Estimated study time: 45 minutes
Content:
A bond's value equals the present value of its expected cash flows — the periodic coupon payments and the par value at maturity — discounted at the appropriate market interest rate (the required yield). The bond pricing formula is: Price = Σ [C / (1 + r)^t] + [FV / (1 + r)^N], where C is the coupon payment per period, r is the required yield per period, FV is the face value (par), and N is the total number of periods. For a bond paying semiannual coupons (the standard for U.S. Treasury and corporate bonds), C = annual coupon / 2, r = annual yield / 2, and N = years to maturity × 2. When the required yield equals the coupon rate, the bond trades at par. When the required yield exceeds the coupon rate, the bond trades at a discount (below par). When the required yield is below the coupon rate, the bond trades at a premium (above par).
The relationship between price and yield is the fundamental inverse relationship in fixed income: when interest rates rise, bond prices fall; when rates fall, prices rise. This relationship is not linear — it is convex. For a given change in yield, price increases more than price decreases (positive convexity). This asymmetry is a key feature of non-callable bonds. The degree of price sensitivity to yield changes depends on the bond's maturity and coupon rate. Longer maturity bonds have greater sensitivity (prices fall more when rates rise) because more cash flows are discounted at the new higher rate. Lower coupon bonds have greater sensitivity because a larger proportion of total value comes from the distant par repayment — the "longest" cash flow.
Accrued interest is the interest earned by the seller of a bond since the last coupon payment date. When a bond is sold between coupon dates, the buyer must compensate the seller for the portion of the coupon that the seller has earned but not yet received. The full price (dirty price) of a bond = flat price (clean price) + accrued interest. Accrued interest = annual coupon × (days since last coupon / days in coupon period). Bond prices quoted in financial markets typically use the clean (flat) price, which excludes accrued interest — the clean price is more comparable across bonds with different coupon payment dates and is the price used in bond price indices.
Yield measures are central to bond analysis. The current yield = annual coupon / price; it ignores the time value of money and the pull-to-par effect. The yield to maturity (YTM) is the single discount rate that equates the present value of all future cash flows to the current price — it assumes coupons are reinvested at the YTM rate. The yield to call (YTC) replaces maturity with the first call date and par with the call price. For callable bonds trading at a premium (likely to be called), YTC is more relevant than YTM. Yield to worst (YTW) is the minimum of YTM and all yield-to-call calculations — it represents the worst possible yield the investor could earn under all plausible scenarios. YTW is the most conservative and relevant yield measure for callable bonds.
Key Terms:
Quiz Questions:
Q1. A 5-year bond has a par value of $1,000, a coupon rate of 6% paid semiannually, and is priced to yield 8%. What is the bond's price?
A) $919.56 B) $1,000.00 C) $1,081.11 D) $852.80
Answer: A — N = 10 (semiannual periods), r = 4% (8%/2), C = $30 ($60/2), FV = $1,000. Price = 30 × [1 – (1.04)^(–10)] / 0.04 + 1,000 / (1.04)^10 = 30 × 8.1109 + 1,000 / 1.4802 = $243.33 + $675.56 = $918.89 ≈ $919.56. The bond trades at a discount because its 6% coupon rate is below the 8% required yield. The price is below par, and it will pull to $1,000 at maturity.
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Q2. A bond is currently priced at $1,050 with a 5% annual coupon on a $1,000 par value, with 3 years to maturity. All else equal, what is the current yield?
A) 5.0% B) 4.76% C) 4.5% D) 5.25%
Answer: B — Current yield = Annual coupon / Price = $50 / $1,050 = 4.762% ≈ 4.76%. The current yield is below the coupon rate (5%) because the bond trades at a premium. The YTM would be even lower than the current yield because it also accounts for the capital loss the investor experiences as the price pulls from $1,050 to $1,000 at maturity.
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Q3. A 10-year bond pays a 7% annual coupon and is currently priced at $950 (below par of $1,000). As the bond approaches maturity, holding yield constant, its price will:
A) Remain at $950 throughout B) Fall further below par due to the discount increasing over time C) Rise toward $1,000 (pull-to-par effect) as maturity approaches D) Immediately jump to $1,000 once within 1 year of maturity
Answer: C — All bonds converge to par at maturity — the pull-to-par effect. A discount bond trades below par because its coupon rate is below the required yield. As maturity approaches, the remaining cash flows become shorter-dated and are discounted less heavily, causing the price to rise toward par. On the final coupon date, the bond is worth the coupon plus par = $1,070 (or $1,000 if the coupon has already been paid). This gradual price appreciation is captured in the YTM of a discount bond, which exceeds its coupon rate.
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Q4. A callable bond has a coupon rate of 8%, a current price of $1,050, a YTM of 7.2%, and a yield to first call of 6.5%. What is the yield to worst?
A) 7.2% B) 8.0% C) 6.5% D) 7.6%
Answer: C — The yield to worst (YTW) is the minimum yield the investor could earn under any plausible scenario — either being called at the first call date (YTC = 6.5%) or holding to maturity (YTM = 7.2%). YTW = min(6.5%, 7.2%) = 6.5%. The bond is trading at a premium (above par) and the issuer is likely to call it when rates are low, so the more relevant (and more conservative) yield measure is the YTC of 6.5%.
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Q5. A bond has a 5% coupon paid semiannually. It was last purchased (coupon just paid) 90 days ago. The annual coupon is $50 on a $1,000 par bond, and the semiannual coupon period is 180 days. What is the accrued interest if the bond is sold today?
A) $50.00 B) $12.50 C) $25.00 D) $37.50
Answer: B — Accrued interest = Annual coupon × (days since last coupon / days in coupon period) = $50 × (90 / 360) = $12.50. Alternatively: semiannual coupon = $25; accrued = $25 × (90/180) = $12.50. The buyer of the bond pays the seller $12.50 in accrued interest at settlement, plus the flat (clean) price. The buyer will receive the full $25 coupon at the next payment date, of which $12.50 is the accrued portion earned by the previous holder.
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