Financial Markets, Financial Terms and Concepts, and Bond Mathematics

The central question is simple: how do we compare money, assets, and risks across time? First understand where trading happens and who trades; then learn to translate future cash flows into today’s prices.

Learning goals

After this lesson, you should be able to:

Distinguish trading venues, asset classes, and participant roles.
Read a bid–ask quote and explain common financial terms.
Convert between compounding conventions and discount a cash flow.
Price a plain fixed-rate bond and distinguish coupon, current yield, and yield to maturity.
Explain yield curves, duration, and convexity without confusing them with forecasts.

1. What financial markets do

Financial markets connect people who need capital with people willing to supply it. They also let participants transfer risk and exchange existing claims.

In the primary market, an issuer raises money by selling newly issued securities, such as shares in an initial public offering (IPO) or a new bond. In the secondary market, investors trade existing securities with one another; the issuer generally receives no proceeds from those trades. Secondary-market liquidity can make primary-market fundraising more attractive.

Trading venues

Venue	How trading works	Example or intuition
Exchange	Organized trading under a venue’s rules, often using an electronic order book	Listed shares on NYSE or Nasdaq
Over-the-counter (OTC)	Counterparties trade through dealer networks or bilateral arrangements, including electronic platforms	Many bonds, FX transactions, and customized derivatives
Electronic communication network (ECN)	An electronic system matches compatible buy and sell orders	Automated matching at specified prices

These descriptions overlap: an exchange can be electronic, and OTC trading can use electronic systems. OTC does not mean informal or necessarily unregulated. The SEC’s market-center explanation describes exchanges, market makers, and ECNs.

Reading a quote

Suppose a dealer quotes 99.80 bid / 100.20 ask per share. You can sell to the dealer at the bid or buy from the dealer at the ask. The spread is 0.40. Buying and immediately selling one share loses 0.40 before fees if prices stay unchanged.

A market order prioritizes execution but does not guarantee the price. A limit order sets the highest purchase price or lowest sale price you accept, but may never execute. Liquidity means being able to transact promptly in meaningful size without moving the price much. A tight spread alone does not guarantee deep liquidity.

2. Instruments: what is being traded?

Asset or instrument	Economic meaning	Main risks to recognize
Equity	Ownership claim on a company; dividends are generally discretionary	Business losses, valuation changes, dilution
FX	Exchange of one currency for another; always think in currency pairs	Exchange-rate movements and leverage
Index / ETF	An index measures a basket; an ETF is a traded fund that can track an index or pursue another strategy	Underlying exposures, tracking differences, fees
Bond	Debt claim with contractual payments subject to its terms and issuer performance	Default, interest rates, inflation, liquidity
Mortgage / asset-backed security (ABS)	Claim supported by a pool of loans or receivables	Defaults, prepayments, structural complexity
Commodity	Physical goods or financial exposure to them, such as oil, gold, or wheat	Supply shocks, storage costs, volatility
Derivative	Contract whose value depends on an underlying asset, rate, or index	Market exposure, leverage, counterparty risk

Cryptocurrencies are digital assets; they should not automatically be treated as fiat money or cash equivalents. A bond’s payment frequency depends on its contract: some pay semiannually, some annually, and zero-coupon bonds make no periodic coupon payments.

Four derivative building blocks

Call / put option: the holder has the right to buy / sell an underlying asset at a strike price, subject to the exercise terms and expiry. The buyer pays a premium; the writer assumes the corresponding obligation if exercised.
Forward: a bilateral commitment to transact at a specified future date and price.
Future: a standardized contract traded on an exchange, typically subject to margin and daily settlement.
Swap: an agreement to exchange cash-flow streams, such as fixed-rate interest for floating-rate interest on a specified notional amount.

Example: an airline worried about higher fuel costs can hedge some of that exposure with an appropriate derivative. A hedge reduces a targeted risk but can introduce basis risk when the hedge and the actual business exposure do not move together.

