Optionality in loans · RiskAtoms Academy

By the end you will be able to

1Measure prepayment with the CPR and SMM, and write the surviving balance as a power of (1 − CPR).
2Apply the Basel 0.8 and 1.2 scenario multipliers and show what faster or slower prepayment does to EVE and to the hedge.
3Explain how early repayment charges work and compare one with the bank's loss when a borrower refinances.
4Recognise pipeline risk and caps as options the bank has sold, and size them on the book.

Why it matters

A UK homeowner with a five-year fix can usually repay early, sell the house, or remortgage, paying at most a few per cent in charges. A bank that has sold millions of such loans has sold millions of small options. They share one feature: borrowers use them when it suits them, which is when it hurts the bank. When rates fall, they refinance and the bank loses an above-market loan it had hedged. When rates rise, they stay, and the bank keeps a below-market loan longer than it planned. On RiskAtoms Bank's book, this behaviour costs EVE £21.4m in parallel up and £23.9m in parallel down.

Core explanation

The options borrowers hold

Illustrative dataRiskAtoms Bank plc's banking book

The same £67bn balance sheet as Module 3.5, on one made-up reporting date, in pounds, with how each line reprices (fixed, floating or administered), its customer rate, a SONIA curve with Bank Rate at 4.00%, a £19.2bn macro hedge on the fixed-rate mortgages and a £16bn structural hedge. Tier 1 capital is taken as £3.5bn. The figures are round illustrative values, not any bank's data. Every lesson in Module 3.6 uses this book, so the figures agree.

The book models prepayment of its £24bn of fixed-rate mortgages at a CPR of 8% a year, scaled by the standard's multipliers in each shock. Early repayment charges, the mortgage pipeline and the cap are extras added for this lesson: they leave the book's totals unchanged. Loans are treated as interest-only to the end of their fix.

L3.6.02: Where interest rate risk in the banking book comes from called this option risk. The Basel standard splits it in two. Automatic options are in the contract and are exercised whenever they pay, such as a cap on a loan rate. Behavioural options depend on what customers do: they tend to repay fixed-rate loans when rates fall, but not all of them, and some repay for reasons that have nothing to do with rates, such as moving house.

Prepayment: CPR and SMM

Prepayment is measured by the conditional prepayment rate (CPR): the share of what is still outstanding that is repaid early in a year. "Conditional" means each year's rate applies to the survivors, so the balance left after $t$ years is a power, $(1-\text{CPR})^t$ . Mortgage data is monthly, so lenders also quote the single monthly mortality (SMM).

Scenario CPR	SMM	Left after 1y	Left after 2y	Left after 3y	Left after 4y	Left after 5y
Rates up (× 0.8): 6.4%	0.550%	93.6%	87.6%	82.0%	76.8%	71.8%
Base: 8.0%	0.692%	92.0%	84.6%	77.9%	71.6%	65.9%
Rates down (× 1.2): 9.6%	0.838%	90.4%	81.7%	73.9%	66.8%	60.4%

The Basel standardised framework does not model why borrowers prepay. It takes the bank's baseline CPR and scales it: × 0.8 in the three scenarios where rates mostly rise (parallel up, short up, steepener), × 1.2 where they fall (parallel down, short down, flattener). The book applies the same rule.

What prepayment does to the hedge and to EVE

The macro hedge pays fixed on 80% of the fixed mortgages' expected profile at an 8% CPR (L3.6.10: Hedging the banking book). The swaps do not shrink when borrowers prepay. After a year at 9.6%, £10.85bn of five-year fixes are left instead of £11.04bn, against £8.83bn of swaps:

Prepayment	Hedge ratio after 1 year	After 3 years
Rates up (× 0.8)	78.6%	76.0%
Base	80.0%	80.0%
Rates down (× 1.2)	81.4%	84.3%

Faster prepayment leaves the bank over-hedged just as rates have fallen and its pay-fixed swaps are losing; slower prepayment leaves it under-hedged as rates rise. In EVE the effect shows up in both directions:

Fixed-rate mortgages, ΔPV	CPR held at 8%	Scenario CPR	Prepayment effect
Parallel up (CPR 6.4%)	−£1,048.2m	−£1,069.6m	−£21.4m
Parallel down (CPR 9.6%)	+£1,126.9m	+£1,103.0m	−£23.9m

This is negative convexity, the mirror of the positive convexity of a plain bond in L1.4.09: Convexity: the borrowers hold the option, so the bank loses on the way down and on the way up. On the whole book the multipliers add −£21.4m to ΔEVE in parallel up and −£23.9m in parallel down (L3.6.05: Economic value of equity sensitivity).

