Mechanics of Price Correction

In earlier lessons, we focused on lending and borrowing at a conceptual level and introduced bonds as an alternative to traditional loans. We now shift from why bonds exist to how they are valued and traded in real markets. This lesson develops three core ideas:

1. How bond prices are determined using the time value of money
2. How yields emerge from market prices
3. How bonds are actually traded, including the role of dealers and bid-ask spreads

By the end of this lesson, you should understand not only how to compute a bond's value, but also why its market price may differ from its face value and how intermediaries profit from facilitating trades.

A bond is not a single payment – it is a sequence of promised cash flows occurring at specific dates. These typically include:

• Periodic coupon payments
• A final repayment of principal at maturity

Because these payments occur at different points in time, they cannot simply be added together. Money received in the future is worth less than money received today, a principle you already know as the time value of money.

To compare cash flows occurring at different times, we convert them into their present value using an appropriate discount rate.

Suppose you expect to receive $1,000 one year from now, and the relevant annual interest rate is 4%. The present value of that future payment is:

If instead you expect to receive $3,000 in two years, discounted at the same rate, its present value is:

This means that receiving $3,000 two years from now is equivalent to receiving approximately $2,775.15 today, given a 4% discount rate.

Each future cash flow has its own discount factor depending on how far into the future it occurs. Bond pricing simply applies this logic repeatedly to every payment the bond makes.

Credit risk is the risk that a lender does not receive the full amount of principal and interest promised under a contract. It arises from uncertainty regarding the borrower’s ability – or willingness – to pay.

Credit risk exists only when there is a contractual obligation. This distinction matters. Equity investors face market risk, not credit risk, because dividends and share prices are not contractually guaranteed.

When credit risk is present, future cash flows become probabilistic rather than certain. Payments may arrive late, arrive partially, or fail altogether. This uncertainty transforms otherwise deterministic valuation problems into problems of expected value and loss.

Consider a bond with the following terms:

• Face value: $1,000,000
• Coupon rate: 4.5%
• Coupon frequency: Annual
• Time to maturity: 5 years
• Current market interest rate: 5%

Step 1: Identify Cash Flows

The bond pays:

$45,000 each year for five years

An additional $1,000,000 at maturity

Step 2: Discount Each Cash Flow

Each payment is discounted at the prevailing market rate of 5%. The bond’s price today is the sum of the present values of all these payments.

When this calculation is carried out, the bond’s price is approximately $978,353.

Because this price is below the bond’s face value, the bond is said to be trading at a discount (or below par).

The reason this bond trades below par is straightforward: its coupon rate (4.5%) is lower than the current market rate (5%). Investors will not pay full face value for a bond that offers lower income than newly issued alternatives.

This relationship generalizes:

• Higher market interest rates → lower bond prices
• Lower market interest rates → higher bond prices

This inverse relationship is one of the most important principles in fixed-income markets.

The yield to maturity answers a different question:

Given the bond’s current price, what return will an investor earn if the bond is held to maturity?

If a bond trades exactly at face value, its YTM equals its coupon rate. However, when a bond trades above or below par, the yield must be inferred from its price.

Example

• Face value: $1,000
• Coupon rate: 4%
• Price today: $950
• Maturity: 4 years

The yield to maturity is the discount rate that makes the present value of all future payments equal to $950. This rate cannot be solved algebraically and is typically found using numerical methods or a spreadsheet. In this case, the YTM is approximately 5.42%, higher than the coupon rate because the bond was purchased at a discount.

Many bonds pay coupons more frequently than once per year, commonly semi-annually. The pricing logic remains unchanged, but the mechanics adjust:

• Coupon and yield are divided by the number of payments per year
• The total number of discount periods increases

While the formula becomes more complex, spreadsheets make these calculations straightforward and are the standard tool used in practice.

Not all bonds make periodic interest payments. Zero-coupon bonds pay no coupons and instead are issued at a discount to face value.

Example:

• Face value: $1,000
• Maturity: 10 years
• Price today: $630

The yield is determined entirely by the difference between purchase price and maturity value. Zero-coupon bonds are especially useful because their yields directly reveal market discount rates for specific maturities. These rates are foundational in constructing yield curves and discounting more complex cash flow streams.

Two bonds with the same maturity can have different yields if their issuers have different probabilities of default. Government bonds issued by stable sovereigns are often treated as risk-free, while corporate bonds carry credit risk.

Investors demand additional yield – called a credit spread – to compensate for this risk. The lower the perceived credit quality of the issuer, the higher the yield investors will require and the lower the bond's price will be.

Financial markets are commonly divided into two groups:

• Buy-side institutions manage money on behalf of clients (e.g., pension funds, mutual funds, asset managers). They decide what to buy and sell based on investment objectives.
• Sell-side institutions act as intermediaries, providing liquidity by standing ready to buy and sell securities.

The sell-side does not primarily invest for long-term returns; its role is to facilitate trading.

A market maker typically quotes two prices:

• Bid: the price at which they will buy
• Ask (offer): the price at which they will sell

The difference between these prices is the bid–ask spread, which compensates the dealer for providing liquidity, bearing inventory risk, and covering transaction costs.

This mechanism is identical to what you observe at foreign exchange kiosks, where currencies are bought at one price and sold at another. In bond markets, bid–ask spreads are the primary source of revenue for sell-side market makers.

Because market makers must always be prepared to sell, even when they lack inventory, they often engage in short selling – selling a bond they do not currently own and borrowing it later to complete delivery.

Understanding short selling completes the picture of how modern bond markets function, and it will be the focus of the next lesson.