# Exam

Consider the following two bond issues.Bond A: 5% 15‐year bond. Bond B: 5% 30‐year bond. Neither bond has an embedded option. Both bonds are trading in the market at the same yield. Which bond will fluctuate *more *in price when interest rates change? Why?All other factors constant, the longer the maturity, the greater the price change when interest rates change. So, Bond B is the answer. What is the quarterly coupon payment for a $1,000 par value bond with a 5.25% coupon rate? $1,000 x 0.0525 = $52.50 per year. To get quarterly, divide by 4 = $13.125/quarter A semi‐annual interest payment of $36 implies what annual coupon rate? (assume par value = $900) $36 semi annually = $72 annually. $72/$900 = 8% What is the value today of a $1 million payment to be paid in 18 months, assuming an interest rate of 3.25% per annum? 1,000,000/(1+0.0325)^1.5 = =$953,158.03 If interest rates are currently at 2.5% and they increase by 25 basis points, what is the new interest rate? 25 basis points = 0.25%, therefore 2.5% + 0.25% = 2.75% Consider the following two bond issues.Bond A: 0% 10‐year bond. Bond B: 10% 10‐year bond. Neither bond has an embedded option. Which bond will fluctuate *more *in price when interest rates change? Why? All other factors constant, the lower the interest rate the greater will be the change in price when interest rates change so A is the answer.A portfolio manager wants to estimate the interest rate risk of a bond using duration. The current price of the bond is 82. A valuation model found that if interest rates decline by 30 basis points, the price will increase to 83.50 and if interest rates increase by 30 basis points, the price will decline to 80.75. What is the duration of this bond? The information for computing duration: price if yields decline by 30 basis points = 83.50 price if yields rise by 30 basis points = 80.75 initial price = 82.00 change in yield in decimal = 0.0030 Then, duration = (83.50 − 80.75) / [2(82.00)(0.0030)] = 5.59 A portfolio manager purchased $8 million in market value of a bond with a duration of 5. For this bond, determine the estimated change in its market value for the change in interest rates shown below: a. 100 basis points b. 50 basis points c. 25 basis points d. 10 basis points Since the duration is the approximate percentage price change for a 100 basis point change in interest rates, a bond with a duration of 5 will change by approximately 5% for a 100 basis point change in interest rates. Since the market value of the bond is $8 million, the change in the market value for a 100 basis point change in interest rates is found by multiplying 5% by $8 million. Therefore, the change in market value per 100 basis point change in interest rates is $400,000. To get an estimate of the change in the market value for any other change in interest rates, it is only necessary to scale the change in market value accordingly. a. for 100 basis points = $400,000 b. for 50 basis points = $200,000 (= $400,000/2) c. for 25 b asis points = $100,000 ($400,000/4) d. for 10 basis points = $40,000 ($400,000/10) Using the hypothetical rating transition matrix shown below, answer the following questions: a. What is the probability that a bond rated BBB will be downgraded? b. What is the probability that a bond rated BBB will go into default? c. What is the probability that a bond rated BBB will be upgraded? d. What is the probability that a bond rated B will be upgraded to investment grade? e. What is the probability that a bond rated A will be downgraded to noninvestment grade? f. What is the probability that a AAA rated bond will not be downgraded at the end of one year? a. The probability that a bond rated BBB will be downgraded is equal to the sum of the probabilities of a downgrade to BB, B, CCC or D. From the corresponding cells in the exhibit: 5.70% + 0.70% + 0.16% + 0.20% = 6.76%. Therefore, the probability of a downgrade is 6.76%. b. The probability that a bond rated BBB will go into default is the probability that it will fall into the D rating. From the exhibit we see that the probability is 0.20%. c. The probability that a bond