Free sample chapter from FRM Level I: Valuation Models.

Chapter 1: Pricing Conventions, Discounting, Arbitrage and Interest Rates

Chapter 1: Pricing Conventions, Discounting, Arbitrage and Interest Rates

Financial Markets and Products II priced Ravensmoor's $350 million government book — one five-year Treasury paying 4.00% twice a year — at 100.9096 per 100 of face value, by discounting each cash flow off the bank's zero curve. That chapter used the curve. This one asks what the curve is, why a bond's price must be the one it gives, and what else the same curve can tell us.

The answer to the second question is the foundation of everything in this course: a bond is a portfolio of zero-coupon bonds, and two portfolios with identical cash flows must have the same price. If they did not, a trader could buy the cheaper, sell the dearer and lock in a profit with no risk. Ravensmoor's bond can be split into eleven separate Treasury zero-coupon securities — STRIPS — and their prices add up to exactly 100.9096.

From one table of discount factors this chapter then reads every kind of interest rate a fixed-income market quotes:

• Spot rates, the yields on zero-coupon bonds, from 3.02% at six months to 3.82% at five years.
• Forward rates, the rates the curve locks in for future periods, rising from 3.00% to 4.07% before dipping at the long end.
• Par rates, the coupons at which new bonds would sell at 100: 3.7993% at five years — which is also the fixed rate of a five-year interest rate swap on the same curve.

Learning Objectives

1.1 Discount Factors and the Law of One Price

Discount factor The price today of $1 to be received with certainty at a future date $t$, written $d(t)$. Viewed as a function of $t$, it is the discount function.

A discount factor turns any certain future cash flow into its present value, and back again:

Discount factors are less than 1 for any future date, because a dollar later is worth less than a dollar now, and they fall with maturity. Ravensmoor's curve, from Financial Markets and Products II:

A bond with several cash flows is priced by applying each date's discount factor to that date's payment and adding:

The law of one price

Law of one price Two securities, or portfolios, that pay exactly the same cash flows at exactly the same times must have the same price.

The argument is an arbitrage argument. If two portfolios with identical cash flows had different prices, an investor could buy the cheaper and sell the dearer. Every future cash flow received on the long side would exactly meet the cash flow owed on the short side, so the investor would keep the price difference today with no risk and no further obligation. Traders seek out such opportunities, and their buying and selling pushes the two prices together until the difference disappears, or shrinks to less than the costs of the trade.

Applied to bonds, the law of one price says that every bond's price must equal the sum of its cash flows times the market's discount factors. Discount factors are therefore not an assumption made by an analyst; they are implied by the prices of traded bonds, and a bond that disagrees with them is mispriced relative to the others.

1.2 Arbitrage and Replication

Arbitrage A trading strategy that requires no net investment, has no chance of loss, and has a positive chance of profit. In fixed income, it usually means buying a security and selling a portfolio with the same cash flows for more, or the reverse.

To test whether a bond is fairly priced, build a replicating portfolio: a combination of other securities whose cash flows match the bond's on every date. Work backward from the last cash flow, because each later security also pays earlier coupons that the earlier securities must then offset.

import numpy as np

d1, d2 = np.exp(-0.030 * 0.5), np.exp(-0.032 * 1.0)  # six-month and one-year discount factors
bill = 100 * d1                                     # six-month bill
bond6 = 3 * d1 + 103 * d2                           # one-year 6% bond
target = 1 * d1 + 101 * d2                          # one-year 2% bond, fair value

x = 101 / 103                                       # match the one-year cash flow
y = (1 - 3 * x) / 100                               # then the six-month cash flow
print(f"hold {x:.6f} of the 6% bond and {y:.6f} of the bill: cost {x * bond6 + y * bill:.4f}")
print(f"fair value {target:.4f}; quoted 98.60 -> buy it, sell the replica, lock in {target - 98.60:.4f}")
# Output:
# hold 0.980583 of the 6% bond and -0.019417 of the bill: cost 98.8043
# fair value 98.8043; quoted 98.60 -> buy it, sell the replica, lock in 0.2043

In practice, apparent arbitrages are usually smaller than the costs of executing them — bid-ask spreads, financing and the cost of borrowing a bond to short it — or reflect something the cash flows do not show, such as a bond's liquidity or its value as collateral. The law of one price holds up to these frictions, not beyond them.

1.3 Treasury Coupon Bonds and STRIPS

A US Treasury note or bond pays a fixed coupon every six months and its face value at maturity. Under the Treasury's STRIPS program (Separate Trading of Registered Interest and Principal of Securities), a dealer can ask for an eligible coupon security to be split into its individual payments, each of which then trades as a separate zero-coupon security:

• Coupon STRIPS (C-STRIPS) — one for each coupon payment.
• Principal STRIPS (P-STRIPS) — one for the face value at maturity.