Early repayment charges

UK lenders charge an early repayment charge (ERC) for repaying a fixed-rate deal early: Lloyds Bank describes it as usually between 1% and 5% of what is still owed, and Moneyfacts notes that on many deals the charge steps down over the deal. Most deals let borrowers overpay up to about 10% of the balance a year without a charge. The book's illustrative schedule for five-year fixes is 5%, 4%, 3%, 2%, 1% in years 1 to 5. The charge compensates the bank for part of what it loses; whether a borrower refinances depends on whether the saving beats it.

One £250,000 loan fixed at 3.8%, rates down 275bp (new five-year fix 2.24%)
Years of fix left	ERC	Borrower's saving	Refinances?	Bank's loss less ERC
4	£10,000	£15,157	Yes	£13,114
3	£7,500	£11,451	Yes	£10,232
2	£5,000	£7,688	Yes	£7,167
1	£2,500	£3,870	Yes	£3,795

The pipeline

A mortgage offer fixes the rate weeks or months before the loan completes; a formal UK offer typically stays valid for three to six months. Until then, the borrower can walk away. That is pipeline risk. Suppose RiskAtoms Bank has £1.5bn of five-year offers at 4.99%, and treasury hedges the expected pull-through of 75% (£1.125bn) with pay-fixed swaps. If rates jump, the cheap offers become attractive and 90% complete; if rates fall, borrowers find better deals and only 55% do (all illustrative):

Parallel move	Completed	Loans ΔPV	Swaps ΔPV	Net
+275bp	£1.35bn	−£160.7m	+£133.4m	−£27.3m
−275bp	£0.825bn	+£112.1m	−£152.4m	−£40.3m

The bank loses both ways: the borrowers hold an option and the swap hedge is linear. An option-like hedge, such as a swaption, matches the shape better, at a premium.

Caps: an automatic option

The book's trackers have no cap. Suppose instead that £1bn of them promised the rate would never exceed 6% (a cap). At Bank Rate + 0.75% that binds once Bank Rate has risen 125bp. Below that it costs nothing; above it the bank gives up the excess on every capped pound:

Rise in Bank Rate	+100bp	+200bp	+275bp	+425bp
Year-1 NII given up	£0.00m	£6.88m	£13.75m	£27.50m

The +425bp column is the overnight end of the short up shock. A cap is automatic: it pays whenever rates are above it, with no customer decision involved, and its value depends on volatility as well as on the level of rates, which is why option pricing models (Level 6) are needed to put it into EVE.

Sort itAutomatic or behavioural?

Sort each option in a bank's book.

A cap at 6% on a tracker mortgage
A homeowner remortgaging when fixed rates fall
A mortgage offer the borrower may not take up
A 0% floor on a corporate loan's SONIA rate
A saver breaking a fixed-term deposit to move to a better rate

The formal version

Prepayment is a repeated percentage, so the surviving balance is a power (M-A03: Exponents).

