Lecture 2 of 15, 45 minute read

Interest rates, bonds and risk

Last time we treated the discount rate as given. Today we ask where it comes from. We start with the simplest risky asset, a bond: how to price it, what its yield means and how much its price moves when rates change. We then read the yield curve, price credit risk and work out the return shareholders demand, and we finish with how all of it feeds into WACC, valuation multiples and the financing of a leveraged buyout.

1. Where interest rates come from

An interest rate is the price of borrowing money for a period of time, set by supply and demand between savers and borrowers. At the short end, one player has outsized influence: the central bank. It sets a policy rate at which banks borrow or deposit overnight: the Federal Reserve targets a range for the federal funds rate, and the European Central Bank sets, among others, its deposit facility rate. Overnight market rates follow; the benchmarks are SOFR, the secured overnight financing rate, in dollars and €STR in euros. Longer rates are set by the market and contain more than the policy rate. A useful way to break any yield into pieces:

The first two pieces together are what the market expects short-term rates to average over the life of the bond. The term premium is the extra return investors want for locking money away longer and bearing the risk that rates move. A government bond in its own currency is treated as free of default risk, so it has no credit spread, and its yield is our risk-free rate. That convention works for the United States, Germany or the United Kingdom; for a government that could plausibly default, analysts strip a default spread out of its yield first.

2. What a bond is

A bond is a loan cut into tradable pieces. The issuer promises two things: a coupon, a fixed interest payment each period, and the face value, usually $1,000, at maturity. Governments issue bonds, and so do companies, from investment-grade giants to small miners funding a new pit.

US Treasuries and most US corporate bonds pay coupons twice a year; German Bunds, French OATs and most euro corporate bonds pay once a year. Prices are quoted as a percentage of face value, so a bond "at 98.5" costs $985 per $1,000. Banks' debt capital markets teams bring new issues to market, as we see in lecture 13.

3. Pricing a bond

A bond's price is the present value of its promised payments, discounted at the return the market currently demands, the yield:

Look at the structure: an annuity, the coupons, plus a single zero-coupon payment, the face value. Every bond calculation you will meet uses one or both of those pieces.

4. Price and yield move in opposite directions

This is the single most important fact about bonds. The coupon is fixed when the bond is issued. If market yields rise, the fixed coupon becomes less attractive, so the price must fall until the bond offers the new yield to a buyer. If yields fall, the price rises.

• Coupon equal to yield: the bond trades at par, its face value.
• Coupon above yield: it trades at a premium.
• Coupon below yield: it trades at a discount, like our example.

Prices also move with time, even when yields do not. As a bond nears maturity, its price converges to face value: the pull to par. If the yield stays at 6%, our discount bond is worth $965.35 a year from now, then $973.27, $981.67, $990.57 and finally $1,000. A premium bond drifts down in the same way. The drift is part of the return. A buyer at $957.88 earns the $50 coupon plus $7.47 of price gain in the first year, $57.47 in total, which is exactly 6% of the price paid.

5. Yield to maturity and current yield

Usually we know the price and want the yield. The yield to maturity (YTM) is the single discount rate that makes the present value of all promised payments equal to the price. You will recognise it: it is the bond's IRR. A coupon bond has no closed-form solution, so you solve by trial or with a spreadsheet's YIELD or RATE function.

Two warnings about YTM. First, like any IRR, it equals your realised return only if you hold to maturity, every payment is made, and every coupon is reinvested at the YTM itself. Second, for a bond that might default, the YTM is a promised yield, not an expected one: a distressed bond yielding 18% is pricing a real chance of not being repaid in full.

Many corporate bonds, especially high-yield ones, are callable: the issuer can repay early at set prices. Investors then compute a yield to call for each call date and quote the lowest of these yields and the YTM, the yield to worst. For a bond trading above its call price, the yield to worst is usually a yield to call, because the issuer will refinance if rates fall.