The process also runs in reverse: a dealer who assembles every coupon STRIP and the principal STRIP of a particular note can reconstitute the original bond. STRIPS and reconstitution are what keep a coupon bond's price tied to the prices of its pieces. If the pieces were worth more than the whole, dealers would buy bonds and strip them; if less, they would buy STRIPS and reconstitute.

import numpy as np

t = np.arange(1, 11) * 0.5
z = np.array([3.00, 3.20, 3.35, 3.45, 3.53, 3.60, 3.66, 3.71, 3.75, 3.78]) / 100
d = np.exp(-z * t)                                  # discount factors

cash = np.full(10, 2.0)
cash[-1] += 100                                     # five-year 4.00% semiannual bond
print("discount factors:", " ".join(f"{x:.6f}" for x in d))
print(f"coupon STRIPS {(2 * d).sum():.4f} + principal STRIP {100 * d[-1]:.4f} = {(cash * d).sum():.4f}")
# Output:
# discount factors: 0.985112 0.968507 0.950992 0.933327 0.915532 0.897628 0.879765 0.862086 0.844720 0.827787
# coupon STRIPS 18.1309 + principal STRIP 82.7787 = 100.9096

P-STRIPS and C-STRIPS

A P-STRIP and a C-STRIP maturing on the same date pay the same $100 at the same time, so by the law of one price they should trade at the same price. In practice they can differ slightly, for two reasons:

• Fungibility. C-STRIPS maturing on a given date are interchangeable regardless of which bond they came from. A P-STRIP is tied to one particular note or bond, and is the only piece that can be used to reconstitute it.
• Reconstitution value and liquidity. Because a P-STRIP is needed to rebuild its specific bond, it can carry a premium when that bond is in demand, for instance as the cheapest bond to deliver into a futures contract or when it trades special in the repo market.

These differences are small, but they are a real example of the frictions that limit the law of one price.

1.4 Clean and Dirty Prices, Accrued Interest and Day Counts

Between coupon dates, a bond's seller has earned part of the next coupon. The buyer will receive the full coupon on the next payment date, so the buyer compensates the seller for the part already earned: accrued interest.

Clean and dirty price The clean (or flat, or quoted) price excludes accrued interest. The dirty (or full, or invoice) price includes it, and is what the buyer actually pays: dirty price = clean price + accrued interest.

US Treasury prices are quoted in 32nds of a point: 98-16 means $98 + 16/32 = 98.50$ per 100.

Markets quote clean prices because dirty prices jump. A dirty price rises steadily as accrued interest builds and then falls by the coupon amount on each payment date, even if nothing about the bond's value has changed. The clean price strips out that saw-tooth pattern and moves only when yields move or the bond pulls toward par.

Day-count conventions

How "days since last coupon" and "days in the period" are counted depends on the market:

Under 30/360, a period from 15 March to 15 September is always 180 days, and from 15 March to 26 May is $2 \times 30 + 11 = 71$ days. Under actual/360, a full year of interest is paid on 365 days at a rate that assumes 360, so the investor earns slightly more than the quoted rate.

1.5 Compounding Frequency

Financial Markets and Products II showed that one rate can be stated at different compounding frequencies. Investing $1 at a rate $R_m$ compounded $m$ times a year for $T$ years, and at a continuously compounded rate $R_c$, gives

Two rates are equivalent if they give the same terminal value:

For a fixed quoted rate, more frequent compounding means a higher future value and a lower present value. $100 invested for five years at 4% grows to $121.67 with annual compounding, $121.90 semiannually, $122.10 monthly and $122.14 continuously. Equivalently, a bond's value falls as the compounding frequency of the rate used to discount it rises, because the same quoted rate is in fact a larger rate.

The practical rule is to discount a cash flow with a rate quoted on the same basis as the curve it came from — and to convert before mixing. Ravensmoor's curve is continuously compounded; US Treasury yields are quoted semiannually.

1.6 Spot Rates and Discount Factors

Spot rate The yield on a zero-coupon bond maturing at a given date: the single rate that discounts a cash flow at that date to its present value. Also called the zero rate. Each maturity has its own.

A spot rate and a discount factor contain exactly the same information:

Ravensmoor's spot rates, restated with semiannual compounding to match Treasury quotation, run from 3.0226% at six months to 3.8159% at five years. Spot rates are found by bootstrapping — solving for the discount factor at each maturity in turn from the prices of coupon bonds, as Financial Markets and Products II did — or read directly from STRIPS prices: a five-year STRIP priced at 82.7787 implies a five-year discount factor of 0.827787.

