The Formula Comes After the Question
Fixed-income questions feel overwhelming when every equation sits in one pile. The better approach is to ask what job the formula is supposed to do. A bond prompt usually asks for one of five tasks:
- Price the promised cash flows.
- Solve the discount rate implied by the price.
- Separate clean price, full price, and accrued interest.
- Estimate percentage price sensitivity to a yield change.
- Convert percentage sensitivity into dollar risk.
The same bond facts can support different formulas. A five-year coupon bond with a quoted price might be used for yield to maturity, modified duration, accrued interest, or convexity. The task word tells you which formula family to use.
Price: Discount the Promised Cash Flows
The basic bond price is the present value of coupons plus principal. For an option-free fixed-rate bond, the structure is straightforward:
Price = sum of PV of coupon payments + PV of principal
Worked Example: Bond Price
Arbor Rail issues a 4-year annual-pay bond with a 5% coupon and par value of 1,000. If the required yield is 6%, the cash flows are:
- Year 1: 50
- Year 2: 50
- Year 3: 50
- Year 4: 1,050
The price is:
50 / 1.06 + 50 / 1.06^2 + 50 / 1.06^3 + 1,050 / 1.06^4 = 965.35
Use this formula family when the prompt gives the required yield and asks for value or price.
Yield: Solve the Discount Rate Implied by Price
Yield to maturity is the internal rate of return that equates the bond's price with its promised cash flows, assuming the bond is held to maturity and payments are made as promised.
If the same Arbor Rail bond trades at 965.35, the yield to maturity is the discount rate that makes the present value of 50, 50, 50, and 1,050 equal 965.35. In this example, that rate is 6%.
Use yield logic when the prompt gives price and cash flows, then asks for the return implied by the price. Do not use duration to solve yield. Duration estimates sensitivity after a yield is known or assumed.
Clean Price, Full Price, and Accrued Interest
Bond quotes often use clean price, but settlement uses full price.
Full price = clean price + accrued interest
Accrued interest compensates the seller for coupon interest earned since the last coupon date. The buyer pays full price because the buyer will receive the next coupon even though the seller owned the bond for part of the coupon period.
Worked Example: Accrued Interest
Beacon Water Authority has a semiannual-pay bond with a 6% annual coupon on par value of 1,000. Each semiannual coupon is 30. If settlement occurs 80 days into a 180-day coupon period:
Accrued interest = 30 x 80 / 180 = 13.33
If the clean price is 101.20 per 100 of par, the full price is:
101.20 + 1.333 = 102.533 per 100 of par
The clean price is useful for quoted market comparison. The full price is the economic settlement amount.
Modified Duration: First-Order Price Sensitivity
Modified duration estimates the approximate percentage price change for a small change in yield:
Approximate percentage price change = -modified duration x change in yield
The yield change must be expressed as a decimal. A 40 basis point increase is 0.0040.
Worked Example: Duration Estimate
Coral Ridge Utilities has a bond priced at 104.00 with modified duration of 5.25. If yield rises by 40 basis points:
Approximate percentage price change = -5.25 x 0.0040 = -0.0210
The bond's price is expected to fall by about 2.10%. The approximate new price is:
104.00 x (1 - 0.0210) = 101.82
Use modified duration when the question asks for a small yield-change estimate and does not introduce changing cash flows or embedded-option behavior.
Convexity: Improve the Estimate for Larger Yield Moves
Duration is a first-order estimate. Convexity adds a second-order adjustment:
Approximate percentage price change = -modified duration x change in yield + 0.5 x convexity x (change in yield)^2
Convexity matters more when the yield change is larger or when the question specifically gives convexity.
Worked Example: Duration Plus Convexity
Meridian Storage owns a bond with modified duration of 6.80 and convexity of 54. The yield decreases by 75 basis points, or -0.0075.
Duration effect:
-6.80 x -0.0075 = 0.0510
Convexity effect:
0.5 x 54 x (-0.0075)^2 = 0.0015
Estimated percentage price change:
0.0510 + 0.0015 = 0.0525, or about 5.25%.
The convexity term is positive for both yield increases and yield decreases when convexity is positive. That is why duration-only estimates understate the price gain when yields fall and overstate the price loss when yields rise for a positively convex bond.
PVBP: Turn Sensitivity Into Dollar Risk
Portfolio managers often need a currency amount, not just a percentage. Price value of a basis point estimates the price change for a one-basis-point yield move.
For a bond priced per 100 of par:
PVBP per 100 par = full price x modified duration x 0.0001
For a portfolio:
Portfolio PVBP = market value x modified duration x 0.0001
Worked Example: Portfolio PVBP
Northstar Income Fund holds 18,000,000 market value of bonds with modified duration of 4.60.
PVBP = 18,000,000 x 4.60 x 0.0001 = 8,280
A one-basis-point rise in yield would reduce value by approximately 8,280, before convexity and spread effects.
Exam Framing: Match Task Words to Formula Families
If the Prompt Says Price or Value
Discount cash flows using the stated required yield or spot rates. If each cash flow has a different spot rate, discount each cash flow separately.
If the Prompt Says Yield
Solve the rate that makes price equal present value of promised cash flows. Treat yield to maturity as an IRR concept.
If the Prompt Says Clean, Full, Dirty, or Settlement
Separate quoted clean price from accrued interest. Settlement uses full price.
If the Prompt Says Approximate Price Change
Use modified duration for a small yield change. Add convexity if the prompt gives convexity or the yield change is large enough that curvature matters.
If the Prompt Says Dollar Value, PVBP, or Hedge Size
Scale duration by market value and the yield change. This is the risk-budgeting version of duration.
Bottom Line
Fixed-income formulas are not random tools. Each formula answers a different question about the same cash flows. Start by naming the task, then choose the formula family. That habit prevents the most common exam error: using a familiar equation for the wrong job.