Foundations

Net present value & basis point value

Learn how net present value (NPV) turns future cash flows into one number. The basis point value, or BPV, is then introduced as the most basic measure of risk when dealing with rate-sensitive instruments such as a bond, a loan or a swap.

1. Introduction

A contract has cash flows on many dates. The net present value (NPV) adds them up as one amount in euros today. Each cash flow is first discounted to today. The mark-to-market (MtM) is the NPV of a contract that already exists.

The basis point value (BPV) measures the risk of a rate change. A basis point (bp) is 0.01%. The BPV is the value today of one basis point of interest, paid every year until the contract ends. Multiply the BPV by a margin, an extra rate quoted in basis points, and you get an amount in euros.

By the end you will be able to:

Read a mark-to-market as a net present value and say where it comes from.
Show a bank’s fee on a derivative as a margin and as a premium paid today.
Say which discount curve to use, with and without credit risk.
Value a fixed rate or a margin on a leg with one multiplication: rate times annuity.
Explain the basis point value (BPV), also called PV01, in two ways.
Turn a margin in basis points into euros, and back.

The chapter builds on Time value of money & discount factors and on Instrument definitions & market conventions. Every example uses one interest leg, the series of interest payments of a contract. The leg can pay a fixed or a floating rate. It has a notional of €10 million, the amount on which interest is calculated. It runs for five years with annual interest payments. The discount curve is flat at 3.00%, so every maturity has the same rate. The numbers are illustrative.

2. Net present value: the method behind mark-to-market

2.1 From present value to net present value

In the chapter Time value of money & discount factors, the value of any instrument was the sum of its discounted cash flows. A contract usually has cash flows in both directions. So we give each cash flow a sign, positive when you receive and negative when you pay. We discount them and add them up. The two directions net into a single number, the net present value (NPV).

$$\text{NPV} = \sum_{i=1}^{n} C_i \times \text{DF}(t_i)$$

$C_i$ is cash flow number $i$, $t_i$ is its date in years and $\text{DF}(t_i)$ is the discount factor for that date. The sign $\Sigma$ means: add up the terms for $i = 1$ to $n$.

It is the same formula as before. The word net reminds you that $C_i$ can be positive or negative. A cash flow today has a discount factor of 1, so it counts in full.

2.2 NPV, mark-to-market, fair value and close-out amount

You will meet several names for closely related ideas. NPV is the method. The other names describe where and how it is applied.

Term	Where you meet it	What it means
NPV	Pricing, risk reports	The sum of the discounted, signed cash flows
Mark-to-market (MtM)	Bank statements, valuation reports	The NPV of an existing contract at today’s mid market rates
Fair value	Financial statements	The value used in financial statements. It is the NPV, plus any adjustments that accounting rules require, for example for credit risk
Close-out amount	Cancelling a contract early	What the bank says you owe, or will pay you, to end the contract today. It is the bank’s own calculation. It can include the bank’s costs of closing its hedges, so it can differ from the MtM

2.3 Sign and perspective

A positive NPV means the contract has a positive value for you, so in principle you would be paid to close it. A negative NPV means you would have to pay. In principle, the other party sees the same number with the opposite sign.

A party that pays a fixed rate has a positive NPV when that rate is below the market rate. It has a negative NPV when the rate is above it. The party that receives the rate sees the opposite.

2.4 The market rate and the starting NPV

Take a contract that pays a fixed rate on a notional. Its market rate is the fixed rate that gives an NPV of zero today, for the same notional profile, dates and conventions. The contract includes its other cash flows, such as the repayment of capital. In a loan, the amount you receive and the repayment of capital have the same value at market rates. Then only the fixed rate can move the NPV away from zero. The discount curve gives this rate. The chapter Yield curves calls it the par rate. At that rate, what you pay and what you receive have the same present value. As market rates move, the NPV of a fixed rate drifts away from zero. A loan with a floating rate and no margin resets every period, so its NPV stays near zero.