rated BBB will be upgraded is equal to the sum of the probabilities of an upgrade to AAA, AA, or A. From the corresponding cells in the exhibit: 0.04% + 0.30% + 5.20% = 5.54%. Therefore, the probability of an upgrade is 5.54%. d. The probability that a bond rated B will be upgraded to investment grade is the sum of the probabilities that the bond will be rated AAA, AA, A or BBB at the end of the year. (Remember that the first four rating categories are investment grade.) From the exhibit: 0.01% + 0.09% + 0.55% + 0.88% = 1.53%. Therefore, the probability that a bond rated B will be upgraded to investment grade is 1.53%. e. The probability that a bond rated A will be downgraded to noninvestment grade is the sum of the probabilities that the bond will be downgraded to below BBB. From the exhibit: 0.37% + 0.02% + 0.02% + 0.05% = 0.46%, therefore, the probability that a bond rated A will be downgraded to noninvestment grade is 0.46%. f. The probability that a bond rated AAA will not be downgraded is 93.2%. What is the annualized yield on a 93‐day T‐Bill issued at a price of 99.458? (Assume Canadian yield calculations) Using the formula, T‐bill yield: Kbey= F-P/P x 365/n we solve for KBEY: (100‐99.458) / (99.458) x(365 / 93) = 2.139%Suppose a portfolio manager purchases $1 million of par value of a Treasury inflation protection security. The real rate (determined at the auction) is 3.2%.Since the inflation rate (as measured by the CPI‐U) is 3.6%, the semiannual inflation rate for adjusting the principal is 1.8%. (i) The inflation adjustment to the principal is $1,000,000 × 0.018% = $18,000 (ii) The inflation‐adjusted principal is $1,000,000 + the inflation adjustment to the principal = $1,000,000 + $18,000 = $1,018,000 (iii) The coupon payment is equal to inflation‐adjusted principal × (real rate/2) = $1,018,000 × (0.032/2) = $16,288.00 b. Since the inflation rate is 4.0%, the semiannual inflation rate for adjusting the principal is 2.0%. (i) The inflation adjustment to the principal is $1,018,000 × 0.02% = $20,360 (ii) The inflation‐adjusted principal is $1,018,000 + the inflation adjustment to the principal = $1,018,000 + $20,360 = $1,038,360 (iii) The coupon payment is equal to inflation‐adjusted principal × (real rate/2) = $1,038,360 × (0.032/2) = $16,613.76 Suppose that the yield on a 10‐year non‐callable corporate bond is 7.25% and the yield for the onthe‐ run 10‐year Treasury is 6.02%. Compute the following: a. the absolute yield spread b. the relative yield spread c. the yield ratio absolute yield spread = 7.25% − 6.02% = 1.23% = 123 basis points b. relative yield spread = (7.25% − 6.02%) / 6.02% = 0.204 = 20.4% c. yield ratio = 7.25% / 6.02% = 1.204 If the swap spread for a 5‐year interest rate swap is 120 basis points and the yield on the 5‐year Treasury is 4.4%, what is the swap rate? The swap rate is the sum of the 5‐year Treasury yield of 4.4% and the swap spread of 120 basis points. The swap rate is therefore 5.6%. Suppose that an institutional investor has entered into an interest rate swap, as the fixed‐rate payer, with the following terms: Term of swap: 2 years Frequency of payments: quarterly Notional amount: $10 million Reference rate: 3‐month LIBOR Swap spread: 100 basis points At the time of the swap, the Treasury yield curve is as follows: 3‐month rate: 4.0% 3‐year rate: 6.5%, 6‐month rate: 4.4% 4‐year rate: 7.1%, 1‐year rate: 4.9% 5‐year rate: 7.8% 2‐year rate: 5.8%, a. What is the swap rate? b. What is the dollar amount of the quarterly payment that will be made by the fixed‐rate payer? c. Compute the quarterly payment that will be received by the fixed‐rate payer if 3‐month LIBOR is 6%? 