6. Clean and dirty prices

Coupons are paid on fixed dates, but bonds trade every day. A buyer between coupon dates receives the whole next coupon, although the seller owned the bond for part of the period. So the buyer pays the seller the accrued interest, the share of the coupon earned since the last payment date:

Bonds are quoted at the clean price, which excludes accrued interest, because the dirty price saw-tooths: it climbs between coupon dates and drops by the coupon on each payment date even when the market is still. The buyer actually pays the dirty price, also called the full or invoice price, and it is the dirty price that equals the present value of the remaining cash flows.

Day counts vary: US Treasuries and most euro-denominated bonds use actual/actual, US corporates 30/360. Use the wrong one and your accrued interest is wrong.

7. Spot rates and forward rates

So far we have discounted every cash flow of a bond at one rate, its YTM. That is a convenient summary, but it is not how the market really prices. A payment due in one year and a payment due in ten years are different products, and each has its own rate. The yield on a zero-coupon bond maturing at \(t\) is the spot rate \(s_t\), and the correct price of any bond discounts each cash flow at the spot rate for its date:

A coupon bond's YTM is then a kind of average of the spot rates for its payment dates, weighted towards the final payment.

Spot rates come from bond prices, by bootstrapping. Suppose the one-year zero yields 4.00% and a two-year bond with a 6% annual coupon trades at 101.91. Its first coupon is worth \(6/1.04 = 5.77\). The rest of the price, 96.14, must be the value of the 106 paid in year 2, so \((1+s_2)^2 = 106/96.14\) and \(s_2 = 5.00\%\). Repeat with a three-year bond to get \(s_3\), and so on along the curve.

Spot rates imply forward rates: the rate for lending between two future dates that you can lock in today. Investing for two years at the two-year spot rate must give the same result as investing for one year and rolling over at the forward rate \(f_{1,2}\); otherwise there would be a free lunch:

Read a forward rate as a break-even, not a forecast. It is the future rate at which you would be indifferent between the long investment and rolling short ones. If investors cared only about expected returns, it would equal the expected future short rate; in reality it also contains a term premium.

8. The shape of the yield curve

Plot the yields of government bonds against their maturities and you get the yield curve. Its shape is one of the most watched indicators in finance.

• Normal, or upward sloping: long rates above short rates. This is the usual shape, because investors want a premium for tying money up longer.
• Flat: short and long rates close together, often a transition between the other shapes.
• Inverted: short rates above long rates. The market expects the central bank to cut, typically because it sees a slowdown coming. The US curve has inverted ahead of almost every recession of the past half-century, which is why the gap between 2-year and 10-year yields is watched so closely. It is a loose signal, not a law: that gap stayed negative from mid-2022 to late summer 2024, the longest inversion on record.
• Humped: medium maturities yield more than both short and long ones, for example when markets expect rates to rise for a while and then fall.

Three theories explain the shape, and a strong interview answer uses all three.

1. Expectations hypothesis. Long rates are the average of expected future short rates. An upward curve means the market expects rates to rise; an inverted one means it expects cuts. On its own this cannot explain why the curve slopes upward most of the time, since rates do not rise most of the time.
2. Liquidity preference, or the term premium. Investors prefer short maturities, which are less exposed to rate moves, so they demand extra yield to hold long bonds. This adds an upward tilt to whatever expectations imply.
3. Market segmentation, and its softer version, preferred habitat. Different investors have natural maturities: money market funds buy short, pension funds and insurers buy long to match their liabilities. Supply and demand within each segment move its yield. Central-bank bond buying works partly through this channel.

9. How central-bank policy reaches long rates

A central bank controls the overnight rate directly. Its influence on a ten-year yield is indirect, through four channels.

