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Lecture 4 2012   valuation and risk
Lecture 4 2012   valuation and risk
Lecture 4 2012   valuation and risk
Lecture 4 2012   valuation and risk
Lecture 4 2012   valuation and risk
Lecture 4 2012   valuation and risk
Lecture 4 2012   valuation and risk
Lecture 4 2012   valuation and risk
Lecture 4 2012   valuation and risk
Lecture 4 2012   valuation and risk
Lecture 4 2012   valuation and risk
Lecture 4 2012   valuation and risk
Lecture 4 2012   valuation and risk
Lecture 4 2012   valuation and risk
Lecture 4 2012   valuation and risk
Lecture 4 2012   valuation and risk
Lecture 4 2012   valuation and risk
Lecture 4 2012   valuation and risk
Lecture 4 2012   valuation and risk
Lecture 4 2012   valuation and risk
Lecture 4 2012   valuation and risk
Lecture 4 2012   valuation and risk
Lecture 4 2012   valuation and risk
Lecture 4 2012   valuation and risk
Lecture 4 2012   valuation and risk
Lecture 4 2012   valuation and risk
Lecture 4 2012   valuation and risk
Lecture 4 2012   valuation and risk
Lecture 4 2012   valuation and risk
Lecture 4 2012   valuation and risk
Lecture 4 2012   valuation and risk
Lecture 4 2012   valuation and risk
Lecture 4 2012   valuation and risk
Lecture 4 2012   valuation and risk
Lecture 4 2012   valuation and risk
Lecture 4 2012   valuation and risk
Lecture 4 2012   valuation and risk
Lecture 4 2012   valuation and risk
Lecture 4 2012   valuation and risk
Lecture 4 2012   valuation and risk
Lecture 4 2012   valuation and risk
Lecture 4 2012   valuation and risk
Lecture 4 2012   valuation and risk
Lecture 4 2012   valuation and risk
Lecture 4 2012   valuation and risk
Lecture 4 2012   valuation and risk
Lecture 4 2012   valuation and risk
Lecture 4 2012   valuation and risk
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Lecture 4 2012 valuation and risk

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  • 1. 固定收益课题 4 : 债券定价和风险 / Valuation and Risk 陈国辉博士 / 南洋理工大学商学院
  • 2. 债券定价模式 <ul><li>C = coupon rate p.a. </li></ul><ul><li>m = number of coupon payment per annum </li></ul><ul><li>y = yield to maturity p.a. </li></ul><ul><li>n = number of period </li></ul><ul><li>P = price of bond </li></ul>
  • 3. 价格和收益率的关系 (Price-Yield Relationship) Price Yield <ul><li>价格和收益率成反向关系 (Inverse relation) </li></ul><ul><li>高收益率使到价格下滑 (High yield causes price to decrease) </li></ul><ul><li>凸性 (convexity) </li></ul>
  • 4. 到期收益率 ( Yield-to-Maturity,YTM) 的局限性 <ul><li>YTM 不是真正能实现的收益率 </li></ul><ul><li>YTM is not the realized rate of return </li></ul><ul><li>实现 YTM 的 2 个条件 (2 conditions for realizing YTM as return) : </li></ul><ul><ul><li>持有债券到到期日 (Hold the security up to maturity) </li></ul></ul><ul><ul><li>利息的再投资率必须等于 YTM (Reinvestment rate has to equal to YTM) </li></ul></ul><ul><li>债券投资者更关心总收益 ( total return) :资本收益 (capital gain or price return) + 票息收益 coupon return) + 再投资收益 ( reinvestment return) </li></ul>