7% 8% d. What is the net payment amount that must be made in the three cases in c. above? a. From the Treasury yield curve, the relevant rate is the 2‐year rate because the swap has a two year term. The swap rate is 6.8%, computing by adding the 2‐year rate of 5.8% and the swap spread of 100 basis points. b. The annual payment made by the fixed‐rate payer of a $10 million notional amount interest rate swap with a swap rate of 6.8% is: $10,000,000 × 0.068 = $680,000. Since the swap specifies quarterly payments, the quarterly payment is $170,000 (=$680,000/4). c. If 3‐month LIBOR is 6%, quarterly payment = $150,000. At 7%, payment = $175,000. At 8%, payment = $200,000. d. The quarterly fixed‐rate payment is $170,000, therefore at 6% the net payment is ‐$20,000 (negative indicates the fixed‐rate payer must make a payment). At 7%, the net payment is $5,000 (positive indicates the amount received by the fixed‐rate payer). At 8%, net payment is $30,000 (received by the fixed‐rate payer). Compute the value of a 5‐year 7.4% coupon bond that pays interest annually assuming that the appropriate discount rate is 5.6%. Calc PV of value 5.6%. Coupon bond/(1+rate)^n, do for all years. 107.6655. Assuming annual interest payments, what is the value of a 5‐year 6.2% coupon bond when the discount rate is (i) 4.5%, (ii) 6.2%, and (iii) 7.3%? Do same thing above. a. 107.4630 b. 100 c. 95.5258. Consider a 4‐year 5.8% coupon bond that pays interest annually. a. What is the price of the 4‐year 5.8% coupon bond when it is selling to yield 7%? b. What is the price of this bond one year later assuming the yield is unchanged at 7%? c. What is the price of this bond one year later if instead of the yield being unchanged the yield decreases to 6.2%? Calculate the pv for the given rate for 7%. Note: Add face value to final year. Same thing above for each rate. a. $95.9353 b. $96.8508 c. $98.9347. Same rate, less years – same rate, more years values. Increase in discount is lower rate value – higher rate value. Total price change is the sum of the two. What is the value of a 5‐year 7.4% coupon bond selling to yield 5.6% assuming the coupon payments are made semiannually? Cash flow/Rate is divided by 2 if semi annual. $107.7561What is the value of a zero‐coupon bond paying semiannually that matures in 20 years, has a maturity of $1 million, and is selling to yield 7.6%. Value=maturity value/(1+yield/2))^n*2. $224,960.29. Suppose that a bond is purchased between coupon periods. The days between the settlement date and the next coupon period is 115. There are 183 days in the coupon period. Suppose that the bond purchased has a coupon rate of 7.4% and there are 10 semiannual coupon payments remaining. a. What is the dirty price for this bond if a 5.6% discount rate is used? b. What is the accrued interest for this bond? c. What is the clean price? Dirty price is the sum of the pv at discount rate for all years. Accrued interest, Cash flow (1-(days/total days in coupon period)) Clean price is dirty- accrued interest. a) w = 115/183 = 0.6284; PV of cash flows = full price (dirty price) = 108.8676. b) accrued interest (AI) = semi‐annual coupon payment x (1‐w) = $1.3749 c) clean price = full price – AI = $108.8676 – $1.3749 = $107.4927. Suppose you see a 5%, 10‐year Treasury bond trading at a price of $113.528 while the value you calculate based on Treasury spot rates is $113.753. a) Is there an opportunity for a riskless profit given these two quotes, and if yes quantify the amount? b) If you think a riskless profit could be made in a), how would you exploit this? Yes an arbitrage opportunity exists of $113.753 ‐ $113.528 = $0.225. b) Using the buy low sell high approach (or the law of one price) a dealer could buy the lower‐priced Treasury security in the market at $113.528, strip it, and sell the resulting package of Treasury strips at a combined total value of $113.753, thus netting a riskless profit of $0.225 per stripped security. Suppose the bond referenced in question 8 was trading at a price of $113.929 and you calculate a price of $113.753 based on Treasury spot rates. a) Is there an opportunity for a riskless profit? b) If yes, how would you exploit this? Yes, an arbitrage opportunity exists of $113.929 ‐ $113.753 = $0.176. b) Using the buy low, sell high approach (or the law of one price) a dealer could follow a reconstitution procedure by buying a package of Treasury strips in the market for a combined total of $113.753 (to create a synthetic Treasury security) and sell short the higher‐priced Treasury security at $113.929, thus netting a riskless profit of $0.176. Note the cash flows from the package of Treasury strips purchased is used to fund the payments required for the shorted Treasury securities. Determine the yield to maturity of a 6.5% 20‐year bond that pays interest semiannually and is selling for $90.68. Set par value of bond ($100) as FV. Set 6.5% of par value ($6.50) as PMT. Set N to 20. Set -$90.68 as PV. CPT I/Y. Ans: 7.4%. **Bonds can have any par value but generally round numbers are used (100, 1000, 5000, 1 million, etc)** a)What would a Certificate of Deposit (CD) be worth that costs $108.32, pays 7% annually (on a bond‐equivalent basis, or 3.5% semiannually) and the interest is compounded semi‐annually over 5 years (ie. 10 six month periods)? b) how much interest is earned over this time period? A) FV = Cost*(1+semi-annual rate)^n = $108.32*(1+.035)^10 = $152.80. B) Interest Earned = Future Value – Cost = $152.80-$108.32 = $44.48. Suppose an investor can purchase a 5‐year 9% coupon bond that pays interest semiannually and the price of this bond is $108.32. The yield to maturity for this bond is 7% on a bond‐equivalent basis. If held to maturity, a) what is the total coupon interest (undiscounted)? b) what is the capital gain/loss? c) what is the total reinvestment income? D) what is the total dollar return? A) Coupon Interest = (New Coupon Rate*Par Value)*Periods = (0.045*100)*10 = $45. Capital Gain/Loss = Par Value – Cost = $100.00-$108.32 = -$8.32(loss). C) Re-Investment Income = (New Coupon Rate*Par Value)*(((1+Semi-AnnualYTM)^Periods-1)/Semi-AnnualYTM)-Coupon Interest = (0.045*100)*(((1+0.035)^10-1)/0.035)-$45.00 = $7.80. D) Total Return = Sum of Coupon Interest + Capital Gain/Loss + Re-Investment Income = $45+(-$8.32)+$7.80 = $44.48. Suppose a 5% coupon 6‐year bond is selling for $105.2877 and is putable in four years at par value. The yield to maturity for this bond is 4% and payments are semi‐annual. What is the yield to put? Set par value of bond ($100) as FV. Set 5% of par value ($5.00) as PMT. Set N to 4. Set -$105.2877 as PV. CPT I/Y. Ans: 3.56% A US Treasury bill with 275 days from settlement to maturity is quoted as having a yield on a discount basis of 3.68%. What is the price of this Treasury bill? (assume US T‐bill calculations). Price = 1-(Discount Basis*(Days/360)) = 1-(0.0368*(275/360)) = $0.971889. Suppose that the annual yield to maturity for the 6‐month and 1‐year Treasury bill is 4.6% and 5.0%, respectively. These yields represent the 6‐month and 1‐year spot rates. Also assume the following Treasury yield curve (i.e., the price for each issue is $100) has been estimated for 6‐month periods out to a maturity of 3 years: Years to maturity; 1.5,2.0,2.5,3.0. Annual yield to maturity (BEY); 5.4%, 5.8%, 6.4%, 7.0%. Compute the 1.5 year, 2 year, and 2.5 year spot rates. Step 1: Compute Cash Flows. 6 month = $2.70 (Par x (BEY1.5YR/2)), 1-year=$2.70(Same as 6 month), 1.5year= $102.70(Same + Par Value). Step 2: Compute PV of Cash Flows. PV = 100 = 6monthCF/(1+(BEY6month/2)) + 1yrCF/(1+(BEY1yr/2))^2 + 1.5yrCF/(1+Z_{3 })^3. PV = 100 = $2.70/(1+(0.046/2)) + $2.70/(1+(0.05/2))^2 + $102.70/(1+Z_{3 })^3. PV = 2.6393 + 2.5699 + $102.70/(1+Z_{3 })^3 = 100. $94.7908= $102.70/(1+Z_{3 })^3. (1+Z_{3 })^3 = $102.70/$94.7908. (1+Z_{3 })^3 = 1.083438. Take cube root. 1+ Z_{3 }= 1.027073. Z_{3} = 0.027073. Take 0.027073 x 100 = 2.7073. Multiply this by 2. 2.7073 x 2 = 5.4146%. This is the spot rate for 1.5 years. Compute the following forward rates: a. the 6‐month forward rate six months from now. b. the 6‐month forward rate one year from now. c. the 6‐month forward rate three years from now. tFm = forward rate notation. t= Number of 6-month periods. m=Period beginning ‘m’ 6-month periods from now. 1f8 = 6-month forward rate, starting four years (eight 6- month periods) from now. f8 = one-year (2 period) forward rate, starting four years (8 periods) from now. 1f0 = z1= current spot rate for one 6-month period. Formula for determining 6-month forward rate: 1Fm = [(1+Z_{m+1 })^m+1/(1+zm)^m]-1. Formula for determining any forward rate. tFm = [[(1+Z_{m+t})^m+t/(1+z_{m})^m]^1/t]-1]. A)Given the following 6‐month forward