Key takeaways (short)
– A discount bond is any bond that sells for less than its face (par) value.
– Causes include higher prevailing interest rates, issuer credit problems, or bond structure (e.g., zero-coupon bonds).
– Yield-to-maturity (YTM) converts price, coupons, and time to maturity into an annualized return estimate.
– Check issuer credit, maturity, coupon type, liquidity, callable features, and tax treatment before buying.
Definition and basic mechanics
– Bond (or fixed‑income security): a loan from investor to issuer. The issuer promises periodic interest (coupon) and repayment of face value (par) at maturity.
– Face value / par: amount returned at maturity (commonly $1,000).
– Coupon: stated periodic interest payment (expressed as a percentage of par).
– Discount bond: a bond trading below par. It can be issued below par or trade below par later in the secondary market.
– Deep‑discount bond: a discount at a large magnitude (commonly described when price is 20% or more below par).
– Distressed bond: a bond trading at a steep discount because the issuer has a high risk of default.
Why bonds trade at a discount
– Interest‑rate moves: bond prices and market interest rates move inversely. If market rates rise above a bond’s coupon rate, that bond’s price will fall below par so its effective yield aligns with current rates.
– Credit concerns: worsening issuer credit or default risk lowers price.
– Structure: zero‑coupon bonds pay no periodic coupons and are issued at large discounts; they accrete in price toward par as maturity approaches.
Yields and key metrics (simple definitions)
– Current yield: annual coupon / current price. Measures income-only return.
– Yield to maturity (YTM): the annualized return if you hold the bond to maturity, assuming coupons are reinvested at the YTM and the issuer repays par. YTM incorporates price, coupon amounts, time to maturity, and par.
– For zero‑coupon bonds, YTM can be computed directly from price and time to maturity since there are no coupons.
Worked numeric example (step‑by‑step)
Example A — coupon bond sold at a discount
– Parameters: face value F = $1,000; price P = $950; coupon = 4% annually (so annual coupon C = $40); years to maturity n = 3.
– Current yield = C / P = 40 / 950 = 0.0421 = 4.21%.
– Approximate YTM (common approximation formula):
YTM ≈ [C + (F − P) / n] / [(F + P) / 2]
Plugging numbers: YTM ≈ [40 + (1000 − 950)/3] / [(1000 + 950)/2]
≈ (40 + 16.6667) / 975
≈ 56.6667 / 975 ≈ 0.0581 = 5.81% (annualized)
– Interpretation: You earn 4.21% in current income, and the remaining gain to reach par at maturity raises your overall expected annual return to about 5.8% (approximation).
Example B — zero‑coupon bond
– Parameters: F = $1,000; P = $800; n = 5 years; no coupons.
– Exact YTM = (F / P)^(1/n) − 1 = (1000/800)^(1/5) − 1 ≈ 1.25^(0.2) − 1 ≈ 0.0456 = 4.56% per year.
– Interpretation: All return is capital appreciation from $800 to $1,000 over 5 years.
Advantages and disadvantages (concise)
Advantages
– Potential capital gain: buy below par and receive full par at maturity (if issuer does not default).
– Higher yield potential versus a comparable premium bond (because price is lower).
– Availability across maturities—can fit short- or long‑term strategies.
– Regular coupon income if not zero‑coupon.
Disadvantages / risks
– Interest rate risk: rising
rising interest rates reduce the market price. This price volatility is larger for longer maturities and for bonds with lower or no coupon (zero‑coupon bonds have the highest interest‑rate sensitivity).
– Reinvestment risk: if the bond pays coupons, you must reinvest those coupon payments. Lower reinvestment rates reduce realized returns. Zero‑coupon bonds avoid coupon reinvestment risk because they have no interim cash flows.
– Inflation risk: fixed nominal payments lose purchasing power if inflation rises.
– Credit/default risk: the issuer might default or be downgraded, reducing both price and recovery at maturity.
– Liquidity risk: some discount bonds trade thinly; selling before maturity can incur wide bid‑ask spreads or lower prices.
– Call risk (if callable): an issuer can redeem the bond early when rates fall, capping upside for the investor.
– Tax/timing complications (U.S. context): original‑issue discount (OID) often creates imputed taxable interest each year even if no cash is received. Consult tax rules for your jurisdiction.
How to calculate returns and yields (practical steps)
1) Current yield — quick measure of income:
– Formula: Current yield = Annual coupon payment / Current price.
– Use: compares coupon income to price; ignores capital gain/loss and time to maturity.
2) Yield to maturity (YTM) — the standard total-return measure:
– Definition: the single annualized discount rate that equates the present value of all promised cash flows (coupons and principal) to the current price.
– Exact formula: solve P = sum_{t=1..n} C / (1 + y)^t + F / (1 + y)^n for y,
where P = price, C = annual coupon, F = face (par) value, n = years to maturity.
– Practical solution: use a financial calculator, Excel/Google Sheets, or numerical solver (no closed‑form for coupons).
Worked numeric example — coupon discount bond
– Parameters: F = $1,000; P = $950; coupon rate = 3% (annual), so C = $30; n = 5 years.
– Approximate
Approximate YTM — use the common shortcut formula that treats coupons and capital gain/loss evenly across the life of the bond:
y ≈ (C + (F − P)/n) / ((F + P)/2)
Plugging our numbers:
– C = $30
– F − P = $1,000 − $950 = $50
– n = 5
– (F + P)/2 = (1,000 + 950)/2 = 975
y ≈ (30 + 50/5) / 975 =
40 / 975 ≈ 0.04103, or about 4.10%.
