MODULE 10: Interest Rate Risk and Return

10 Interest Rate Risk and Return

Core Flow

1. This module asks one question: you bought a fixed-rate bond, so what return do you actually earn? The answer is not automatically yield-to-maturity.
2. What is yield-to-maturity: the single discount rate that prices the bond today, assuming promised cash flows are received and reinvestment works as assumed.
3. A fixed-rate bond has three return sources: scheduled coupon and principal payments, reinvestment income on coupons, and capital gain or loss if sold before maturity.
4. Think of a bond like a rental flat. Rent checks are coupons, investing the rent is reinvestment income, and selling the flat creates the capital gain or loss. Your total result is all three.
5. Horizon yield is the annualized return over your actual holding period:

1. Notation in simple language:
2. \(FV_{\text{coupons}}\): future value of coupons reinvested until the horizon date.
3. \(P_{\text{sale or maturity}}\): sale price if sold before maturity, or redemption value if held to maturity.
4. \(PV_{\text{purchase}}\): purchase price at the start.
5. \(T\): investment horizon in years.
6. Example: buy at 100, coupons grow to 27.203, sell at 100 after 4 years. The realised return is:

1. That 6.20% equals the original yield-to-maturity because coupons were reinvested at 6.20% and the sale price stayed on the constant-yield price path.
2. What is constant-yield price path: the bond's carrying value path if the market yield stays equal to the original yield-to-maturity.
3. Carrying value = purchase price plus accumulated discount amortization, or minus accumulated premium amortization. Discount bonds get pulled up toward par; premium bonds get pulled down toward par.
4. The big rate-move rule is simple: rising rates help coupon reinvestment but hurt sale price; falling rates hurt coupon reinvestment but help sale price.
5. Reinvestment risk is the risk that coupons are reinvested at worse rates than expected. Price risk is the risk that the bond must be sold at a worse price than expected.
6. A buy-and-hold investor has no realised price risk if there is no default, but still has reinvestment risk. An investor who sells before the first coupon has price risk but no reinvestment risk.

1. Source example: all investors buy a 10-year, 6.2% annual coupon bond at par. If rates rise to 7.2%, the 10-year investor's horizon yield rises to 6.43% because reinvestment improves and there is no sale loss.
2. The 4-year investor in the same bond gets hurt when rates rise. Coupons reinvest better, but the sale price drops to 95.263, so horizon yield falls to 5.28%.
3. If rates fall to 5.2%, the 10-year investor's horizon yield falls to 5.98% because coupons reinvest at lower rates.
4. The 4-year investor benefits when rates fall because the sale price rises to 105.043; horizon yield rises to 7.16%.
5. Same bond, same rate move, different investor. Long horizon fears falling rates. Short horizon fears rising rates.

1. Interest income is the return from time passing: coupons, reinvestment income, and discount or premium amortization. Capital gain or loss is the return from the bond's market value changing because yield-to-maturity changed.
2. Investment horizon is how long the investor expects to hold the bond. This is the key variable that decides whether reinvestment risk or price risk matters more.
3. Macaulay duration is the holding period where price risk and reinvestment risk roughly offset for a one-time instantaneous parallel yield-curve shift.
4. What is Macaulay duration: the present-value-weighted average time to receive the bond's cash flows.
5. Why Macaulay duration matters: if your horizon equals Macaulay duration, the bad effect of a rate move is roughly balanced by the good effect.
6. Duration gap is:

1. If duration gap is positive, Macaulay duration is greater than the horizon. Price risk dominates. The bad scenario is rising rates.
2. If duration gap is zero, investment horizon equals Macaulay duration. Price risk and reinvestment risk roughly offset.
3. If duration gap is negative, Macaulay duration is less than the horizon. Reinvestment risk dominates. The bad scenario is falling rates.
4. Source example: Macaulay duration 9.85, investor horizon 7.0 years:

1. Positive duration gap means price risk dominates; this investor is mainly worried about rates rising. P for positive. P for price.
2. Example: BRWA's 10-year, 6.2% bond has Macaulay duration 7.7429 years. The 4-year investor is short-horizon and price-risk-heavy; the 10-year investor is long-horizon and reinvestment-risk-heavy; the 8-year investor is close to balanced.
3. Romanian 30-year, 5.61% eurobond example: Macaulay duration is 15.16 years. The 15-year investor stays near 5.61% whether rates move up or down by 50 basis points because horizon is close to duration.

1. Macaulay duration is calculated as a weighted-average time to cash flows:

1. What is full price: the invoice price including accrued interest. It is used because the weights must add up to the actual present value paid for the bond.
2. Macaulay duration is the bond's cash-flow center of gravity. A zero-coupon bond has all cash flow at maturity, so Macaulay duration equals maturity.
3. A coupon bond usually has Macaulay duration below maturity because coupons arrive earlier. Higher coupons pull more value earlier and usually reduce duration.
4. Do not confuse Macaulay duration with modified duration. Macaulay duration asks, "On average, when do I get my money back?" Modified duration asks, "How much does price move when yield changes?"
5. Source example: a 10-year, 6.2% annual coupon bond priced at par has Macaulay duration 7.7429 years, not 10, because coupons arrive before final maturity.
6. In that example, the final cash flow of 106.2 is about 58% of present value, so duration is still closer to maturity than to the early coupon dates.
7. If a bond pays semiannual coupons, the table gives Macaulay duration in periods; divide by 2 to annualize. If it pays quarterly, divide by 4.
8. Between coupon dates, the first cash flow arrives in less than one full period:

1. Here, \(t\) is days since the last coupon and \(T\) is days in the coupon period. Each later cash flow is one period after the previous one.
2. Source example: 57 days after issuance on a 30/360 annual coupon basis, the first cash flow arrives after 0.8417 years, and Macaulay duration falls from 7.7429 to 7.5845.
3. Key property: Macaulay duration is not static. As time passes, remaining cash flows get closer, so duration normally falls.
4. Source self-assessment: a 2-year, 4% semiannual coupon bond at 4.60% yield is 63 days into a 180-day period. First cash flow timing is:

1. The weighted sum is 3.5331 periods, so annualized Macaulay duration is:

1. Source question-set example: a 3-year, 4% semiannual coupon bond priced at 100 has period-based Macaulay duration 5.71346 periods:

1. Negative yields do not break Macaulay duration. A 5-year, 0% coupon bond with yield-to-maturity of -0.38% still has Macaulay duration 5.0000 years because there is only one cash flow.
2. A 5-year, 0.5% annual coupon bond with yield-to-maturity of -0.20% has Macaulay duration 4.9515 years; if it is 91 days into the first coupon period, duration falls to 4.6987 years.
3. The source also gives a closed-form Macaulay duration equation, but the exam mechanics are usually easier from the cash-flow table or spreadsheet function. Spreadsheet shortcut:

1. Frequency is 1 for annual, 2 for semiannual, 4 for quarterly. Basis is the day-count convention: 0 or omitted for 30/360, 1 for actual/actual, 2 for actual/360, and 3 for actual/365.

1. Exam trap: if the investor holds to maturity, do not say price risk dominates. No sale means no realized price risk, assuming no default.
2. Exam trap: if the investor sells before the first coupon, do not say reinvestment risk dominates. There is no coupon to reinvest.
3. Exam trap: duration gap uses Macaulay duration minus investment horizon, not maturity minus horizon.
4. Exam trap: for semiannual bonds, annualize Macaulay duration by dividing the period result by 2.
5. Final mental picture: rates rise, price falls now, coupons reinvest better later. Rates fall, price rises now, coupons reinvest worse later. Macaulay duration is the balancing point between those two forces.