1 Learning roadmap
How much should you pay today for a stream of future bond payments? Start with the cash flows, discount them consistently, then ask which yield describes the return and how that price changes when yields move.
This guide uses the supplied bond-prices workbook as teaching notes. All dollar examples are hypothetical, before fees and taxes unless explicitly stated. The regular-period models assume settlement on a coupon date, no accrued interest, no default and payment at the end of each period.
2 Inputs, cash flows and conventions
| Symbol | Meaning | Workbook pricing example |
|---|---|---|
| F | Face value | $1,000 |
| c | Annual coupon rate | 5.50% |
| m | Payments per year | 2 |
| T | Years to maturity | 10 |
| n = mT | Number of periods | 20 |
| C = Fc/m | Coupon each period | $27.50 |
| y | Nominal annual yield, compounded m times | 4.75% |
| y/m | Discount rate per period | 2.375% |
The first 19 payments are $27.50. The twentieth is $1,027.50, combining coupon and redemption. Coupon rate applies to face value; required yield discounts the payments. They play different roles.
3 Price as present value
Discount every payment to the same valuation date. For regular equal periods:
For $1,000 face, a 5.50% coupon, ten years and semiannual payments, a 4.75% required yield gives $1,059.16. Excel: =-PV(4.75%/2,20,27.5,1000).
At y = 0, the price is simply nC + F; do not divide by zero in an annuity shortcut. Negative yields can also occur, provided the periodic discount factor remains positive. The interactive model below supports nominal yields above −5% for illustration.
4 Premium, par and discount
| Regular fixed-rate bond redeeming at par | Price | Yield ordering |
|---|---|---|
| Coupon rate > YTM | Premium: P > F | YTM < current yield < coupon rate |
| Coupon rate = YTM | Par: P = F | All three equal |
| Coupon rate < YTM | Discount: P < F | Coupon rate < current yield < YTM |
These standard orderings assume positive yields, positive coupons, the same annual quotation convention and the regular-period setup. They are not universal rules for all embedded-option or irregular-payment instruments.
A $1,000 ten-year 5% semiannual bond is worth $1,081.76 at a 4% yield, $1,000.00 at 5%, and $925.61 at 6%. Unchanged promised payments become less valuable when their discount rate rises.
As maturity approaches, price tends toward the redemption amount if the yield and credit assumptions remain consistent. A premium’s amortisation or a discount’s accretion is only one component of total return.
5 Coupon yield, current yield and YTM
| Measure | Calculation | What it omits |
|---|---|---|
| Coupon / nominal yield | Annual coupon ÷ face value | Purchase price and capital gain or loss |
| Current yield | Annual coupon ÷ quoted market price | Timing, redemption and reinvestment |
| Yield to maturity | IRR equating price to promised payments | Default and actual reinvestment outcomes |
The pricing example’s current yield is $55 / $1,059.16 = 5.1928%, while its contractual coupon is 5.50% and its YTM is 4.75%.
YTM is the discount rate implied by the price and promised cash flows. It is not a guaranteed realised return. Receiving all payments and reinvesting interim coupons at that yield are needed to obtain the same compounded terminal wealth over the full horizon.
6 Yield to call and yield to worst
YTC uses a specified call date and redemption price. Evaluate each relevant contractual call scenario, rather than assuming the earliest date is always the worst. YTW is the lowest yield among the applicable redemption scenarios, conventionally including maturity.
Workbook callable example: pay $1,060 for $1,000 face, 6% coupon, semiannual payments, ten-year maturity, and a call in five years at $1,030. YTM = 5.2221%; five-year YTC = 5.1570%; the lower of these two is 5.1570%. The last call-date receipt includes the $30 coupon plus $1,030 redemption.
A lower YTC does not establish that the issuer will call. Exercise depends on refinancing conditions, costs and contract terms. YTW is not a default-loss floor and does not protect against losses on an early sale.
Excel: =RATE(20,30,-1060,1000)*2 for YTM; =RATE(10,30,-1060,1030)*2 for this call. Review callable-bond mechanics.
