25400 · Financial Literacy
Bonds & Interest-Rate Risk
Week 5 (first half) prices fixed-income securities. A coupon bond is the present value of its coupon annuity plus its face value, PB = C·[1 − (1 + i)^−n]/i + F(1 + i)^−n, with coupon C = F × CR and i the market yield; a zero-coupon bond is simply F(1 + i)^−n. You compute the current yield (C/P0), the capital-gain yield (CGY = YTM − CY) and the yield to maturity, and you show why bond prices move inversely to interest rates — the essence of interest-rate risk. Bond pricing is a reliable structured quiz item.
What this chapter covers
- 01Coupon amount C = F × CR (face value times coupon rate)
- 02Coupon bond price PB = C·[1 − (1 + i)^−n] / i + F·(1 + i)^−n, discounting at the market yield i
- 03Zero-coupon bond price PB = F·(1 + i)^−n
- 04Yield to maturity (YTM): the single rate i that equates price to discounted cash flows (Excel RATE)
- 05Current yield CY = C / P0 (annual coupon over current price)
- 06Capital-gain yield CGY = (P1 − P0)/P0, and the identity CGY = YTM − CY
- 07The inverse price-yield relationship: rates up → price down, and interest-rate risk
- 08Discount vs premium vs par bonds (coupon rate below, above or equal to the yield)
Interest-rate risk: re-pricing a bond when yields rise
- +1Coupon is fixed: C = F × CR = 1,000 × 5% = $50 per year, regardless of the market yield. At a 5% yield the price equals par ($1,000) because the coupon rate equals the yield.
- +1Re-price at the new 7% yield: PB = 50·[1 − 1.07^−4]/0.07 + 1,000·1.07^−4. The annuity factor is (1 − 0.76290)/0.07 = 3.38721 and 1.07^−4 = 0.76290.
- +1So PV of coupons = 50 × 3.38721 = $169.36 and PV of face = 1,000 × 0.76290 = $762.90, giving PB = 169.36 + 762.90 = $932.26.
- +1The price fell from $1,000 to $932.26, a capital loss of $67.74 (−6.8%). Because the coupon is fixed, a higher required yield can only be delivered by a lower price — this inverse price-yield relationship is interest-rate risk, and it is larger for longer-maturity bonds.
Key terms
- Coupon (C)
- The periodic interest payment on a bond, C = F × CR, where F is the face value and CR the coupon rate. It is contractually fixed for a standard fixed-coupon bond, whatever happens to market yields.
- Bond price
- The present value of the bond's cash flows discounted at the market yield: PB = C·[1 − (1 + i)^−n]/i + F(1 + i)^−n for a coupon bond, or F(1 + i)^−n for a zero-coupon bond.
- Yield to maturity (YTM)
- The single discount rate that makes the present value of a bond's coupons and face value equal to its price — its internal rate of return if held to maturity. Found with a financial calculator or Excel RATE.
- Current yield (CY)
- Annual coupon divided by current price, CY = C/P0. It captures the income return but ignores any capital gain or loss to maturity.
- Capital-gain yield (CGY)
- The price-change component of return, (P1 − P0)/P0, equal to YTM − CY. It is positive for a discount bond (price rises to par) and negative for a premium bond.
- Interest-rate risk
- The risk that a bond's price falls when market interest rates rise, arising from the inverse price-yield relationship. It is greater for longer-maturity and lower-coupon bonds.
Bonds & Interest-Rate Risk FAQ
Do I discount at the coupon rate or the market yield?
Always the market yield (YTM). The coupon rate only sets the size of the fixed coupon payment C = F × CR; the discount rate that prices those cash flows is the yield the market currently demands. Confusing the two is the single most common bond-pricing error.
How do I know if a bond trades at a discount, premium or par?
Compare the coupon rate with the market yield. Coupon rate below the yield → the bond is a discount (price under par); coupon rate above the yield → a premium (price over par); equal → par. This is a five-second sanity check on any price you compute.
Why do current yield and capital-gain yield add to the YTM?
Total return has two parts: the income you collect (current yield = coupon/price) and the price change to maturity (capital-gain yield). Together they equal the yield to maturity, so CGY = YTM − CY. A discount bond earns a positive capital-gain yield as its price pulls up toward par.
Why do bond prices move inversely to interest rates?
The coupons are fixed. If required yields rise, the only way a fixed stream of coupons can deliver that higher yield is for its price to fall; if yields fall, the price rises. Longer maturities amplify the effect, which is the core of interest-rate risk.
Assessment move
Anchor on one master formula — price = PV of the coupon annuity + PV of the face value — and reuse the Week-2 annuity factor for the coupons. Discipline yourself to discount at the market yield every time, and run the discount/premium/par check as a built-in sanity test. Practise decomposing return into current yield and capital-gain yield (they must sum to the YTM), and re-price a bond after a yield change so the inverse relationship and interest-rate risk become intuitive. Know the zero-coupon shortcut F(1 + i)^−n and the Excel PV/RATE functions. Rehearse a full pricing-plus-yield item under time pressure for Quiz 1, and ask Sia to set fresh coupon and zero-coupon bonds and check your working.
Working through Bonds & Interest-Rate Risk in 25400? Sia is AskSia’s AI Business and Economics tutor — ask any 25400 Bonds & Interest-Rate Risk question and get a clear, step-by-step explanation grounded in how 25400 is taught and assessed. Read this chapter free, then take your hardest questions to Sia.