1.7 Forward Rates

Forward rate The rate, agreed today, for borrowing or lending over a future period. It is implied by the spot curve: investing to a later date must give the same result as investing to an earlier date and then rolling over at the forward rate.

With continuous compounding the forward rate for the period from $T_1$ to $T_2$ is

and in terms of discount factors, $d(T_2) = d(T_1)\, e^{-F (T_2 - T_1)}$. With other compounding, the same no-arbitrage logic gives, for example, $(1 + R_2)^{T_2} = (1 + R_1)^{T_1} (1 + F)^{T_2 - T_1}$ with annual rates.

The curve's six-month forward rates rise from 3.00% to 4.07% for the period from four to four and a half years, then dip to 4.05% for the last six months. A forward rate below the one before it means the spot curve is bending over: its slope is starting to flatten.

1.8 Par Rates, and How Spot, Forward and Par Relate

Par rate The coupon rate at which a bond of a given maturity would be priced at exactly its face value. It is the yield to maturity of a bond that trades at par.

For a bond paying a semiannual coupon $c$ for $n$ periods to be worth 100,

The sum of the discount factors in the denominator is the present value of receiving 1 every six months for the life of the bond — the semiannual annuity factor. Per $1 a year paid semiannually, it is half that sum: on Ravensmoor's curve, 9.065454 ÷ 2 = 4.5327.

How the three curves relate

(The forward rates in the table are restated semiannually; their continuous values are those in §1.7.)

The pattern is general:

• When the spot curve slopes upward, forward rates lie above spot rates, and spot rates lie above par rates. A par bond's yield is a weighted average of the spot rates for all its cash flows, and the early coupons are discounted at the lower short rates, so the par rate sits below the spot rate for the final maturity.
• When the spot curve slopes downward, the order reverses: forward rates below spot rates, and spot rates below par rates.
• When the curve is flat, all three are equal.

The gaps are small at short maturities and widen as the coupons' weight in the bond grows.

import numpy as np

t = np.arange(1, 11) * 0.5
z = np.array([3.00, 3.20, 3.35, 3.45, 3.53, 3.60, 3.66, 3.71, 3.75, 3.78]) / 100
d = np.exp(-z * t)

r = z[-1]
print(f"3.78% continuous = {2 * (np.exp(r / 2) - 1):.4%} semiannual = {np.exp(r) - 1:.4%} annual")
fwd = (z * t - np.concatenate([[0], z[:-1] * t[:-1]])) / 0.5
print("six-month forwards:", " ".join(f"{f:.2%}" for f in fwd))
print(f"one-year forward from 2 to 3 years: {z[5] * 3 - z[3] * 2:.4%}")
for n in (2, 4, 6, 10):
    print(f"{n / 2:.0f}-year par rate: {2 * (1 - d[n - 1]) / d[:n].sum():.4%}")
# Output:
# 3.78% continuous = 3.8159% semiannual = 3.8524% annual
# six-month forwards: 3.00% 3.40% 3.65% 3.75% 3.85% 3.95% 4.02% 4.06% 4.07% 4.05%
# one-year forward from 2 to 3 years: 3.9000%
# 1-year par rate: 3.2241%
# 2-year par rate: 3.4744%
# 3-year par rate: 3.6231%
# 5-year par rate: 3.7993%

1.9 Maturity and Price

Hold a bond's yield fixed and let time pass. A bond's price at a given yield depends on how many coupons remain, so its price changes even when the market does not:

• A premium bond — coupon above its yield — falls in price as maturity approaches, because the excess coupon has fewer periods left to be worth anything.
• A discount bond — coupon below its yield — rises in price as maturity approaches.
• Both converge to par at maturity. This is the pull to par.

At a constant yield of 3.7985%, Ravensmoor's 4.00% bond is worth 100.9098 with five years left, 100.5663 with three and 100.1959 with one. A 3.00% bond at the same yield rises from 96.3947 to 97.7560 to 99.2237. (The five-year figure is 0.0002 above the book's price of 100.9096 only because the yield is rounded to four decimals; at the unrounded 3.79855% it is exactly 100.9096.)

The same logic answers a related question: how does a bond's price depend on its maturity, holding coupon and yield fixed? For a premium bond, a longer maturity means a higher price, because the excess coupon is received for longer. For a discount bond, a longer maturity means a lower price. A bond whose coupon equals its yield is priced at par whatever its maturity.

In reality, yields do not stay fixed while time passes, and the return a bond earns over a period combines this pull to par with movement along, and of, the curve. Chapter 2 separates the pieces.

1.10 Flattening and Steepening Trades

The slope of the curve is the difference between a long-term rate and a short-term one, often the 10-year rate minus the 2-year rate. The curve steepens when that difference widens and flattens when it narrows. Either can happen with rates rising or falling:

A trader who expects a change in slope but has no view on the level of rates builds a DV01-neutral spread trade: positions in a short and a long bond sized so that a parallel move in rates leaves the portfolio's value unchanged.