7 Zero-coupon and effective annual yields
A zero-coupon bond has a single future receipt: P = F/(1+y/m)mT. Its effective annual yield is (1+y/m)m − 1.
For $1,000 due in ten years at a 5% nominal yield compounded semiannually, price = $610.27. The effective annual yield is 1.025² − 1 = 5.0625%, not 5.00%. Both describe the same discounting with different conventions.
Compare investments only after aligning compounding, day-count, currency, maturity, options and credit. A money-market discount quotation on face value is not directly comparable with an investment yield on purchase price.
8 Real yield and tax-equivalent yield
For a consistent nominal return r and inflation rate π over the same period, real return = (1+r)/(1+π) − 1. Subtracting inflation is a useful approximation, not the exact Fisher relation.
At a 5.10% nominal return and 3.00% inflation, exact real return is 1.051/1.03 − 1 = 2.0388%. The subtraction approximation gives 2.10%.
Under a simple single-tax comparison, taxable-equivalent yield = exempt yield / (1−t). A hypothetical 3.80% fully exempt yield and 35% marginal rate give 5.8462%. This assumes the specified exemption actually applies and otherwise comparable risk.
Do not automatically combine tax rates using a deductibility formula. Actual exemptions, deduction limits, investor status and jurisdiction matter. The workbook’s federal/state calculation is a conditional teaching assumption, not a universal tax rule.
9 Clean price, dirty price and holding return
Dirty settlement price = clean quoted price + accrued interest. On a coupon date immediately after coupon payment, accrued interest is normally zero. Between dates, the market’s day-count and ex-coupon rules must be applied.
If a regular $25 semiannual coupon has accrued for 60 of 180 days under an assumed straight-line convention, accrued interest is $8.33. A $980 clean quote implies a $988.33 dirty price. The ratio 60/180 is an explicit illustration, not a universal convention.
Holding-period return = (cash income received + ending dirty value − beginning dirty value) / beginning dirty value, before costs and taxes. Include reinvestment income when relevant and align treatment of coupon-date cash.
Pay $980, receive $50 over one year and sell for $990 with zero accrued interest at both endpoints: holding-period return = ($50+$990−$980)/$980 = 6.1224%. It need not equal the original YTM.
10 Interactive pricing and yield laboratory
Change the regular-period inputs below. Outputs recompute from the cash flows; rates are nominal annual percentages at the selected frequency. The chart shows exact prices for yields from 0% to 10%.
11 Duration, convexity and DV01
Macaulay duration is the present-value-weighted average time to payment. Modified duration is Macaulay duration divided by (1+y/m). It measures local proportional price sensitivity to a change in the quoted yield. Convexity adds the curvature term.
DV01 ≈ P × Dmod × 0.0001
A basis point is 0.01 percentage point: +100 bps is Δy = +0.01. DV01 is often reported as a positive sensitivity magnitude; a positive yield shock usually produces a negative price change for an option-free bond.
Use the derivative convention matching the quoted annual yield. Duration has units of years and convexity years squared under this convention. Scalar duration assumes a common yield shift; non-parallel curve moves require more detailed measures such as key-rate durations.
12 Short-cut repricing method
Reprice around the base yield with a small symmetric bump h. Let V− be price at y−h, V₀ at y and V+ at y+h.
C ≈ (V− + V+ − 2V₀) / (V₀h²)
For the workbook’s separate $1,000 ten-year 5% semiannual bond at 5%, use h = 0.0025 (25 bps). V− = $1,019.7185; V₀ = $1,000; V+ = $980.7417. Repriced duration = 7.795371; convexity = 73.633054.
The workbook calls these effective measures. For fixed cash flows bumped by a single YTM, they are finite-difference approximations to modified duration and yield convexity. True option-sensitive effective measures require a valuation model that allows cash flows or exercise decisions to respond to curve shocks. A plain PV formula cannot supply that optionality model.