1. Expectations. The ten-year yield reflects the expected path of short rates over ten years. A single hike matters less than what it signals about the next ones, which is why markets move on central bankers' guidance.
2. Inflation credibility. If investors believe the bank will hold inflation near its target, usually 2%, expected inflation in long yields stays anchored. If that credibility slips, long yields can rise even while the bank cuts.
3. The balance sheet. Through quantitative easing (QE), a central bank buys long bonds, pushing their prices up and yields down, mostly by compressing the term premium. Quantitative tightening (QT) works the other way.
4. Floating-rate loans. Many loans, from mortgages in much of Europe to the term loans in a leveraged buyout, are priced at a benchmark such as SOFR or EURIBOR plus a margin. A policy move reaches these borrowers at the next reset date.

Short and long rates can therefore move in opposite directions. Between September and December 2024 the Federal Reserve cut its policy rate by a full point, while the ten-year Treasury yield rose by close to a point, as markets priced a stronger economy, fewer future cuts and a larger term premium. "The Fed cut, so long rates fall" is a common and wrong interview answer.

10. Duration: how sensitive is the price?

Not all bonds react equally to a change in rates. A one-year bond barely moves; a thirty-year bond can swing violently. Duration measures that sensitivity. The Macaulay duration is the average time at which you receive your cash, each payment weighted by its share of the bond's present value:

The working version is modified duration, Macaulay duration divided by \(1+y\). It gives the approximate percentage price change for a small move in yield:

DV01. Traders think in money, not percentages. The DV01, the dollar value of one basis point, is the price change for a 0.01% move in yield: modified duration × price × 0.0001. For our bond, \(4.28 \times 957.88 \times 0.0001 = \$0.41\) per $1,000 of face value. A $10 million position in a bond trading at par with a modified duration of 7 has a DV01 of $7,000: every basis point makes or loses $7,000.

Rules of thumb: duration rises with maturity and falls as the coupon or the yield rises. A zero-coupon bond's Macaulay duration equals its maturity. A perpetuity's Macaulay duration is \((1+y)/y\): 21 years at a 5% yield, although it never matures. For a one-point rise in yields from 5% to 6%, a 10-year par bond loses 7.4%, a 10-year zero 9.0%, a 30-year par bond 13.8% and a 30-year zero 24.8%.

11. Convexity

Duration draws a straight line, the tangent, through a price–yield curve that is actually bent. The bend is convexity, and adding it gives a better estimate:

Positive convexity is valuable: whichever way rates move, a convex bond gains more or loses less than duration predicts. All plain bonds have it. Callable bonds and mortgage-backed securities can have negative convexity: when rates fall, the issuer or the homeowner refinances, which caps the price gain. For these, analysts use effective duration, measured by repricing the bond under shifted yields.

The interactive bond pricer for this lecture runs in the course. Open it in the interactive course.

Duration beyond bonds. Any asset whose cash flows arrive far in the future has a long duration. A growing perpetuity has a modified duration of about \(1/(r-g)\), so a share valued at \(r = 9\%\) and \(g = 3\%\) behaves like a bond with a duration of nearly 17 years. Raise the discount rate by one point and its value falls by 14.3%, from \(C/0.06\) to \(C/0.07\); with 5% growth the fall is 20%. A start-up that will only make money in ten years is an even longer-duration asset, which is why its valuation collapses when rates rise.

12. Credit risk, ratings and spreads

A company is not a government. It can default. Investors demand compensation for that risk in the form of a credit spread over the risk-free rate:

Spreads are quoted in basis points, hundredths of a percent, usually over a government bond of similar maturity or over the swap curve. The credit default swap (CDS) market prices the same risk directly. Rating agencies grade issuers on scales that run from the strongest to default:

Each grade has notches: A+, A and A− at S&P and Fitch, A1, A2 and A3 at Moody's. BBB−, or Baa3, is the last rung of investment grade; below it is high yield, also called junk. Many funds may only hold investment-grade bonds, so a downgrade across the line, a fallen angel, forces selling. Historically, single-B issuers have defaulted at a few percent a year on average, investment grade at a small fraction of a percent. Ratings map loosely onto leverage: investment-grade industrials tend to run below about 3x net debt to EBITDA, while leveraged buyouts usually carry single-B ratings at 5x or more.