• A steepener profits if the curve steepens: buy the short maturity and sell the long maturity, each sized to the same DV01.
• A flattener profits if the curve flattens: sell the short maturity and buy the long maturity.

The DV01 — the dollar value of a one-basis-point change in yield — is the subject of Chapter 3. Here it is enough that a longer bond's price moves more for each basis point. Per 100 of face value, with $\Delta y$ the change in yield as a decimal and $\Delta P$ the resulting change in price,

so a bond whose price falls by 0.045 per 100 when its yield rises one basis point ($\Delta y = 0.0001$) has a DV01 of 0.045. The validation below bumps each yield one basis point down and one up, $\Delta y = 0.0002$, and takes half the price difference.

def price(c, y, n):                                 # semiannual bond, coupon and yield in %
    return sum(c / 2 / (1 + y / 200) ** k for k in range(1, n + 1)) + 100 / (1 + y / 200) ** n

def dv01(c, n):                                     # par bond: priced at its own coupon
    return (price(c, c - 0.01, n) - price(c, c + 0.01, n)) / 2

d2, d5 = dv01(3.4744, 4), dv01(3.7993, 10)
face2 = 10e6 * d5 / d2
print(f"DV01 per 100: two-year {d2:.6f}, five-year {d5:.6f}")
print(f"buy {face2 / 1e6:.4f}M two-year against 10M five-year")
pnl = face2 / 100 * d2 * 10 + 10e6 / 100 * d5 * 5   # 2y yield -10bp, 5y yield +5bp
print(f"steepening P&L: {pnl:,.0f}")
# Output:
# DV01 per 100: two-year 0.019161, five-year 0.045150
# buy 23.5637M two-year against 10M five-year
# steepening P&L: 67,724

The validation's $67,724 differs from the hand figure by a dollar because it uses unrounded DV01s. Two caveats apply to any such trade: a DV01-neutral position is neutral only to parallel moves and only to first order, and it earns or pays the difference in carry between its two legs while it is held. Chapter 4 shows how to measure exposure to every point on the curve, not just two.

1.11 Swaps and the Par Swap Curve

In a plain-vanilla interest rate swap, two parties exchange payments on a notional principal: one pays a fixed rate, the other a floating rate — today usually compounded SOFR — for the life of the swap. No principal changes hands. The fixed rate is set so that the swap is worth zero when it is agreed.

The floating side of a swap can be valued simply: a floating-rate note that resets to the market rate each period is worth par on every reset date. So a swap's floating leg plus a notional principal at maturity is worth par, and for the swap to be worth zero, the fixed leg plus the same notional must also be worth par. That is exactly the definition of a par bond.

The swap rate for a given maturity is the par rate of the swap market's discount curve. If the swap market's discount factors were Ravensmoor's Treasury curve, the five-year swap rate with semiannual payments would be 3.7993%. (USD SOFR swaps conventionally pay the fixed leg annually; semiannual payments are a simplification here, chosen to match the Treasury bond.) In practice the swap curve is built from SOFR-based swaps and sits a little away from the Treasury curve, and the difference between the two — the swap spread — reflects the credit, liquidity and funding differences between Treasuries and swaps. But the method is the same: the market quotes swap rates, the swap rates are par rates, and bootstrapping them gives the swap discount curve used to value every other swap and derivative.

Because swap rates are quoted directly for a full range of maturities and the market is extremely liquid, the par swap curve is often the most reliable source of a term structure, and the swap curve and the Treasury curve are the two benchmark curves of the fixed-income market.

1.12 Putting It All Together: One Curve, Every Rate

Ravensmoor's ten discount factors were inherited from Financial Markets and Products II. In this chapter they have produced:

Every one of these follows from the law of one price. None depends on a model of how rates will behave; they follow from the requirement that identical cash flows have identical prices. That is why discount factors come first in fixed-income valuation, and why every later chapter of this course — returns, durations, key rates, and even the option models — begins from them.

Chapter Summary

What's Next

This chapter priced bonds. Chapter 2 asks what they earn. A bond's yield to maturity compresses the whole curve into one number, its spread measures what the market charges above a benchmark, and its realised return over a holding period depends on coupons, reinvestment and how prices move. Ravensmoor's government book earns 2.0194% over the next six months if the curve stays where it is, and exactly the six-month rate of 1.5113% if the forward rates of this chapter come true — and the difference has a name.

Concept Checks

Exam-Style Practice

Ten standalone questions in the style of the FRM Level I exam. Allow 24 minutes — the exam's pace of 2.4 minutes a question.

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