13 Separate duration and convexity effects
Keep the three steps visible: duration contribution, convexity contribution, then their sum. Multiply the total percentage change by the starting price to estimate the new value.
| Step | Workbook base case, +100 bps |
|---|---|
| Duration contribution | -7.795371% |
| Convexity contribution | 0.368165% |
| Total estimated change | -7.427206% |
| Estimated new price | $925.7279 |
| Exact new price | $925.6126 |
| Error: estimated − exact | $0.1153 |
The following results use the current laboratory bond, which defaults to the 5.5% coupon pricing example. Set coupon and yield to 5% to reproduce the table above.
Approximation error grows for large shocks. For a conventional option-free bond, positive convexity improves the local estimate. Callable bonds may show negative convexity in relevant regions; do not apply this fixed-cash-flow model to them without adjustment.
14 Yield curves and spread measures
| Curve | Description | Interpretation limit |
|---|---|---|
| Upward-sloping | Long yields exceed short yields | Expectations and term premiums both matter |
| Inverted | Some shorter yields exceed longer yields | A slowdown signal in some settings, not a timed forecast |
| Flat | Similar yields across the compared maturities | Depends on which curve and maturities are measured |
| Humped | Intermediate yields exceed both ends | A single 10Y−2Y number cannot describe the full shape |
Distinguish a par-yield curve, a spot curve and a forward curve. Cash-flow valuation with a term structure discounts each payment at its matching spot rate, rather than assuming one YTM applies economically to every horizon.
| Spread | Reference | Limit |
|---|---|---|
| G-spread | Government benchmark yield | Align maturity and conventions |
| I-spread | Interpolated swap rate | Swap benchmarks bring their own market influences |
| Z-spread | Constant spread applied to a spot curve | Convention-dependent; not a pure default-risk measure |
| OAS | Model-based spread allowing for embedded options | Depends on volatility, exercise model, credit and liquidity |
OAS is not always obtained by a simple arithmetic subtraction of a universal “option cost.” See Spot Rates vs Forward Rates for the discount-factor foundation.
15 Excel blueprint and knowledge check
| Task | Excel expression | Interpretation |
|---|---|---|
| Price | = -PV(4.75%/2,20,27.5,1000) | Regular semiannual periods |
| YTM | =RATE(20,30,-1060,1000)*2 | Nominal annual quotation |
| YTC | =RATE(10,30,-1060,1030)*2 | Five-year call scenario |
| Effective annual yield | =(1+5%/2)^2-1 | 5.0625% |
| Duration contribution | =-Duration*Shock | Shock entered as decimal |
| Convexity contribution | =0.5*Convexity*Shock^2 | Use matching annual-yield convention |
| Estimated price | =BasePrice*(1+DurationEffect+ConvexityEffect) | Both effects are fractional changes |
| Estimation error | =EstimatedPrice-ExactPrice | Keep the sign explicit |
Why does price fall when yield rises?
The same future payments are discounted more heavily.
Can current yield measure the full maturity return?
No. It omits redemption value, timing and reinvestment.
Is yield to worst a guaranteed minimum return?
No. Default or an early sale can produce worse outcomes.
Does 5% semiannual yield equal a 5% effective annual yield?
No. Its effective annual rate is 5.0625%.
Why compare estimated and exact prices?
Duration and convexity are local approximations; repricing reveals their error.
Can one average the YTMs of several bonds to get portfolio IRR?
Not generally. Portfolio IRR requires aggregated dated cash flows and the total investment, not a simple average of yields.
16 References and next steps
Teaching reference: Bond_prices_and_yields_Excel_workbook_request-Genspark_AI_Sheets-20260912_1743.xlsx, particularly Overview & Guide; Pricing & Yield Calculators; Price-Yield & Convexity; and Yield Curves & Markets. Numerical examples are recalculated from the stated inputs. Workbook activity logs are not part of this lesson.
Issuer-specific examples and undated market ranges in the notes are not represented here as current quotes, ratings or investment recommendations.
Continue with Types of Bonds, Duration and Convexity, or return to Fixed Income Securities.