the annuity structure affects how price responds to yield changes.

## **Perpetuities and growing cash flows**

### Level perpetuity

A **perpetuity** pays a constant amount $C$ every period forever, starting at the end of the first period. With per-period discount rate $r$:

$$PV_{\text{perpetuity}}=\frac{C}{r}$$

> **Key Term: Perpetuity**  
> An infinite level-payment stream. Its value is finite when the discount rate is positive.

### Growing perpetuity

If cash flows grow at a constant rate $g$ per period, with first cash flow $C_1$ at the end of period 1, and $r>g$:

$$PV_{\text{growing perpetuity}}=\frac{C_1}{r-g}$$

This formula is widely used in equity valuation (dividend discount models) and can also approximate long-horizon liabilities when growth and discounting assumptions are stable.

### Worked Example 5.1

1. A preferred share pays a constant \$5 per year forever. Required return is 7% per year. Value the preferred share.
2. A stock is expected to pay a dividend of \$2 next year, growing at 3% per year forever. Required return is 8%. Value the stock using the growing perpetuity model.

> **Answer:**
>
> 1. Level perpetuity:
>
> $$PV=\frac{5}{0.07}\approx 71.43$$
>
> Value is approximately \$71.43.
>
> 1. Growing perpetuity:
>
> $$PV=\frac{2}{0.08-0.03}=\frac{2}{0.05}=40$$
>
> Value is \$40.

## **Connecting discounting to bond pricing and arbitrage**

### Discount factors and the law of one price

Bond valuation follows a simple three-step discipline:

1. Estimate cash flows (coupons and principal).
2. Determine the appropriate discount rates (risk-free for Treasuries; risk-free plus risk premium for risky issuers).
3. Multiply each cash flow by its discount factor and sum.

If two portfolios have identical cash flows at each maturity date, they must have the same price; otherwise, an arbitrage trade is available (buy the cheap cash-flow stream, sell the rich one).

### Clean and dirty prices as “timing correction”

Bonds typically trade between coupon dates. The full value including interest earned since the last coupon is the **dirty price**; the quoted price excluding accrued interest is the **clean price**:

$$\text{Dirty price}=\text{Clean price}+\text{Accrued interest}$$

Accrued interest depends on the day-count convention (for example, actual/actual or 30/360). On the exam, errors often come from ignoring the fact that the buyer compensates the seller for coupon interest that has accrued.

### Discounting in derivatives models

In risk-neutral valuation, the present value of a derivative payoff $X_T$ at time $T$ is often written as:

$$V_0 = e^{-rT}\, \mathbb{E}^{\mathbb{Q}}[X_T]$$

Even if the exam question is “about options,” discounting is still time value of money: compute an expected payoff and discount it at the risk-free rate under the model’s convention.

## **Summary**

Time value of money is the shared language across bonds, derivatives, and project valuation: cash flows are valued by discounting them to a common time using a discount factor function. Getting compounding conventions correct is not cosmetic; it changes discount factors, prices, and risk measures.

NPV and IRR are two ways to summarize the value of a cash-flow stream. NPV measures value added in currency units and aligns naturally with term-structure discounting; it is generally the safer ranking tool when projects are mutually exclusive or when discount rates vary across maturities. IRR is a yield-like summary that can be useful, but it can fail when cash flows are non-conventional or when ranking projects of different scale.

Annuities and perpetuities are valuation building blocks that appear in loan payments, fixed legs of swaps, coupon streams in bonds, preferred shares, and growing dividend models. For FRM Part 1, speed and accuracy often come from recognizing these structures and applying the correct formula with the correct timing and compounding.

## **Key Point Checklist**

_This article has covered the following key knowledge points:_

- Discount factors $d(t)$ convert future cash flows into present values: $PV=\sum CF_t d(t)$
- Compounding conventions matter; discounting may use annual, $m$-period, or continuous compounding
- NPV is the sum of discounted cash flows; accept if $NPV>0$ at the appropriate discount rate/curve
- IRR solves $NPV=0$ but can be non-unique and can conflict with NPV rankings
- Ordinary annuity and annuity due present values differ by a factor of $(1+r)$
- Perpetuities value level infinite streams as $C/r$, and growing perpetuities as $C_1/(r-g)$ when $r>g$
- Bond and derivatives valuation apply the same discounting logic, often justified by the law of one price

## **Key Terms and Concepts**

- Time value of money
- Present value (PV) and future value (FV)
- Discount factor and discount function
- Nominal rate, effective annual rate (EAR), compounding frequency
- Net present value (NPV)
- Internal rate of return (IRR)
- Ordinary annuity and annuity due
- Perpetuity and growing perpetuity
- Law of one price (arbitrage-free pricing idea)
- Clean price, dirty price, accrued interest


Time value of money and discounting - NPV, IRR, annuities, and perpetuities