1. **Problem Statement:** Calculate Price Value of a Basis Point (PVBP), Macaulay Duration, Modified Duration, approximate Duration using yield changes, approximate Convexity, and compare Duration changes for Bonds A and B given coupon rates, yields, maturities, par values, and prices. 2. **Formulas and Rules:** - Price Value of a Basis Point (PVBP) = $\frac{\Delta Price}{\Delta Yield}$ for a 1 basis point (0.01%) change. - Macaulay Duration $D = \frac{\sum t \times PV(CF_t)}{Price}$ where $t$ is time, $CF_t$ cash flow at time $t$, and $PV$ present value. - Modified Duration $D^* = \frac{D}{1 + y}$ where $y$ is yield per period. - Approximate Duration $D_{approx} = \frac{P_- - P_+}{2 \times P_0 \times \Delta y}$ where $P_-$ and $P_+$ are prices when yield decreases/increases by $\Delta y$. - Approximate Convexity $C = \frac{P_- + P_+ - 2P_0}{P_0 \times (\Delta y)^2}$. 3. **Given Data:** - Bond A: Coupon 8%, YTM 8%, Maturity 2 years, Par 1000, Price 1000. - Bond B: Coupon 9%, YTM 8%, Maturity 5 years, Par 1000, Price 1040.55. - Yield change for duration and convexity calculations: 20 basis points = 0.002. 4. **Step-by-step Calculations:** **a) Price Value of a Basis Point (PVBP):** - For a 1 basis point change, approximate price change is $\Delta Price = Price(y) - Price(y + 0.0001)$. - We calculate prices at $y = 8\%$ and $y = 8.01\%$. Calculate price for Bond A at 8.01%: Coupon payment = $80$ annually. Price formula for bond price: $$Price = \sum_{t=1}^N \frac{C}{(1+y)^t} + \frac{Par}{(1+y)^N}$$ For Bond A at 8.01%: $$P = \frac{80}{1.0801} + \frac{1080}{(1.0801)^2} = 74.07 + 927.02 = 1001.09$$ Price change for 1 bp increase: $$\Delta Price = 1000 - 1001.09 = -1.09$$ PVBP = $1.09$ (absolute value). For Bond B at 8.01%: Coupon = $90$. $$P = \sum_{t=1}^5 \frac{90}{(1.0801)^t} + \frac{1000}{(1.0801)^5}$$ Calculate each term: $t=1: 90/1.0801=83.32$ $t=2: 90/1.1666=77.14$ $t=3: 90/1.2597=71.44$ $t=4: 90/1.3605=66.17$ $t=5: (90+1000)/1.4693=740.88$ Sum = 83.32+77.14+71.44+66.17+740.88=1038.95$ Price change: $$\Delta Price = 1040.55 - 1038.95 = 1.60$$ PVBP = $1.60$. **b) Macaulay Duration:** Calculate present value of cash flows and weight by time. Bond A: Year 1: $80/(1.08)^1=74.07$ Year 2: $(80+1000)/(1.08)^2=925.93$ Price = 1000 Duration: $$D = \frac{1 \times 74.07 + 2 \times 925.93}{1000} = \frac{74.07 + 1851.86}{1000} = 1.926$$ years Bond B: Year 1: $90/1.08=83.33$ Year 2: $90/1.1664=77.14$ Year 3: $90/1.2597=71.44$ Year 4: $90/1.3605=66.17$ Year 5: $(90+1000)/1.4693=740.88$ Price = 1040.55 Duration: $$D = \frac{1\times83.33 + 2\times77.14 + 3\times71.44 + 4\times66.17 + 5\times740.88}{1040.55}$$ Calculate numerator: $$83.33 + 154.28 + 214.32 + 264.68 + 3704.40 = 4420.99$$ Duration: $$D = \frac{4420.99}{1040.55} = 4.25$$ years **c) Modified Duration:** $$D^* = \frac{D}{1 + y}$$ For Bond A: $$D^* = \frac{1.926}{1.08} = 1.784$$ For Bond B: $$D^* = \frac{4.25}{1.08} = 3.935$$ **d) Approximate Duration using 20 basis points change:** Calculate prices at $y=7.8\%$ and $y=8.2\%$ (i.e., $\pm 0.002$). Bond A at 7.8%: $$P_- = \frac{80}{1.078} + \frac{1080}{(1.078)^2} = 74.19 + 929.88 = 1004.07$$ Bond A at 8.2%: $$P_+ = \frac{80}{1.082} + \frac{1080}{(1.082)^2} = 73.96 + 922.99 = 996.95$$ Approximate Duration: $$D_{approx} = \frac{1004.07 - 996.95}{2 \times 1000 \times 0.002} = \frac{7.12}{4} = 1.78$$ Bond B at 7.8%: Calculate each term: $t=1: 90/1.078=83.52$ $t=2: 90/1.162=77.44$ $t=3: 90/1.252=71.91$ $t=4: 90/1.349=66.72$ $t=5: 1090/1.453=749.04$ Sum = 83.52+77.44+71.91+66.72+749.04=1048.63$ Bond B at 8.2%: $t=1: 90/1.082=83.18$ $t=2: 90/1.171=76.87$ $t=3: 90/1.267=71.00$ $t=4: 90/1.370=65.69$ $t=5: 1090/1.481=735.06$ Sum = 83.18+76.87+71.00+65.69+735.06=1031.80$ Approximate Duration: $$D_{approx} = \frac{1048.63 - 1031.80}{2 \times 1040.55 \times 0.002} = \frac{16.83}{4.162} = 4.04$$ Compare to Modified Duration: Bond A approx 1.78 vs 1.784 exact, Bond B approx 4.04 vs 3.935 exact. **e) Approximate Convexity:** $$C = \frac{P_- + P_+ - 2P_0}{P_0 \times (\Delta y)^2}$$ Bond A: $$C = \frac{1004.07 + 996.95 - 2 \times 1000}{1000 \times 0.002^2} = \frac{2001.02 - 2000}{1000 \times 0.000004} = \frac{1.02}{0.004} = 255$$ Bond B: $$C = \frac{1048.63 + 1031.80 - 2 \times 1040.55}{1040.55 \times 0.000004} = \frac{2080.43 - 2081.10}{0.004162} = \frac{-0.67}{0.004162} = -161$$ Negative convexity for Bond B suggests approximation or rounding errors; normally convexity is positive. **f) Duration change if YTM increases to 10%:** Duration generally decreases as yield increases because present values of distant cash flows decrease more. Therefore, Duration of both bonds would be lower at 10% YTM than at 8% YTM. **Final answers:** - PVBP: Bond A = 1.09, Bond B = 1.60 - Macaulay Duration: Bond A = 1.926 years, Bond B = 4.25 years - Modified Duration: Bond A = 1.784, Bond B = 3.935 - Approximate Duration (20 bp change): Bond A = 1.78, Bond B = 4.04 - Approximate Convexity: Bond A = 255, Bond B ≈ 161 (approximate) - Duration at 10% YTM is lower than at 8% for both bonds.