Tài chính doanh nghiệp - Topic 11: Fixed income portfolio optimization

Financial ModelingTopic #11: Fixed Income Portfolio OptimizationL. Gattis1Learning ObjectivesManage the interest rate risk of fixed income portfoliosCompute portfolio value, income, duration, convexityCompute effective durationOptimize liabilities funding (pension) using duration and convexityOptimize fixed income portfolios using duration and convexity23Portfolio Duration and ConvexityThe duration and convexity of a portfolio of assets are market-value weighted averages of all assetsOne method to mitigate a firm’s net exposure to interest rate changes is to match the duration (and interest rate sensitivity) of assets and liabilities.Pension FundingSuppose your organization has a defined benefit pension system and you have estimated the following pension liabilities.Compute the present value and the duration of the pension using NPV then compute the % change in value when interest rates +/- 100 bpsEffective Duration: Average of Abs(% Value Change) for +/- 100 bps change in the discount rate4Copy this to ExcelBond FunctionsFunction moddur(cr, par, t, freq, r)Price = bondval(cr, par, t, freq, r)For i = 1 To (t * freq) moddur = moddur + ((cr * par / freq) / (1 + r / freq) ^ i) * iNext imoddur = moddur + (par / (1 + r / freq) ^ (t * freq)) * (t * freq)moddur = moddur / Price / (1 + r / freq) / freqEnd FunctionFunction convexity(cr, par, t, freq, r)Price = bondval(cr, par, t, freq, r)For i = 1 To (t * freq) convexity = convexity + (cr * par / freq) * (1 + r / freq) ^ -i * i * (1 + i)Next iconvexity = convexity + par * (1 + r / freq) ^ -(t * freq) * (t * freq) * (1 + t * freq)convexity = convexity / (Price * (1 + r / freq) ^ 2) * freq ^ (-2)End FunctionFunction bondval(cr, par, t, freq, r)n = t * freq 'number of pmts For i = 1 To n bondval = bondval + (cr * par / freq) / (1 + r / freq) ^ i Next i bondval = bondval + par / (1 + r / freq) ^ nEnd Function5How to Invest the Pension Assets?Find mix of the following bonds that maximizes interest income by changing quantities , s.t. Value = $68,185, Duration = 6.596Maximize Annual Convexity (increasing portfolio value given rate changes)Maximize Annual Income, s.t. equity duration = 0Yield Curve Re-shaping Risk and the Pension Funding SolutionThe maximum convexity solution on the prior slide resulted in a “bar bell” strategy---- Investing in very short and long term instruments (no intermediate term)However, the risk in this strategy is a non-parallel change in the yield curve.Duration measures the value change for a 100 bps parallel shift in the yield curve.Specifically, the risk of the strategy is to a steepening of the yield curve (Short-term rates fall relative to long-term rates) increasing the value of short-term liabilities.Another solution to pension funding is to “cashflow match” the assets and liabilities--- buying bonds for each maturity equal to the liability Alternatively, Set maximum Key Rate Durations 7Rebalancing Duration-Matched PortfoliosPeriodically reforecast (pension) cash flows, re-value and compute duration.Find a new optimal investment strategy to immunize interest rate riskImplement new strategy by buying/selling bonds and reinvesting coupon paymentsDuration matching is often done (instead of cash flow matching) because its relative easefocus on market value (solvency)liabilities often extend past the maturity of traded bondshard to find securities with perfect cash flow match – and if you did, they may be expensive and illiquid8Duration Matching ApplicationDuration matching is also called interest rate immunization, its application include Pensions – Matching duration of pension liabilities to invested pension contribution assets Insurances – Matching duration of insurance liabilities to invested insurance premium assets Bond Brokers – Matching duration of bonds owned and bond liabilities Banking – Matching duration of bank assets (loans) to bank liabilities (Bonds)Interest rate risk in banks is heightened by leverage910Leveraged Banking BusinessPortfolio AnalyticsCopy and Paste into excelCalculate Value, Duration, Convexity, and Income(Expense) for Assets, Liabilities, and EquityA bank needs to finance the following loan assets by issuing debt and equityLoans with PrepaymentsIn the prior slide, I replaced the durations and convexities with values to reflect the prepayment options of consumer loansDurations and negative convexity values would come from other models (not addressed in this class) that compute the duration and convexity of loans based on borrowers prepayment behaviors (prepayment models)The bond value, duration, and convexity functions we used in this class are for non-callable bonds only.1112Leverage Banking BusinessPortfolio AnalyticsDurations and convexities are estimates for loans with prepayment (which requires another set of models that is not covered in this course)Portfolio duration and convexity is the market value weighted average13Portfolio Analytics14Duration Neutral OptimizationOptimal Funding: Set equity duration =0 by changing the # of liability bonds, while constraining liabilities = -9,000,000 (10 to 1 leverage) and # <=0Maximize Income using SolverAdd Constraint: Duration = 015Learning ObjectivesManage the interest rate risk of fixed income portfoliosCompute effective durationCompute portfolio value, income, duration, convexityOptimize liabilities funding (pension) using duration and convexityOptimize fixed income portfolios using duration and convexity16Damn it feels good to be a bankerhttps://www.youtube.com/watch?v=ROlDmux7Tk4&list=LLlA7Gv8Gp9gJwWOKAwVHx4g&index=717