Now suppose the contract pays more than the market rate, and you are the one paying it. The difference is a margin that the bank has built into the rate. On a floating leg, the margin is the amount added to the floating rate. The contract then starts with a negative NPV for you. That negative amount on day one is the value of the margin. On a derivative, it is the bank’s fee in euros. The chapter How banks calculate fees on interest rate derivatives lists what that fee covers. The margin on a loan is built in a different way.

2.5 Worked example: paying 20bp above the market rate

A contract pays interest of 3.20% a year on a notional of €10 million for five years. The market rate for the same dates is 3.00%. You pay the interest.

Compared with the market rate, you pay €20,000 more interest every year, which is 0.20% of €10 million. The difference of 20 basis points (20bp) is invented to keep the arithmetic easy. It is not a market level. Each yearly extra payment is discounted back to today.

Year	Extra interest (€)	Discount factor	Present value (€)
1	−20,000	0.9709	−19,417
2	−20,000	0.9426	−18,852
3	−20,000	0.9151	−18,303
4	−20,000	0.8885	−17,770
5	−20,000	0.8626	−17,252
Total	−100,000		−91,594

Each discount factor is $\frac{1}{1.03^t}$, where $t$ is the year. This is annual compounding at 3.00%. The chapter Yield curves writes the discount factor with continuous compounding instead. The discount factors differ slightly between the two conventions. The factors are shown to four digits, but the present values use the exact factors.

Where an NPV comes from

A contract pays 3.20% when the market rate is 3.00%. That costs 20bp more than the market every year: €10,000,000 × 0.20% = €20,000. Discounted to today, the five yearly extra payments add up to the NPV of the contract.

Discounted extra interest (present value)
Undiscounted extra interest
NPV today

This example uses a €10 million notional, 5 years, annual interest and a flat 3.00% curve. Each yearly payment is discounted with its own discount factor.

Today the five extra payments are worth €91,594, even though they total €100,000. You pay them, so the NPV is −€91,594. It is the cost of the extra 20bp in today’s money.

Two ways to charge a derivative fee

On a derivative, the extra 20bp is the bank’s fee. The bank can charge it in two ways.

As a margin in the rate: you pay 3.20% instead of the market rate of 3.00%. The fee comes as extra interest, every year.

As a spot premium: you pay the market rate of 3.00% and a premium on the spot date, the usual start date. This chapter counts it as today. The premium is the present value of the margin, €91,594.

The two are the same fee in two forms. The BPV, introduced in section 4, converts between them.

2.6 Which curve to discount with

The examples in this chapter discount with one flat rate of 3.00%. In practice, each cash flow is discounted with a discount factor from a discount curve. The chapter Yield curves explains what that curve is. The upcoming chapter Curve construction & bootstrapping builds it step by step. This section explains which curve to use.

Situation	Discount curve
A euro mark-to-market with no probability of default on either side	The €STR curve
A valuation that also includes credit risk	The €STR curve plus the credit spreads of both parties, or separate value adjustments

The €STR is the euro short-term rate, the overnight rate of the euro area. The chapter Interest rate benchmarks describes it. The €STR curve is the standard proxy for a risk-free euro rate. The chapter Yield curves describes an OIS curve built from overnight index swaps referencing the €STR. The chapter Interest rate swaps explained says in its section 6 why it is the standard discount curve.

The €STR curve is the right curve when you ignore the chance that either side fails to pay. It is also the curve market practice uses for contracts backed by collateral. Collateral is an asset that one party pledges to cover what it owes. It removes most of the credit risk, and cash collateral earns the €STR.

Credit risk lowers the value of what the other party owes you. Your own credit risk lowers the value of what you owe. In a simple case, you add credit spreads to the curve. If a contract can change sign in value, banks calculate separate value adjustments instead. A credit spread is the extra rate that pays for the chance that a party fails to pay. Both parties carry that risk. If you owe money, your own credit spread matters. If the other party owes you, its credit spread matters. A full valuation takes both into account. Funding and capital costs add further adjustments. The upcoming chapter xVA: value adjustments covers this in detail.

The curve you choose also decides what a margin contains. With the €STR curve, a margin covers everything the bank charges above the risk-free rate, including credit risk. With a curve that already includes your credit spread, the margin is left with the bank’s costs and profit.