rates, compute the forward discount factor for each period. Take annual forward rate (BEY) and divide by 2 for all periods. These are your semi-annual rates. Then take 1/(1+semi-annual rate for period 1). This is your discount factor (0.980392). For period 2, you do 1/(1+semi-annual rate for period 1)*(1+semi-annual rate for period 2) = 0.959288. This is your discount factor for period 2 etc. Compute the value of a 3‐year 8% coupon bond using the forward rates computed in a) above. Calculate your cash flow. Take the coupon rate (8%)/2 = 4% (to reflect semi-annual) and x $100 (par value) = $4. These are your cash flows for all 6 periods. (3yrs*2) Take your discount factors from part A) and multiply them by the cash flows. This is your PV of the cash flow for each period. In the last period (6), you add back the par value ($4 + $100 = $104) and discount $104 by the discount factor. Total Ans = Sum of Discounted Cash Flows = $107.75281.

Chapter 7:

Duration= V_ – V+/ 2(Vo)(change in y) Where • Δy = change in yield in decimal • V0 = initial price • V- = price when yields decline by Δy • V+ = price when yields increase by Δy . The formula for the convexity adjustment to the percentage price change is: Convexity adjustment (%) = C x (Δy)2 x 100. Convexity Adjustment is : C= V+ + V_ – 2Vo/2Vo(change in y)^2 Using the duration computed in question 1, compute the approximate percentage price change using duration for the two 8% coupon bonds assuming that the yield changes by 10 basis points (Δy∗= 0.0010). 8%, 4‐year bond: 10 basis point increase: approximate percentage price change = −3.44 × (0.0010) × 100 = −0.34%; 10 basis point decrease: approximate percentage price change = −3.44 × (−0.0010) × 100 = +0.34%. 8%, 25‐year bond: 10 basis point increase: approximate percentage price change = −12.94 × 0.0010) × 100 −1.29%; 10 basis point decrease: approximate percentage price change = −12.94 × (−0.0010) × 100 = +1.29%. Using the value for C computed in question 2, compute the convexity adjustment for the two 25‐ year bonds assuming that the yield changes by 200 basis points (Δy∗ = 0.02). For the 5% 25‐year bond: C = 141.68 Δy∗ = 0.02; convexity adjustment to percentage price change = 141.68 × (0.02)2 × 100 = 5.67%; For the 8% 25‐year bond: C = 121.89; convexity adjustment to percentage price change = 121.89 × (0.02)2 × 100 = 4.88%. The Basis Point is 25. (0.0025) (Change in Y 0.0025)

A 9% 20-year bond is currently trading at 134.6722. • A 20 basis point decrease in yield causes the bond to rise in price to 137.5888 • A 20 basis point increase in yield causes the bond to fall in price to 131.8439. • What is the duration of the bond? • What does it mean? • Δy = .002, V0 = 134.6722, V- = 137.5888, and V+ = 131.8439 • Duration = (137.5888 – 131.8439)/2(134.6722)(.002) = 10.66 • A 100 bps change in yield causes the bond price to change by 10.66% Price Change Using Duration: Approx % price change = -duration x Δy x 100. Example: Consider the 9%, 20-year bond with duration of 10.66. What is the approximate % price change for a 10 basis point increase in yield? Δy = +0.001; % price chg = -10.66 x 0.001 x 100 = – 1.066% The formula for the convexity adjustment to the percentage price change is: Convexity adjustment (%) = C x (Δy)2 x 100. Convexity Adjustment is : C= V+ + V_ – 2Vo/2Vo(change in y)^2 Where: • C is the change in price not explained by duration • Δy = change in yield in decimal • V0 = initial price • V- = price if yields decline by Δy • V+ = price if yields increase by Δy. Example • Yield change = 20 bps (Δy = 0.002) • Original price (V0 ) = 134.6722, • Price if yields decline by 20 bps (V- ) = 137.5888 • Price if yields increase by 20 bps (V+ ) = 131.8439 • Calculate “C” • Find the convexity adjustment for a 200bps yield change? C = (131.8439 +137.5888 – 2 (134.6722) / 2(134.6722)(.002)2 = 81.95 • Convexity Adjustment = 81.95 x (.02)2 x 100 = 3.28%

Complete the amortization schedule for the first three (3) months only on a 30-year fully amortizing mortgage loan with a mortgage rate of 7.25% where the amount borrowed is $150,000. The monthly mortgage payment is $1,023.26.

