Interest-rate risk and duration
Interest-rate risk is the loss in value that comes when yields change. Duration measures it: the average time to a bond's payments, weighted by their present value, and a guide to how far its price falls when yields rise. This lesson computes duration for a small bond, compares two real Treasury funds, and then turns to a bank, where assets and liabilities have different durations. A calculator revalues an invented bank when yields move, and the case of SVB Financial Group in 2022 shows how large the effect can be.
- Level
- Beginner
- Reading time
- About 27 minutes
- Know this first
- Bond pricing and yield
By the end you will be able to:
- compute Macaulay and modified duration for a bond
- use duration to estimate a price change and see where the estimate falls short
- find the duration of a portfolio from its parts
- explain the duration gap between a bank's assets and liabilities
- describe why held-to-maturity accounting hid a large loss at SVB
When yields move
A bond promises fixed payments. When market yields rise, those payments are worth less today, and the bond’s price falls; when yields fall, the price rises. The lesson on bond pricing and yield showed this and introduced duration as a rough guide to how far. This lesson takes duration further. It shows what the number is made of, how to combine it across a portfolio, and what it means for an institution that owns long assets and owes short liabilities, such as a bank.
The risk is called interest-rate risk. It differs from the credit risk of the expected loss lesson: the borrower can pay every coupon on time and the holder can still lose money, because the bond is now worth less than it cost. The loss is real if the bond has to be sold, and it can be hidden in the accounts if it does not.
Duration as an average time
Macaulay duration is the average time until a bond’s payments arrive, where each payment is weighted by its share of the bond’s price. Take an invented three-year bond with $1,000 face value and a 5% annual coupon, priced at a 4% yield. The payments are $50, $50 and $1,050. Discount each at 4%, divide by the price to get its weight, and multiply by the year in which it arrives.
| Year | Cash flow, $ | Present value at 4%, $ | Share of price | Year × share |
|---|---|---|---|---|
| 1 | 50.00 | 48.08 | 4.68% | 0.0468 |
| 2 | 50.00 | 46.23 | 4.50% | 0.0900 |
| 3 | 1,050.00 | 933.45 | 90.82% | 2.7247 |
| Total | 1,150.00 | 1,027.75 | 100.00% | 2.8615 |
The price is $1,027.75. The last payment, which includes the face value, is about 91% of the price and arrives in year 3, so it pulls the average toward 3. The Macaulay duration is 2.86 years. A bond that pays nothing until maturity, a zero-coupon bond, has a duration equal to its maturity. Coupons bring payments forward and shorten the duration.
To turn this into price sensitivity, divide by one plus the yield. The result is the modified duration: here 2.86 ÷ 1.04 = 2.75. It says that a one-percentage-point rise in yield lowers the price by about 2.75%. Check it: at a 5% yield the same bond is worth exactly $1,000.00, because its coupon equals the yield. That is a fall of 2.70% from $1,027.75, close to the 2.75% that the rule gave. The rule is a straight-line estimate of a curved relationship, so it is slightly off, and the error grows with the size of the move.
Duration of a portfolio
The duration of a portfolio is the average of the durations of its holdings, weighted by their market values. A portfolio that is 60% in a bond with a duration of 2 years and 40% in one with a duration of 10 years has a duration of 0.6 × 2 + 0.4 × 10 = 5.2 years. Two real Treasury funds show how much the figure can differ. One holds Treasury bonds with more than 20 years left, the other Treasury notes with one to three years left.
| Fund | Effective duration, years | 2021 | 2022 | 2023 |
|---|---|---|---|---|
| 20+ year Treasury fund | 15.31 | −4.76% | −31.41% | 2.96% |
| 1-3 year Treasury fund | 1.87 | −0.73% | −3.90% | 4.15% |
In 2022, when yields rose sharply, the long fund returned −31.41% and the short fund −3.90%. The long fund lost 8.1 times as much, and its duration is 8.2 times as long. The match is only rough: the durations in the table are for June 2026, not 2022, and yields of different maturities do not move by the same amount. But the order of magnitude is what duration predicts. Both funds also gained in 2023, which is the same risk working in the other direction.