3. Participants and their objectives

Role	Main purpose	Example
Hedger	Reduce an existing exposure	Exporter hedging foreign-currency receipts
Market maker	Provide buy and sell quotes and manage inventory	Dealer quoting two-way bond prices
Risk seeker / proprietary trader	Accept selected risks to pursue returns	Trader taking a view on falling interest rates

Market makers seek spread revenue, but it is not guaranteed profit. Inventory movements, adverse selection, hedging costs, and counterparty failures can exceed the spread. Fund managers can hedge, provide liquidity, or take active risks; their strategies are not all directional.

Commercial banks accept deposits and lend. Investment banks help raise capital, advise on transactions, and operate markets businesses; organizational divisions vary by firm. Asset and wealth managers invest for clients, while hedge funds use a range of active strategies. Corporates, central banks, and retail investors participate for different purposes. A central bank pursues policy objectives rather than simply maximizing trading returns.

In a typical private-fund structure, limited partners (LPs) supply capital and a general partner (GP) manages the fund under its governing terms. Do not confuse this use of LP with the abbreviation for liquidity provider.

4. Essential vocabulary and the role of mathematics

Term	Meaning
Price vs. value	Price is an observed transaction or quote; value is an estimate based on assumptions
Long / short	A long benefits from a price rise; a short generally benefits from a fall, with instrument-specific obligations
Return	Change in wealth relative to the initial investment, including distributions
Basis point (bp)	0.01 percentage point; 100 bp equals 1 percentage point
Notional / principal	Reference amount for a contract / amount lent or scheduled for repayment; these need not equal market value
Leverage	Exposure financed or created relative to capital; it magnifies losses as well as gains
Credit / counterparty risk	Risk that an issuer or contractual counterparty fails to perform
Beta / alpha	Exposure to a benchmark or risk factor / performance beyond that explained by the chosen model
Arbitrage	A strategy with no net initial investment and no possible loss, with a positive payoff in at least one relevant state

If you buy at 100, receive a dividend of 3, and sell at 105, your holding-period return before costs is (105 − 100 + 3) / 100 = 8%. Price return alone is 5%.

Mathematics helps turn assumptions into prices and measurable risks. Black–Scholes–Merton is a foundational option-pricing framework under restrictive assumptions; it is not a universal pricing formula. Portfolio optimization weighs expected returns against risks and constraints, but its output depends heavily on estimated inputs. A mathematically precise answer can still rest on a poor model.

Strategy choices include public versus private assets, direct management versus funds of funds, passive versus active investing, systematic versus discretionary decisions, trend following versus mean reversion, and short versus long horizons. Value and growth are investment styles with overlapping economic drivers, not guaranteed formulas for returns.

5. Interest: moving money forward in time

Use r for an annual rate, t for time in years, and m for compounding periods per year. Enter rates as decimals: 5% means 0.05.

Simple interest and annual compounding

Simple interest on an initial amount A gives FV = A(1 + rt). Annual compounding gives FV = A(1 + r)ᵗ for whole-year periods. Both turn 1 into 1 + r after one year, so a one-year example cannot distinguish them.

At 5% over two years, 1,000 becomes 1,100 with simple interest and 1,102.50 with annual compounding. The extra 2.50 is interest earned on earlier interest.

More frequent compounding

For a nominal annual rate r compounded m times per year:

FV = A(1 + r/m)ᵐᵗ

The effective annual rate is (1 + r/m)ᵐ − 1. At a 5% nominal rate compounded monthly, 1,000 becomes approximately 1,051.16 after one year, an effective rate of 5.116%.

As m tends to infinity, (1 + r/m)ᵐ → eʳ, giving continuous compounding:

FV = Aeʳᵗ

At a continuously compounded rate of 5%, 1,000 becomes approximately 1,051.27 in one year. The equivalent continuous rate for an effective annual rate R is ln(1 + R). Always compare rates under matching conventions.