Consider the following information: Mortgage balance in month 42 = $260,000,000 Scheduled principal payment in month 42 = $1,000,000 Prepayment in month 42 = $2,450,000 a) What is the Single Monthly Mortality (SMM) amount for month 42? b) What is the Conditional Prepayment Rate (CPR) for month 42? SMM = prepayment in month t/ beginning mortgage balance for month t-scheduled principal payment in month t. SMM = $2.45M / ($260M – $1M) = 0.009459 = 0.9459% b) CPR = 1-(1-SMM)12 = 1-(1-0.009459)12 = 0.107790 = 10.78%. Which of the following represents the fastest level of prepayments for a Mortgage Backed Security (MBS)? A) 50 PSA b) 100 PSA c) 200 PSA d) 300 PSA. Answer is d. If 100 PSA or 100% PSA represents the standard benchmark level of prepayments, then 300 PSA would be a 3x faster level (ie. a greater level) of prepayments. Suppose that a Planned Amortization Class (PAC) bond is created using prepayment speeds of 90 PSA and 240 PSA and the average life is 5 years. Will the average life for this PAC tranche be shorter than, longer than, or equal to 5 years if the collateral pays at 140 PSA over its life? Since the prepayments are assumed to be at a constant prepayment rate over the PAC tranche’s life at 140 PSA which is within the PSA collar in which the PAC was created, the average life will be equal to five years. Suppose that the structure for an asset‐backed security transaction is as follows: senior tranche $220 million, subordinate tranche 1 $50 million, subordinate tranche 2 $30 million, and that the value of the collateral for the structure is $320 million. Subordinate tranche 2 is the first loss tranche. a. How much is the overcollateralization in this structure? b. What is the amount of the loss for each tranche if losses due to defaults over the life of the structure total $15 million? c. What is the amount of the loss for each tranche if losses due to defaults over the life of the structure total $35 million? d. What is the amount of the loss for each tranche if losses due to defaults over the life of the structure total $85 million? e. What is the amount of the loss for each tranche if losses due to defaults over the life of the structure total $110 million? 1. a. The amount of overcollateralization is the difference between the values of the collateral, $320 million, and the par value for all the tranches, $300 million. In this structure it is $20 million. b. If the losses total $15 million, then the loss is entirely absorbed by the overcollateralization. No tranche will realize a loss. c. through e below: Given the following 2 companies, Company A and Company B, compute the Current Ratio and the Total Debt/Capitalization ratio for both companies. Based on these two ratios, which company represents a lower credit risk? Ratios are presented below. Since Company A has a higher Current Ratio and a lower Total Debt/Capitalization, it would appear to be a much lower credit risk. Current Ratio= current assets/current liabilities. Current ratio for company A is 2.17 and company B is 0.84. Total Debt/Capitalization for Company A is 0.84 and company B is 12.86. Total Debt to Capitalization = Current iabilites+LTD/LTD+CL+Shareholder’s Equity. If a company’s Earnings Before Interest and Taxes (EBIT) is $175,000 and current debt outstanding is $800,000 with an annual interest rate of 5.25%, what is the EBIT Interest coverage ratio? b) If the EBIT Interest coverage ratio in the following year is 2.3x, has the risk level for lending to this company increased or decreased? EBIT = $175,000; Annual Interest Expense = $800,000 X 5.25% = $42,000; EBIT/Interest = $175k/$42k = 4.17x b) If the EBIT/Interest coverage ratio the following year is 2.3x, this would indicate a lower amount of coverage or “cushion” to cover interest expenses, therefore the risk level for a lender has increased. EBIT Interest Coverage= EBIT/Annual Interest Expense. EBITDA Interest Coverage= EBITDA/Annual Interest Expense