The book's numbers. With the baseline CPR of 8% and its two scenario values:

\begin{aligned} S(5) &= (1 - 0.08)^5 = 0.6591 \\ \text{SMM} &= 1 - 0.92^{1/12} = 0.692\% \\ \text{CPR}_{\text{down}} &= \min(1,\ 1.2 \times 8\%) = 9.6\%, \quad S(5) = 0.904^5 = 0.6037 \\ \text{CPR}_{\text{up}} &= \min(1,\ 0.8 \times 8\%) = 6.4\%, \quad S(5) = 0.936^5 = 0.7184 \end{aligned}

The hedge ratio as a power. The swaps follow the expected balance at the baseline CPR, 80% of $B_0 (1-\text{CPR}^0)^t$ for the cohorts not yet at the end of their fix. If the actual CPR is $\gamma\,\text{CPR}^0$ , the same cohorts remain, so their amounts cancel and only the powers are left:

\begin{aligned} h(t) &= \frac{0.8\,B_0\,(1-\text{CPR}^0)^t}{B_0\,(1-\gamma\,\text{CPR}^0)^t} = 0.8\left(\frac{1-\text{CPR}^0}{1-\gamma\,\text{CPR}^0}\right)^{t} \\ h(3)\big|_{\gamma = 1.2} &= 0.8\left(\frac{0.92}{0.904}\right)^{3} = 84.32\% \end{aligned}

The cap. With the tracker resetting a month after Bank Rate moves by $\Delta$ : $\text{NII given up} = A \times \max\big(0,\ r_0 + \Delta - K\big) \times \tfrac{11}{12}$ , where $r_0 = 4.75\%$ and $K = 6\%$ : the payoff of a call on the rate, which the bank has sold.

Try it

Change the five-year fixes' base prepayment rate and pick a scenario. Watch the mortgages run off faster or slower than the swaps. The value change of mortgages plus hedge is mostly the 20% left unhedged; compare it with and without the shock, and see how a higher base CPR shifts the hedge ratio.

Try itPrepayment and the macro hedge

Base CPR of the five-year fixes8% a year

Scenario

Five-year fixes still on their fixMacro hedge swaps still running

CPR in the scenario9.6%

Monthly rate (SMM)0.838%

Hedge ratio after 1 year81.4%

Hedge ratio after 3 years84.3%

Δ value, five-year fixes+£869.7m

Δ value, their macro hedge−£655.2m

Mortgages + hedge+£214.4m

The swaps were sized at 80% of the expected profile at an 8% CPR and do not shrink when borrowers prepay. The base case of the value change keeps your base CPR; the shock applies the scenario's multiplier. Illustrative book.

Worked example

One borrower refinances

A borrower owes £250,000 on one of the book's five-year fixes at 3.8%, with 3 years of the fix left (year 3 of the deal). Rates fall 275bp in parallel. Interest-only for simplicity; the ERC schedule is illustrative.

Does the borrower go, and what does it cost the bank?

Step 1 of 5

1
The borrower's new rate
The book writes new five-year fixes at the five-year par swap rate plus 0.70%: 4.99% today. After the shock: 2.24%, against 3.80% on the old loan.

Scaling up

Across the book, the 1.2 multiplier in parallel down takes £23.9m off the gain in EVE. The ERCs collected would offset part of that in practice; the book's EVE leaves them out.

Regulatory lens

The 2016 Basel standard sets out how prepayment enters the standardised framework: a baseline CPR for each portfolio, scaled by 0.8 or 1.2 by scenario, with prepayments added to the scheduled cash flows (paragraphs 121 to 124). Term deposits that may be redeemed early get their own scalars, and sold automatic options are revalued under each shock with implied volatility raised by 25% in relative terms. The application guidance (SRP98) notes that borrowers tend to repay fixed-rate borrowings when rates fall and calls loan prepayment the most complex area of behavioural analysis. The rules in detail are in M4.5: Liquidity and IRRBB regulation; prepayment and deposit models are in M6.7: Liquidity, ALM and behavioural models, and option pricing in Level 6.

Spark Atoms

Spark AtomWhen the pipeline stops

After the UK government's September 2022 mini-budget, the 10-year gilt yield rose to levels not seen since 2008. Virgin Money and Skipton Building Society paused mortgage offers to new customers and Halifax said it would withdraw products with fees (CNBC, 27 September 2022). Within a week the number of residential mortgage products on sale had fallen by more than 1,600 (Moneyfacts). Offers already made were left with fixed rates that markets had just repriced.

Spark AtomLosing both ways

On the book, prepayment costs EVE £21.4m in parallel up and £23.9m in parallel down. A position that loses on both sides has sold an option. The premium is in the mortgage rate and the early repayment charge, not in a separate fee.