6. Present value and no-arbitrage

Discounting reverses compounding. Suppose a claim pays a certain 1 in one year and comparable borrowing and lending are available at the same annual rate r. Its price x must satisfy:

x(1 + r) = 1, so x = 1 / (1 + r).

If the claim were cheaper, borrow its price and buy it: the payoff exceeds the loan repayment. If it were more expensive, selling the claim and investing the proceeds gives the reverse trade, assuming that sale is feasible. No-arbitrage equates the cost of replicating a payoff with the price of that payoff.

This argument assumes matching currency and timing, certainty of payment, feasible borrowing and short selling, and no taxes or transaction costs. An ordinary risky corporate promise cannot simply be discounted at a risk-free rate without adjusting for its risk.

For a cash flow N due in t years:

Annual compounding at a constant rate: PV = N / (1 + r)ᵗ.
Nominal rate compounded m times per year: PV = N / (1 + r/m)ᵐᵗ.
Continuous compounding: PV = Ne⁻ʳᵗ.

The factor multiplying N is the discount factor. For example, 1,000 payable in two years at a 5% annually compounded discount rate is worth 1,000 / 1.05² = 907.03 today.

7. Bond pricing: a worked cash-flow example

Consider a non-callable bond with face value F = 1,000, annual coupon rate c = 5%, and n = 3 years remaining. Its annual coupon C = cF = 50. Assume valuation immediately after a coupon payment, no default, annual payments, and an annual discount rate of y = 4%.

Year	Coupon	Principal	Total cash flow	Present value at 4%
1	50	0	50	48.08
2	50	0	50	46.23
3	50	1,000	1,050	933.45
Total	150	1,000	1,150	1,027.75

Totals use unrounded values. The bond is worth more than face value because its 5% coupon is attractive relative to the 4% required yield.

For an annual-pay bond:

P = Σ from k = 1 to n [C / (1 + y)ᵏ] + F / (1 + y)ⁿ

Summing the coupon geometric series, for y ≠ 0:

P = C[1 − (1 + y)⁻ⁿ] / y + F(1 + y)⁻ⁿ

At y = 0, use P = nC + F. For m payments per year and T years remaining, replace C with cF/m, y with y/m, and n with mT, where y is now the nominal annual yield compounded m times per year. This assumes a whole number of regular periods.

Between coupon dates, the dirty price is the full settlement price used in present-value calculations. The clean price excludes accrued interest: dirty price = clean price + accrued interest. Actual pricing also requires the bond’s day-count and settlement conventions.

8. Coupon rate, current yield, and yield to maturity

The coupon rate determines contractual coupon payments relative to face value. Current yield is annual coupon income divided by the current price. Yield to maturity (YTM) is the single discount rate that makes the present value of scheduled payments equal to the full market price under a specified compounding convention.

For the example bond:

Coupon rate: 50 / 1,000 = 5%.
Current yield: 50 / 1,027.75 ≈ 4.865%.
YTM: 4%, because that rate reproduces the price from all cash flows.

YTM usually requires numerical root-finding rather than a simple rearrangement. It includes the effect of the price moving toward principal repayment, unlike current yield. It is a contractual cash-flow internal rate of return, not a guaranteed realized return: default, early sale, costs, and coupon reinvestment outcomes matter. Realizing the quoted yield as a compounded terminal-wealth return requires coupons to be reinvested at that yield.

Why price and yield move in opposite directions

Annual YTM	Price of the same three-year 5% bond
4%	1,027.75
5%	1,000.00
6%	973.27

A larger discount rate reduces every positive future cash flow’s present value. For a standard fixed-rate bond valued on a coupon date, coupon rate above YTM means a premium; coupon rate equal to YTM means par; coupon rate below YTM means a discount. This comparison requires matching payment and yield conventions.