Assets, liabilities and the duration gap
For a bank, what matters is not the value of the assets alone but the value of assets less liabilities, which is equity. If assets and liabilities moved together, a change in yields would leave equity unchanged. They do not. A bank often owns long assets, such as bonds and fixed-rate loans, and owes short liabilities, such as deposits. When yields rise, the long assets lose more value than the short liabilities, and the difference comes out of equity.
The duration gap measures this mismatch. It is the duration of the assets less the duration of the liabilities scaled by the ratio of liabilities to assets. A positive gap means equity falls when yields rise. A bank can reduce the gap by buying shorter assets, lengthening its funding, or using derivatives. The lesson on how banks work describes the balance sheet this applies to.
SVB Financial Group, 2022
SVB Financial Group, the parent of Silicon Valley Bank, grew quickly: the Federal Reserve’s review reports that its assets grew 271% from the end of 2018 to the end of 2021, against 29% for the banking industry. In the Federal Reserve’s comparison, securities were 55% of its assets, against 25% for large banking organizations, and 78% of those securities were classified as held-to-maturity. Its held-to-maturity securities rose from $16.6 billion at the end of 2020 to $98.2 billion at the end of 2021.
| Item | End of 2021, $ billion | End of 2022, $ billion |
|---|---|---|
| Held-to-maturity securities, amortized cost | 98.2 | 91.3 |
| Unrealized loss on them | 1.0 | 15.2 |
| Loss as a share of amortized cost | 0.99% | 16.59% |
| Weighted-average duration, years | 4.1 | 6.2 |
| Total deposits | 189.2 | 173.1 |
Held-to-maturity securities are reported at amortized cost, and unrealized losses on them are not included in earnings but only disclosed in the notes. At the end of 2021 the loss on SVB’s held-to-maturity portfolio was under 1% of its cost. A year later, with yields much higher and the portfolio’s weighted-average duration longer at 6.2 years, it was $15.2 billion, or 16.59%. SVB’s stockholders’ equity at the end of 2022 was $16.0 billion, so the loss on the held-to-maturity portfolio alone equalled about 94.7% of it. Subtracting that loss from reported equity, and ignoring tax and hedges, would leave about $0.9 billion. The reported figure did not show this, because the loss sat in the notes. Deposits also fell by $16.1 billion during 2022, from $189.2 billion to $173.1 billion, so cash had to come from somewhere, and selling securities means selling them at market value.
Revalue a bank when yields move
The calculator uses an invented bank with $100 of assets, $92 of liabilities and $8 of equity, with every balance valued at a 3% yield. Set the duration of its assets and of its liabilities, then move yields up or down. It revalues each balance exactly, as a single payment due after its duration, and compares the result with the duration rule.
A bank with $100 of assets at a duration of 6.0 years, $92 of liabilities at 1.5 years and $8 of equity faces a change in yields of +1.0 points from 3%. Its assets become $94.4 and its liabilities $90.7, so its equity becomes $3.7, a change of −$4.3 on the $8 it started with. The duration rule, which ignores curvature, puts the change in equity at −$4.5.
Five experiments. First, leave the settings alone: assets with a duration of 6 years, liabilities with 1.5 years and a one-point rise in yields. Assets become $94.4, liabilities $90.7, and equity $3.7, a loss of $4.3 out of $8. Assets lose 5.63%, against 5.83% by the duration rule. The duration gap is 4.62 years. Second, shorten the assets to 2 years. Equity becomes $7.4, a loss of only $0.6. A shorter asset duration shrank the gap. Third, set yields to a fall of 2 points. Equity rises to $17.7, a gain of $9.7: the same gap that hurts when yields rise helps when they fall. Fourth, set the rise to 3 points. Equity becomes −$3.9, so the liabilities exceed the assets. The duration rule gives a change of −$13.5, while the exact change is −$11.9: the straight line overstates the loss from a rise, because the price-yield curve bends. Fifth, set both durations to 3 years with a one-point rise. Equity becomes $7.8, a loss of $0.2. When the two sides have almost the same duration, the gap is near zero and the bank is barely affected.