3. Valuing a constant rate on an interest leg

3.1 Interest leg and annuity

An annuity is a series of payments of the same kind made at equal time intervals, usually over a finite term. The interest payments of a contract fall at equal intervals, so they form such a series. Their amounts can differ. We call it an interest leg, or a leg for short.

Each payment is interest on a notional. The notional is the nominal amount on which interest is calculated. It takes the repayment of capital into account, so it falls when capital is repaid. We write $\text{Notional}_i$ for the notional in period $i$.

The year fraction $\text{yf}$ is the length of a period in years, measured with the day count convention. It is about 1 for an annual payment and about 0.25 for a quarterly payment. The chapter Instrument definitions & market conventions explains the day count conventions. The example leg uses 30/360, so every year fraction is exactly 1.

The interest paid in period $i$ is the notional times $\text{yf}$ times the rate. A fixed leg pays the same rate $K$ in every period. A floating leg pays a rate that is reset every period. The present value of a leg is the sum of its discounted payments:

$$\text{PV} = \sum_{i=1}^{n} \text{Notional}_i \times \text{yf}_i \times \text{rate}_i \times \text{DF}(t_i)$$

3.2 A constant rate: taking K out of the sum

To see what a constant rate $K$ does, write the definition out for a leg with three payments:

$$\begin{aligned} \text{PV} &= \text{Notional}_1 \times \text{yf}_1 \times K \times \text{DF}(t_1) \\ &\quad + \text{Notional}_2 \times \text{yf}_2 \times K \times \text{DF}(t_2) \\ &\quad + \text{Notional}_3 \times \text{yf}_3 \times K \times \text{DF}(t_3) \end{aligned}$$

$K$ appears in every term. A number that appears in every term of a sum can be taken out of the sum. This is the distributive rule. Example: $3 \times 2 + 3 \times 5 + 3 \times 4 = 3 \times (2 + 5 + 4)$. Both sides equal 33. The order of factors does not matter, so $K$ can go first. Applied to the leg, $K$ moves in front of a bracket:

$$\begin{aligned} \text{PV} &= K \times \big( \text{Notional}_1 \times \text{yf}_1 \times \text{DF}(t_1) \\ &\qquad\qquad + \text{Notional}_2 \times \text{yf}_2 \times \text{DF}(t_2) \\ &\qquad\qquad + \text{Notional}_3 \times \text{yf}_3 \times \text{DF}(t_3) \big) \end{aligned}$$

The same step works for any number of payments:

$$\begin{aligned} \text{PV} &= \sum_{i=1}^{n} \text{Notional}_i \times \text{yf}_i \times K \times \text{DF}(t_i) \\ &= K \times \sum_{i=1}^{n} \text{Notional}_i \times \text{yf}_i \times \text{DF}(t_i) \end{aligned}$$

The sum on the right does not contain $K$. It only depends on the notional, the year fractions and the discount factors. It is the present value of the leg if the rate were 100%. Rates markets call this number the annuity too, and write it $A$. Here it includes the notional, so it is an amount in euros. On a constant notional, dividing by the notional gives the annuity per unit of notional, 4.5797 here.

$$A = \sum_{i=1}^{n} \text{Notional}_i \times \text{yf}_i \times \text{DF}(t_i)$$

So the present value of a fixed rate $K$ on the leg is the rate times the annuity:

$$\text{PV} = K \times A$$

The annuity depends on the notional, the dates and the discount factors. It does not depend on the rate. One annuity values any constant rate on the same leg.