Key takeaways

Borrowers hold options: to prepay, to walk away from an offer, to benefit from a cap. Automatic options follow the contract; behavioural ones follow customers.
CPR is the annual share of the outstanding balance prepaid; the surviving share is (1 − CPR)^t and SMM = 1 − (1 − CPR)^(1/12).
The Basel standardised framework scales the baseline CPR by 0.8 when rates rise and 1.2 when they fall.
Prepayment costs RiskAtoms Bank EVE in both parallel shocks (−£21.4m and −£23.9m) and leaves its non-amortising macro hedge over- or under-hedged.
Early repayment charges recover part of the loss; pipelines and caps are further sold options, best seen on the book as extras.

Reaction

Check what stuck

Question 1 · objective 1

A mortgage book prepays at a CPR of 10% a year. What share of it is still outstanding after three years, in %?

%

Question 2 · objective 1

What monthly prepayment rate (SMM) corresponds to a CPR of 12%, in %?

%

Question 3 · objective 2

A bank's baseline CPR is 15%. Under the Basel standardised framework, what CPR applies in the parallel down scenario, in %?

%

Question 4 · objective 3

A borrower owes £300,000 in year 2 of a five-year fix whose early repayment charge falls 5%, 4%, 3%, 2%, 1%. What is the charge for repaying now, in £?

£

Question 5 · objective 4

Treasury hedges 75% of a £1.5bn mortgage pipeline with pay-fixed swaps. Why does it lose whether rates rise or fall 275bp?

AMore borrowers complete when rates rise (under-hedged) and fewer when they fall (over-hedged): the borrowers hold the optionBSwaps always lose money in large movesCThe pipeline is hedged at the wrong swap rateDBecause the bank charges no early repayment charge

Glossary

Option risk: Rate risk from options: either written into a contract (automatic) or exercised by customers through their behaviour (behavioural).
Automatic option: An option the contract exercises by itself when it pays, such as a cap on a loan rate.
Behavioural option: An option customers may or may not use, such as repaying a mortgage early; not always when it pays.
Prepayment: Repaying a loan, or part of it, before its scheduled date.
Conditional prepayment rate (CPR): Conditional prepayment rate: the share of the balance still outstanding that is repaid early in a year.
Single monthly mortality (SMM): Single monthly mortality: the monthly version of the CPR, 1 − (1 − CPR)^(1/12).
Early repayment charge (ERC): Early repayment charge: a fee, usually a percentage of the balance, for repaying a fixed-rate mortgage during its deal period.
Pipeline: Loans offered at a fixed rate but not yet completed. The borrower may take the loan or walk away, so the amount that completes depends on rates.
Pull-through: The share of pipeline offers that turn into completed loans.
Cap: A ceiling on a floating loan rate. The bank has sold the borrower an option that pays when rates rise above it.
Negative convexity: The value of a position falls more when rates move against it than it gains when they move in its favour, because someone else holds an option.

Deep Dives & Origins

Deep Dives

Basel Committee on Banking Supervision, Interest rate risk in the banking book, BIS, April 2016: the standardised framework for behavioural and automatic options. Advanced.
Fannie Mae, Benchmark CPR Methodology Overview, February 2024: how CPR and SMM are used in practice. Intermediate.

Origins

Basel Committee on Banking Supervision, Standards: Interest rate risk in the banking book, BIS, April 2016, paragraphs 121 to 130 and Tables 3 and 4.
Basel Committee on Banking Supervision, Basel Framework, SRP31.2 and SRP98.33 to 98.35, BIS.
Fannie Mae, Benchmark CPR Methodology Overview, February 2024 (SMM and CPR conversion).
Lloyds Bank, What is an early repayment charge?; Moneyfacts, Mortgage early repayment charges explained, August 2025.
CNBC, UK lenders halt mortgage deals to customers after market chaos, 27 September 2022; Moneyfacts, Which lenders have removed their mortgages thus far?, 29 September 2022; NationalWorld, How long does a mortgage offer last?, October 2022.

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