How to read the results
Duration is a single number for a sensitivity, and its usefulness is in comparison. It lets a risk manager say that one portfolio is about eight times as exposed as another, or that a balance sheet will lose a given share of its equity if yields rise by one point. The estimate is best for small moves, because it is a straight line through a curve, as the fourth experiment shows. It also treats a change in yields as the same for every maturity. The lesson on the yield curve shows that short and long yields can move by different amounts. Finally, an exposure that is invisible in the accounts is still an exposure. Held-to-maturity accounting changes what is reported, not what the securities are worth if they must be sold. The earlier lesson on value at risk measured the loss that history shows can occur in a bad year; duration measures the loss a stated change in yields would cause.
What this lesson leaves out
- An invented bank. Its balance sheet, durations and 3% yield are made up. Real banks hold many kinds of assets and liabilities.
- Deposits are not bonds. The calculator values liabilities as fixed payments. Real deposits can be withdrawn at will, and their cost and size change with rates and with depositor behaviour.
- One yield for everything. The calculator moves one yield. Real yield curves twist as well as shift.
- No hedges. Banks use derivatives to reduce a gap, and the SVB estimates exclude their effect.
- Annual compounding and simple cash flows. The bond example pays annual coupons and the bank balances are treated as single payments. Real instruments differ.
- Fund durations are from 2026. The funds’ effective durations are for June 2026, while the returns are for calendar years 2021 to 2023.
- No advice. The lesson explains a measure. It does not recommend a fund, a bank or a position.
Common mistakes
Reading duration as a number of years to repayment
It is a weighted average time, shorter than maturity for a coupon bond, and it is used as a sensitivity.
Treating the rule as exact
The rule is a straight line. For the bank, a rise of 3 points gave a loss of $11.9 against the rule’s $13.5.
Checking credit quality and stopping
Government bonds have almost no default risk, yet the long Treasury fund lost 31.41% in 2022.
Looking only at assets
Equity depends on both sides. Short liabilities can make a bank more exposed, not less.
Trusting reported equity
SVB reported equity of $16.0 billion while the unrealized loss on its held-to-maturity securities, $15.2 billion, was almost as large.
Try it yourself
1. A bond has a Macaulay duration of 2.86 years and a yield of 4%. What is its modified duration, and by about how much does its price fall if the yield rises 1 point?
Show the working
2.86 ÷ 1.04 = 2.75 years, so the price falls by about 2.75%.
2. A portfolio holds $60 of bonds with a duration of 2 years and $40 with a duration of 10 years. What is its duration?
Show the working
0.6 × 2 + 0.4 × 10 = 5.2 years.
3. A portfolio has a modified duration of 6. By about how much does its value fall if yields rise 2 points?
Show the working
6 × 2 = 12%. For a move this size the straight-line estimate overstates the loss a little, because the price-yield curve bends.
4. The long Treasury fund lost 31.41% in 2022 and the short one 3.90%. How many times as large was the first loss?
Show the working
31.41 ÷ 3.90 = 8.1. The ratio of the durations is 15.31 ÷ 1.87 = 8.2.
5. The invented bank has assets of $100 with a duration of 6 years and liabilities of $92 with a duration of 1.5 years. Using the duration rule, what is the change in equity if yields rise 1 point from 3%?
Show the working
The duration-weighted exposure is 100 × 6 − 92 × 1.5 = 462. Then −462 ÷ 1.03 × 0.01 = −$4.5. The exact revaluation gives −$4.3.
Where this is used
- A bank’s treasury measures the duration gap and sets limits on how much equity a rate move may cost.