A leg can also pay a floating rate plus a margin $s$. Write $\text{float}_i$ for the floating rate of period $i$. Split each payment in two parts. The margin part has the same $s$ in every period, so $s$ comes out of its sum:

$$\begin{aligned} \text{PV} &= \sum_{i=1}^{n} \text{Notional}_i \times \text{yf}_i \times (\text{float}_i + s) \times \text{DF}(t_i) \\ &= \sum_{i=1}^{n} \text{Notional}_i \times \text{yf}_i \times \text{float}_i \times \text{DF}(t_i) \\ &\quad + s \times \sum_{i=1}^{n} \text{Notional}_i \times \text{yf}_i \times \text{DF}(t_i) \\ &= \sum_{i=1}^{n} \text{Notional}_i \times \text{yf}_i \times \text{float}_i \times \text{DF}(t_i) + s \times A \end{aligned}$$

3.3 Valuing the example leg at 3.00% and 3.20%

Add the five exact discount factors behind the table in section 2.5. The sum is 4.5797072. All five notionals are €10 million and all five year fractions are 1. So the annuity is €10 million × 4.5797072, which is €45,797,072. Now value the interest at different rates:

Interest rate	Calculation	Present value of the interest (€)
3.00% (market)	3.00% × 45,797,072	1,373,912
3.20% (the contract)	3.20% × 45,797,072	1,465,506
Difference	0.20% × 45,797,072	91,594

The difference is the cost we found in section 2.5. As an NPV it is −91,594, because you pay it. Paying 0.20% more on every notional costs 0.20% of the annuity.

Example: a loan at 3.00% and at 3.20%

Take a loan of €10 million over five years, with all the capital repaid at the end. You receive the €10 million today.

At 3.00%, the interest is worth €1,373,912. The repayment of capital is worth €10 million × 0.8626088, which is €8,626,088. Together they are worth exactly €10 million. The NPV is zero.

At 3.20%, the interest is worth €1,465,506. Together with the capital, your payments are worth €10,091,594 for the €10 million you received. The NPV is −€91,594.

Now take a floating loan that pays the floating rate plus 20bp. Take the floating rate from the discount curve. The floating interest and the capital are worth exactly €10 million. A loan linked to EURIBOR takes its floating rates from the EURIBOR curve, not the discount curve. Its value is then close to €10 million, not equal. The chapter Interest rate benchmarks describes EURIBOR. The 20bp adds €91,594, so the NPV is again −€91,594.

3.4 Bullet and amortising notionals

The notional of a leg can stay the same until the end. This is a bullet profile. Or it can fall by an equal amount every year. This is an amortising profile. Bullet and amortising are two repayment profiles.

A third profile has level total payments, interest plus capital. We do not use it here.

4. Basis point value (BPV)

4.1 First reading: one basis point a year

A basis point (bp) is one hundredth of a percentage point. It is 0.01%, or 0.0001 as a decimal. On a notional of €10 million, one basis point is €1,000 a year.

The basis point value (BPV) is the value today of one basis point a year on the notional of a leg. It is the value of a fixed leg with a rate of one basis point. By section 3.2, a fixed rate $K$ is worth $K$ times the annuity. With $K = 0.0001$, the BPV is:

$$\text{BPV} = A \times 0.0001$$

For the example leg, $45{,}797{,}072 \times 0.0001 \approx$ €4,580. The exact value is €4,579.71, and we round it in the text. One basis point a year, for five years, on €10 million is worth about €4,580 today.

The figure below shows how it builds up year by year. You can change the notional, the maturity, the rate and the repayment profile.

How a BPV builds up, year by year

The BPV is the value today of one basis point a year on the notional. Each bar shows what one basis point of interest paid in that year is worth today. The bars add up to the BPV.

Notional at startNotional€10m

Maturity5 years

Flat market rateRate3.00%

Repayment profile

Bullet bars, for comparisonBullet bars

BPV, per bp
Annuity
Share of the bullet BPV

Each bar = notional × yf × DF × 0.0001. BPV = annuity × 0.0001.

A higher rate means heavier discounting, so every bar gets shorter and the BPV falls. Repayments of capital are not part of the BPV, but the repayment profile sets the notional each year.

In the figure, a higher rate means heavier discounting. A payment one year from now is then worth less today. Every discount factor falls, every bar is shorter, and the BPV is lower. The later years lose the most in proportion, because they are discounted for longer. Section 4.4 gives the sizes.