Exact YTM (numerical solution)
– The shortcut above gives a quick approximation. To get the exact yield-to-maturity (YTM) you must solve the bond pricing equation for y:
P = C * [1 − (1 + y)^(−n)] / y + F * (1 + y)^(−n)
where P = 950, C = 30, F = 1,000, n = 5. There is no closed-form algebraic solution when coupons exist, so use a numerical solver, financial calculator, or spreadsheet.
Worked numeric solution (iterative/Excel)
1. Spreadsheet (Excel/Google Sheets): use the RATE function.
– Example: =RATE(5, 30, -950, 1000)
– This returns y ≈ 0.04128 (about 4.13% annual).
2. Financial calculator: enter N = 5, PMT = 30, PV = −950 (cash outflow), FV = 1000; compute I/Y ⇒ ≈ 4.13%.
3. Quick manual check (interpolation): at y = 4.1025% PV ≈ 951.2; at y = 4.15% PV ≈ 949.0, so true y ≈ 4.13%.
Interpretation and comparisons
– YTM ≈ 4.13% is the annualized internal rate of return if you buy at 950, hold to maturity, and receive promised coupons and par.
– Current yield = C / P = 30 / 950 ≈ 3.16%. Current yield measures only coupon income and ignores capital gain/loss (here, the capital gain of 50 when principal rises from 950 to 1,000).
– Because coupon rate (3.00%) 1; you must use a root-finding method, financial calculator, or spreadsheet.
– Quick approximation: use the bond-yield approximation
y_approx ≈ (C + (F − P)/n) / ((F + P)/2)
This gives a fast, reasonably accurate estimate (best when price is near par and n is not very large).
– Period adjustment: if coupons are semiannual, solve for the semiannual yield y_s and annualize:
y_annual = 2 × y_s (for nominal APR convention) or use (1+y_s)^2 − 1 for effective annual yield.
3. Calculator / spreadsheet recipes (worked example)
Assumptions carried forward: F = 1,000 (par), P = 950 (clean price), annual coupon rate = 3% → C = 30, remaining n = 5 years.
A. Exact (financial calculator)
– Enter N = 5, PV = −950 (money paid), PMT = 30, FV = 1000, compute I/Y → I/Y ≈ 4.13% (annual).
B. Excel (RATE function — returns periodic rate)
– Formula: =RATE(n, -PMT, -PV, FV)
– Example: =RATE(5, -30, -950, 1000) → ≈ 0.0413 → annual YTM ≈ 4.13%.
C. Approximation (quick check)
– y_approx = (C + (F − P)/n) / ((F + P)/2)
– Plug numbers: (30 + 50/5) / ((1000 + 950)/2) = (30 + 10) / 975 = 40 / 975 ≈ 0.04103 ≈ 4.10%.
– Note: approximation (4.10%) is close to the exact 4.13% and is useful for sanity checks.
4. Practical checklist when reporting or using YTM
– Confirm whether yield is nominal (APR) or effective annual yield.
– Confirm coupon periodicity (adjust n, C, and y accordingly).
– Ensure consistent sign convention in calculators/spreadsheets (cash paid is negative, cash received is positive).
– Use clean price for market-quoted yields; add accrued interest if you need the cash settlement (dirty price).
– Remember YTM assumes reinvestment of coupons at the same YTM and no default; realized return can differ.
5. Interpreting a discount bond
– Price below par indicates market yield > coupon rate.
– Sources of discount: higher prevailing interest rates since issue, increased credit risk, or longer duration until maturity.
– An investor buying at a discount expects a capital gain (par − price) plus coupon income if bond is held to maturity and issuer does not default.
– Key risks: interest-rate risk (price sensitivity), reinvestment risk (coupons reinvested at lower/higher rates), and credit/default risk.
6. Common pitfalls and quick troubleshooting
– Using current yield (C/P) as a substitute for YTM: current yield ignores capital gain/loss and term structure.
– Forgetting semiannual compounding when coupon payments are semiannual — this leads to mis-stated yields.
– Mixing clean and dirty prices when computing yields or comparing quotes.
– Relying solely on approximate formulas for large discounts/premiums or long maturities — use an exact solver.
Summary numeric check (ties back to earlier numbers)
– Given P = 950, C = 30, F = 1000, n = 5:
– Current yield = 30 / 950 ≈ 3.16% (coupon income only).
– Approximate YTM ≈ 4.10% (quick formula).
– Exact YTM (solve PV equation) ≈ 4.13% (annualized internal rate of return if held to maturity and coupons reinvested at YTM).
Further reading (reputable sources)
– Investopedia — Discount Bond: https://www.investopedia.com/terms/d/discountbond.asp
– U.S. Securities and Exchange Commission — Bonds: https://www.sec.gov/reportspubs/investor-publications/investorpubsbondsguidehtm.html
– U.S. Department of the Treasury
– U.S. Department of the Treasury — https://home.treasury.gov/
– FINRA — Bonds: https://www.finra.org/investors/learn-to-invest/types-investments/bonds
– TreasuryDirect — Individual investors: https://www.treasurydirect.gov/indiv/products/prod_bonds.htm
– Federal Reserve Education — Teaching materials on bonds and interest rates: https://www.federalreserveeducation.org/
Educational disclaimer: This information is educational only and not individualized investment advice. It explains concepts and sources to help you learn about discount bonds and yield calculations; it is not a recommendation to buy or sell any security. Consult a licensed financial professional for advice tailored to your situation.