The spread compensates first for expected loss: the probability of default times the loss given default, which is one minus the recovery rate. An issuer with a 2% annual chance of default and a 40% expected recovery has an expected loss of \(2\% \times 60\% = 1.2\%\) a year, or 120 bp. Observed spreads are larger than expected losses, often several times larger for investment grade, because investors also charge for uncertainty, for the way defaults cluster in recessions, and for poorer liquidity than government bonds.

13. Risk and return for shareholders

Shareholders face more risk than lenders, so they want more return. But which risk? Here finance makes a subtle and powerful point.

A share's price swings for two kinds of reasons. Some are specific to the company: a mine floods, a CEO resigns. That is idiosyncratic risk, and an investor holding many shares can diversify it away, because one company's bad news is offset by another's good news. Other swings hit everything at once: recessions, interest rates, a market crash. That is systematic risk, and no amount of diversification removes it.

Because investors can diversify away the first kind for free, markets only pay a premium for the second. We measure a share's exposure to systematic risk with beta: a beta of 1 moves with the market, 1.5 moves 50% more, 0.5 moves half as much. Beta is estimated by regressing the stock's returns on the market's, usually over two to five years of weekly or monthly data. Equivalently, \(\beta = \rho\,\sigma_i/\sigma_m\): the correlation with the market times the ratio of volatilities. A stock with 40% volatility and a 0.6 correlation with a market of 18% volatility has a beta of \(0.6 \times 40/18 = 1.33\). A volatile stock can still have a low beta if its swings are mostly its own; gold miners are a classic case.

The capital asset pricing model turns beta into a required return:

The term in brackets is the equity risk premium, the extra return investors expect from the market as a whole over government bonds, often taken somewhere around 5% to 6%.

14. Why equity costs more than debt

Lenders are paid first, receive a contractual amount and often hold security over assets. Shareholders are paid last, from whatever is left, with no promise at all. Their returns are more volatile and they lose everything first in a bankruptcy. So equity always demands a higher return than the same company's debt. Debt has a second advantage: interest is usually tax-deductible, which lowers its cost further. A 6% pre-tax cost of debt at a 25% tax rate costs \(6\% \times (1 - 25\%) = 4.5\%\) after tax. Hold onto both facts; they drive capital structure, WACC and the whole logic of a leveraged buyout.

15. From rates to valuation and deals

Everything in this lecture lands in three places later in the course.

WACC. The risk-free rate enters twice: in the cost of equity, through CAPM, and in the cost of debt, as the base under the credit spread. Take a company financed 70% by equity and 30% by debt, with a 25% tax rate. If government yields rise one point while spreads and the equity risk premium stay put, the cost of equity rises one point and the after-tax cost of debt 0.75 points, so WACC rises by \(0.7 \times 1 + 0.3 \times 0.75 = 0.925\) points.

Multiples. A growing perpetuity is a multiple: enterprise value is roughly next year's free cash flow divided by \(WACC - g\). With an 8% WACC and 2.5% growth, that is 18.2 times next year's cash flow. Add 0.925 points to WACC and the multiple drops to 15.6 times, a 14.4% fall in value with no change in the business. This is the mechanism behind "rising rates compress multiples", and it bites hardest where growth is high and the denominator small.

LBO financing. Leveraged loans pay floating rates, so higher policy rates raise interest costs at the next reset: on a $500m term loan at SOFR plus 3.5%, a move in SOFR from 1% to 5% lifts annual interest from $22.5m to $42.5m. Lenders also cap debt by interest coverage. If they require EBITDA to cover interest at least 2.5 times, a company with $100m of EBITDA can carry at most $40m of interest. At a 7% all-in rate that supports $571m of debt, 5.7x EBITDA; at 10%, only $400m, 4.0x. The sponsor must fill the gap with equity or pay a lower price, and either way returns fall. Sponsors often hedge part of the floating exposure with interest rate swaps, paying fixed and receiving floating.