4.2 Second reading: a floating leg and one basis point

There is a second way to read the number. It uses a floating leg. A floating leg pays the market rate on the notional, and the rate is reset every period. The chapter Yield curves explains where the market rate of each future period comes from. The value of the floating leg is again the sum of its discounted payments. The chapter Interest rate swaps explained shows floating legs at work.

Now move the market rate of every period by one basis point. Keep the discount factors where they are. Each payment grows by $\text{Notional}_i \times \text{yf}_i \times 0.0001$. The value of the floating leg changes by:

$$\Delta\text{PV} = \sum_{i=1}^{n} \text{Notional}_i \times \text{yf}_i \times 0.0001 \times \text{DF}(t_i) = A \times 0.0001 = \text{BPV}$$

Moving the market rate by one basis point adds one basis point a year to every payment of the floating leg. That is reading 1 again. The two readings give the same number for the same dates, notional and day count. This holds for rates not yet fixed. A cap or a floor breaks it, and so does compounding in arrears. The upcoming chapter Introduction to options explains caps and floors. The chapter Interest rate benchmarks explains compounding in arrears.

Check it on the example leg. Every payment of the floating leg rises by 0.01% of €10 million, which is €1,000. The five extra payments, discounted with the same factors, are worth €4,579.71. This is the BPV.

Two readings of the BPV

Reading 1: the value of a fixed leg with a rate of one basis point.

Reading 2: the change in value of a floating leg when market rates move one basis point, discount factors unchanged.

Both add one basis point a year to every payment of the leg. So they give the same number when dates, notional and day count match and the rates are not yet fixed.

4.3 BPV, PV01, DV01 and PVBP

This chapter leads with BPV, the basis point value. PV01, DV01 and PVBP are other names you will meet. Reports do not always use them in the same way, so check the unit, the sign and the instrument on any report. In this chapter the BPV is a positive amount. The sign belongs to the position. Some reports use a larger move, such as ten basis points.

Some reports also move the discount factors together with the market rates. Others keep them constant, as in the second reading above. On the example leg, moving the whole curve changes the interest of the floating leg by €4,186 instead of €4,580. The repayment of capital loses the same €4,186, so a floating loan with its capital does not move on this flat curve. A fixed leg at 3.00% moves by only about €392 when the whole curve shifts by one basis point. Only its discount factors change. A figure defined as a curve shift is therefore not this chapter’s BPV, whatever name a report gives it.

Section 11 of the chapter Bonds & basic bond pricing also uses PV01. There it is the change in a bond’s price for one basis point of yield. A bond’s price includes its final repayment of capital, while the BPV in this chapter counts interest only. That chapter finds about €4,570 for a five-year bond of €10 million with a 3.00% coupon and a 3.04% yield. This is close to the €4,580 found here. The two are almost equal for a bond priced at par. A 6% coupon at the same yield gives about €4,960.

4.4 What changes the BPV

If this changes	The BPV	Why
Notional doubles	Doubles	Twice the notional carries twice the interest
Maturity gets longer	Rises	There are more years of interest to discount
Rates rise	Falls	Every future payment is discounted more heavily. Around a 3% rate, one extra percentage point cuts the BPV by about 3% on 5 years and 12% on 30 years
The leg amortises	Falls	The notional is smaller in the later years
Two legs are combined	Adds up	The BPV is a sum of discounted payments, so BPVs add up on one curve

5. Turning a margin into euros

5.1 A margin or a spot premium

On a derivative, a bank bases its fee on the risk behind the instrument. It often quotes the fee as a margin in basis points. A margin is built into the rate of the leg. On a fixed leg the rate that the client pays is the market rate plus the margin. On a floating leg it is the floating rate plus the margin. Either way, the client pays the margin in every period as extra interest on the notional. The bank can also charge the same fee as a spot premium. The client then pays the market rate and a premium on the spot date.

A margin is a fixed rate paid on the notional. By section 3.2, it is worth the margin times the annuity. This holds on a fixed leg and on a floating leg alike. It needs the margin to accrue on the same notional and year fractions. So the BPV converts between the two:

$$\text{Spot premium} = \text{Margin in bp} \times \text{BPV} \qquad\qquad \text{Margin in bp} = \frac{\text{Spot premium}}{\text{BPV}}$$

Take an invented margin of 5bp on the example leg. Its spot premium is 5 × €4,579.71, which is about €22,899. This is the present value of the margin. It is a value today, and it is how a margin becomes an amount you can compare and negotiate.