9. Yield curves: different dates, different rates

A yield curve relates a specified yield measure to maturity. Distinguish a spot curve of zero-coupon rates, a par curve of coupon rates that price hypothetical bonds at par, and a curve of observed bond YTMs. They are not interchangeable.

With annual spot rates sₖ, value certain annual cash flows using:

P = Σ CFₖ / (1 + sₖ)ᵏ

Equivalently, P = Σ CFₖ × D(0,k), where D(0,k) is the discount factor for year k. A bond’s YTM compresses these differently discounted payments into one number; it does not imply the whole curve is flat.

An upward-sloping curve has higher longer-term yields; an inverted curve has lower longer-term yields over the maturities being compared. Expectations about future short rates and term premia both influence the shape. Central banks strongly influence short-term rates, but do not mechanically set all bond yields.

Inversion has historically been studied as a recession indicator, especially for US Treasury spreads, but it gives neither a certain outcome nor an exact date. The Federal Reserve Bank of New York’s yield-curve research explains the probabilistic interpretation.

10. Duration and convexity: measuring price sensitivity

Keep the example bond and treat P as a function of its annual yield y. These measures assume fixed cash flows and changes in the bond’s own yield.

Duration: the slope

Modified duration Dₘₒd = −(1/P) × dP/dy

For a small yield change Δy:

ΔP/P ≈ −Dₘₒd × Δy

The minus sign captures the inverse relationship. A modified duration of 2.75 means a 1 percentage point yield rise implies approximately a 2.75% price fall. Use Δy = 0.01, not 1.

Macaulay duration is the present-value-weighted average time to cash flows:

Dₘₐc = Σ [k × CFₖ/(1 + y)ᵏ] / P

For annual payments, Dₘₒd = Dₘₐc/(1 + y). With m payments a year and nominal annual yield y, the relationship is Dₘₒd = Dₘₐc/(1 + y/m), measuring payment times in years.

Convexity: the curvature

Convexity K = (1/P) × d²P/dy²

Adding the second-order term gives:

ΔP/P ≈ −Dₘₒd × Δy + ½K(Δy)²

For annual fixed cash flows, compute K = Σ [k(k + 1)CFₖ/(1 + y)ᵏ⁺²] / P. Convexity describes curvature; it is not simply the derivative of normalized duration, because the normalization by P also changes with yield.

At a 4% yield, the example bond has Macaulay duration about 2.861 years, modified duration about 2.751, and convexity about 10.409. A rise from 4% to 5% gives:

Duration-only estimate: −2.751%.
Duration plus convexity: −2.751% + 0.052% ≈ −2.699%.
Exact repricing: 1,000 / 1,027.75 − 1 ≈ −2.700%.

For ordinary option-free bonds with positive cash flows, positive convexity makes the price gain from a yield fall larger than the loss from an equal yield rise, locally. Callable bonds can exhibit negative convexity. Non-parallel curve changes, credit-spread movements, and changing cash flows require more detailed risk measures. See CFA Institute’s duration overview.

11. Check your understanding

A dealer quotes 50.00 / 50.10. What happens if you buy and immediately sell 20 shares at those unchanged quotes?
What is the present value of 1,000 payable in two years at a 5% annual discount rate?
A one-year zero-coupon bond pays 1,000 and costs 950. What is its annual YTM?
Why does a 5% coupon bond trade above par when its matching YTM is 4%?
A bond has modified duration 6. Estimate the price effect of a 25 bp rise in yield, ignoring convexity.
Does an inverted yield curve guarantee recession? Does a market maker eliminate all trading risk?

Answers

You pay 1,002 and receive 1,000: a 2 loss before fees.
907.03, from 1,000 / 1.05².
5.263%, from 1,000 / 950 − 1.
Its contractual coupon exceeds the required yield, so investors pay a premium for those cash flows.
Approximately −1.5%, from −6 × 0.0025.
No to both. Inversion is a probabilistic signal; dealers still face inventory and other risks.