- A bond fund reports its duration so that investors can compare funds and judge how they react to rate changes.
- A pension fund or insurer matches the duration of its assets to that of the benefits it owes.
- A supervisor looks at unrealized losses and the duration of securities when judging a bank’s resilience.
- A company with floating-rate debt or a large fixed-income portfolio estimates what a rate change does to its value.
Key terms
- Interest-rate risk
- The risk that a change in yields lowers the value of an asset or of equity.
- Macaulay duration
- The average time to a bond’s payments, weighted by their present values.
- Modified duration
- Macaulay duration divided by one plus the yield; the approximate percentage price change for a one-point change in yield.
- Duration gap
- The difference between the duration of a bank’s assets and that of its liabilities, adjusted for their sizes.
- Held-to-maturity
- A security a bank intends to keep until it is repaid, reported at amortized cost.
- Unrealized loss
- A fall in market value that has not been turned into a loss by a sale.
- Effective duration
- A duration measured from how a bond or fund’s value responds to a change in yields.
- Equity
- Assets less liabilities, the cushion that absorbs losses.
Questions people ask
Why does a longer bond lose more when yields rise?
Its payments come later on average, so each is discounted for longer, and a change in the discount rate compounds over more years.
Can a bank lose money from interest rates even if no borrower defaults?
Yes. If its assets are long and its funding is short, a rise in yields cuts the value of the assets more than the liabilities, and equity falls. No default is needed.
Why did held-to-maturity accounting matter at SVB?
Those securities are reported at amortized cost, so the unrealized loss of $15.2 billion did not reduce reported equity. It still reduced what the bank would receive if it had to sell them.
Does a low duration mean no risk?
No. It means a small sensitivity to yields. Other risks, such as credit risk or liquidity risk, remain.
Sources and review
- Board of Governors of the Federal Reserve System, Evolution of Silicon Valley Bank and the accessible version of the figures, from the Review of the Federal Reserve’s Supervision and Regulation of Silicon Valley Bank (April 2023), for SVB Financial Group’s growth, its securities as a share of assets, the balances of its securities (figure 8) and the Federal Reserve’s estimates of unrealized gains and losses on them (figure 9), which exclude hedging, tax and transfers of available-for-sale securities.
- SVB Financial Group, Fourth quarter and full year 2022 earnings release (Form 8-K, Exhibit 99.1), for stockholders’ equity of $16.0 billion, the weighted-average duration of the held-to-maturity portfolio (6.2 years at 31 December 2022 and 4.1 years a year earlier) and total deposits.
- William M. VanDenburgh and Philip J. Harmelink, The CPA Journal, Bank Failures Highlight the Shortcomings of Held-to-Maturity (HTM) Accounting (8 April 2024), for the rule that held-to-maturity securities are reported at amortized cost without unrealized losses in earnings, and for SVB’s held-to-maturity amortized cost and unrealized loss in 2021 and 2022, taken by the article from SVB’s 2022 annual report.
- BlackRock, iShares 20+ Year Treasury Bond ETF fact sheet (as of 30 June 2026), for the effective duration of 15.31 years and the calendar-year returns of −4.76% (2021), −31.41% (2022) and 2.96% (2023).
- BlackRock, iShares 1-3 Year Treasury Bond ETF fact sheet (as of 30 June 2026), for the effective duration of 1.87 years and the calendar-year returns of −0.73% (2021), −3.90% (2022) and 4.15% (2023).
How the numbers were checked: the bond example, its present values, the Macaulay and modified durations and the exact repricing at a 5% yield were recomputed in plain code. The bank revaluations and the five experiments were recomputed separately, and the calculator, which runs on its own, was compared with that code for every combination of settings it allows. The SVB figures are quoted from the cited documents, including the loss as a share of cost, which is the loss divided by amortized cost, and the share of equity, which divides the held-to-maturity loss by stockholders’ equity. The bank in the calculator, its durations and its 3% yield are invented.