Use the BPV of the leg that carries the margin. It depends on the curve, the notional profile, the payment dates and the day count. Two legs of one contract can have different BPVs. The conversion assumes the contract runs to maturity. The chapter How banks calculate fees on interest rate derivatives shows how a bank builds its fee from three components.

5.2 Effect of the repayment profile on the BPV

To turn a margin into euros, you take the BPV of the leg. The BPV counts the interest only. Repayments of capital are left out. But the repayment profile still matters, because interest is charged on the notional of each period.

The figure below compares two versions of the same €10 million, five-year leg. In the first, the notional stays at €10 million until the end (bullet). In the second, it falls by €2 million every year (amortising).

The amortising leg has a smaller notional in every year after the first, so one basis point on it is worth less. Its BPV is about €2,802 against €4,580, which is 61% of the bullet figure.

Same leg, two repayment profiles, two BPVs

BPV is measured on the interest only. Repayments of capital are not part of it. But the repayment profile sets the notional that interest is charged on. So it changes the BPV.

Bullet (capital repaid at the end) BPV €4,580
Amortising (equal capital) BPV €2,802 (61% of bullet)

Notional during the year (€m)

Value today of 1bp on that notional (€)

This example uses a €10 million notional, 5 years, annual interest and a flat 3.00% curve.

6. The limits of BPV

BPV assumes a parallel shift, which means every rate on the curve moves by the same amount. Real curves rarely move this way.

BPV also assumes a small change in rates. It measures the effect of 1bp, not of a big move. A move of 1%, which is 100bp, equals four consecutive hikes of 25bp. For a move that large, the value no longer follows the rate in a straight line. A BPV multiplied by 100 is then only an estimate of the change in value when the market rate moves 100bp. A margin of 100bp does not move the curve. So 100 times the BPV is still the exact value of a 100bp margin.

This is especially a problem for instruments that contain options. Their sensitivity can change sharply when rates move, so the BPV measured today can be far off. The upcoming chapter Introduction to options covers them.

7. Key takeaways

The net present value (NPV) is the sum of the discounted, signed cash flows of a contract. The mark-to-market (MtM) is the NPV of an existing contract at today’s market rates.
Positive means the contract has a positive value for you, and negative means it costs you. A contract at the market rate starts at zero. A rate worse than the market rate starts negative. That negative amount is the value of the margin. On a derivative, it is the bank’s fee in euros.
On a derivative, the bank can charge its fee in two ways. A margin is built into the rate, and the client pays it in every period. A spot premium is the NPV of that margin, paid today.
Use the €STR curve for a euro mark-to-market without default risk. Add the credit spreads of both parties when credit risk is included.
The notional is the nominal amount on which interest is calculated, after any repayment of capital. The annuity is the present value of a leg’s payments at a rate of 100%. A fixed rate, or a margin on a fixed or floating leg, is worth the rate times the annuity.
The basis point value (BPV) is the annuity times 0.0001. It is also called PV01, DV01 or PVBP, although reports do not always use these names in the same way.
BPV reads two ways. It is the value of a fixed leg with a rate of one basis point. It is also how much a floating leg gains when market rates rise one basis point, with the discount factors fixed. The two are the same number when dates, notional and day count match and the rates are not yet fixed.
BPV grows with notional and maturity. It falls with amortisation. It also falls with higher rates, because heavier discounting makes later payments worth less. It adds up across legs on the same curve.
Spot premium = margin in bp × BPV, and the reverse also holds. It is a value today, and it assumes one discount curve and an instrument that runs to maturity.
To turn a margin into euros, use the BPV of the leg, without the repayments of capital. The repayment profile still sets the notional each year.
The margins in this chapter are invented. They are not market levels.
BPV assumes a parallel shift and a small change in rates. It is least reliable for a big move, and for instruments that contain options.