Suppose a portfolio manager observed the following two quotes: 1 year T‐bill: 3.3% BEY, 6 month T‐bill: 3.0% BEY. What is the “break‐even” interest rate for the 6‐month term starting in 6 months where the manager is indifferent between these two investment alternatives? What should the manager do if her view is different from the market expectations? . We need to calculate the 6‐month rate, starting in 6 months, or 1f1 = (1+z2) 2 / (1+z1) ‐1 z1 = 6 month rate = 3.0% / 2 = 1.5%; z2 = 1 year rate expressed as a 6 month interval (ie. semi‐annually) = 3.3% / 2 = 1.65% 1f1 = (1+0.0165)2 / (1+0.015) – 1 = 0.01800 or 3.6% on a BEY basis (0.0180 x 2) Therefore if the 6 month rate starting in 6 months is 3.6%, the manager is indifferent between the two alternatives since they will both produce the same investment return. If the manager believes that the 6 month rate starting in 6 months will be lower than 3.6%, she should lock in the 1 year rate of 3.3%. If the manager believes the 6 month rate starting in 6 months will higher than 3.6%, she should buy the 6 month T‐

bill at 3.0%, with the expectation of buying a 6 month T‐bill in 6 months at the new higher rate. In this manner the portfolio manager can add value through an active strategy and her beliefs about what is priced into the market expectations. Suppose that the present value of the liabilities of a financial institution is $6 billion and the economic surplus is $8 billion. The duration of the liabilities is equal to 5. The duration of the assets is 6. a. What is the market value of the asset portfolio? b. What is the dollar duration per 100 par value for the asset portfolio? c. What is the dollar duration per 100 par value for the liabilities? d. Suppose that interest rates increase by 50 basis points. What is the approximate new value for the economic surplus? e. Suppose that interest rates decrease by 50 basis points. What is the approximate new value for the economic surplus? The market value of the asset portfolio is $14 billion. ($14 billion assets minus $6 billion liabilities gives a surplus of $8 billion.) b. Since the duration of the assets is 6, then for a 100 basis point change in interest rates the change in the value of the asset portfolio will be approximately 6%. The dollar duration per 100 par value is then $6. c. A duration of 5 for the liabilities means that if interest rates change by 100 basis points, the present value of the liabilities will change by approximately 5%. The dollar duration per 100 par value is then $5. d. If interest rates increase by 100 basis points, because the market value of the assets is $14 billion and the dollar duration per 100 market value is $6, the market value will decrease by approximately $840 million ($14 billion x

6%). For a 50 basis point increase in rates, the market value of the assets will decrease by half that amount, $420 million. The present value of the liabilities will decrease by approximately $150 million for a 50 basis point change (or half of $300 million for a 100 bps change, calculated as $6 billion x 5%). Thus, a 50 basis point increase in interest rates decreases the assets by $420 million and decreases the liabilities by $150 million. The net effect on the surplus is a decline of $270 million. Since the initial surplus is $$8 billion, the surplus after a 50 basis point rate increase would be $7.73 billion ($8 billion minus $270 million). e. If interest rates decrease by 50 basis points, the value of the assets will increase by $420 million to $14.42 billion and the value of the liabilities will increase by $150 million to $$6.15 billion. Hence, the surplus will increase to $8.27 billion ($14.42 billion minus $6.15 billion = $8.27 billion. Alternatively, $8 billion plus $270 million = $8.27 billion). Interest Rate Risk: Economic Surplus=Market Value of Assets-Present Value of Liabilities