  • 5. 总收益 Total Return <ul><li>假设一投资者的投资期限是 2 年。再假设他买了 9 %票息的 2 年期债券,价格为 98.26 (ytm = 10%) </li></ul><ul><li>请问此投资的总收益是多少 , 假设其再投资收益率是 10 %? </li></ul><ul><ul><li>Future value of his investment </li></ul></ul><ul><ul><ul><li>First coupon: $9(1.1) </li></ul></ul></ul><ul><ul><ul><li>Second coupon $9 </li></ul></ul></ul><ul><ul><ul><li>Maturity value = 100 </li></ul></ul></ul><ul><ul><ul><li>Total future value = 9(1.1) + 9 + 100 = 118.9 </li></ul></ul></ul><ul><ul><li>Today ’s investment = 98.26 </li></ul></ul><ul><ul><li>Total return: </li></ul></ul><ul><ul><ul><li>118.9/(1+R) 2 =98.26 </li></ul></ul></ul><ul><ul><ul><li>R = 10% </li></ul></ul></ul><ul><li>如果再投资收益率是 9 %,那总收益是多少? </li></ul><ul><li>如果再投资收益率是 11 %,那总收益是多少? </li></ul>Reinvestment Risk
  • 6. 总收益 Total Return <ul><li>假设一投资者的投资期限是 1 年。再假设他买了 9 %票息的 2 年期债券,价格为 98.26 (ytm = 10%) </li></ul><ul><li>请问他的总收益率是多少? </li></ul><ul><ul><li>假设他卖出债券时的价格的 YTM 是 10%, ie., 109/(1+0.1) or 99.0909 </li></ul></ul><ul><ul><li>或假设以低于 99.0909 的价格卖出? </li></ul></ul><ul><ul><li>或假设以高过 99.0909 的价格卖出? </li></ul></ul>Price Risk
  • 7. 平均到期期限 Average Maturity <ul><li>投资者在乎到期日吗? </li></ul><ul><li>假设 3 个债券 </li></ul><ul><ul><li>Debt A: Pay $10,000 in 5 years </li></ul></ul><ul><ul><li>Debt B: Pay $1 in 0.25 year and $9,999 in 5 years. </li></ul></ul><ul><ul><li>Debt C: Pay $9,999 in 0.25 year and $1 in 5 years. </li></ul></ul><ul><li>注意付息的时间和额度 </li></ul>
  • 8. 久期 Duration <ul><li>为了更准确的计算平均到期期限,我们可以考虑用时间来权重每个现金流。得到的结果叫久期 ( duration) 或麦考利久期 Macaulay Duration 。 </li></ul><ul><ul><li>假设债券每年付 $C , YTM=y 。 先假设每年复利一次,那么久期: </li></ul></ul><ul><ul><li>因为其分母是债券的价值,所以我们也可以用以下公式表示: </li></ul></ul>
  • 9. 久期:债券价格对利率变动的敏感度 <ul><ul><li>假设年票息 </li></ul></ul><ul><ul><li>dp / dy : </li></ul></ul>
  • 10. 久期:债券价格对利率变动的敏感度 <ul><li>dp/dy =实值久期 ( Dollar Duration) (Bloomberg calls it RISK.) </li></ul><ul><li>如果价格变化是以百分比来表示,那么叫修正久期 ( modified duration) : </li></ul>
  • 11. 国债的久期 在付息期间 ( w =从今天到下个付息日的部分时间): 在付息日( w = 1 ):
  • 12. 久期 Duration <ul><li>假设以下 3 个债券: </li></ul>Bond Yr 1 Yr 2 Yr 3 Yr 4 Yr 5 Duration A 5 5 5 5 105 4.5 Yr B 20 20 20 20 120 3.8 Yr C 0 0 0 0 100 5.0 Yr
  • 13. 久期的特性 ( 1 ) <ul><li>给予既定的到期期限,票息越高,久期越短 ( Holding maturity unchanged, increasing the coupon reduces duration) </li></ul><ul><ul><li>这是因为更高的现金流在短期内实现 </li></ul></ul><ul><ul><li>This follows because more cash flows are paid sooner. </li></ul></ul><ul><li>给予既定的到期期限和票息, 到期收益率越高,久期更短 ( Holding maturity and coupon unchanged, increasing yield leads to lower duration) </li></ul><ul><ul><li>这是因为高 YTM ,意味着高折现率,更远的现金流的现在值就越低了 </li></ul></ul><ul><ul><li>This follows because higher yield discount distant cash payment more heavily. </li></ul></ul><ul><li>给予既定的票息和 YTM ,更长的到期期限,更长的久期 ( Holding coupon and yield unchanged, increasing maturity leads to higher duration) </li></ul><ul><ul><li>看似很容易理解,但不是所有的债券都有此特性 </li></ul></ul><ul><ul><li>Seems clear but not true all the time. It only applies to premium bonds (see next page). </li></ul></ul>
  • 14. Zero coupon bond Discount bond Par bond Premium bond Perpetual bond Maturity Duration
  • 15. 久期的特性 ( 2 ) <ul><li>随着到期日的贴近,久期下滑 ( Duration declines as bond approaches maturity ) </li></ul><ul><ul><li>开始时慢速下滑 </li></ul></ul><ul><ul><li>期限低过 5 年后,下滑的速度加快。 </li></ul></ul><ul><ul><li>最后以 1 对 1 的比例下滑 </li></ul></ul><ul><li>成锯齿状变化 ( Sawtooth fluctuation ) </li></ul><ul><ul><li>形状来自于定期付票息: </li></ul></ul><ul><ul><ul><li>在付息日,久期上升 ( On coupon payment date, duration increases due to payment of the near-term coupon) </li></ul></ul></ul><ul><ul><ul><li>期限更长的债券,其上升的幅度更大 ( Long term bond has bigger jump (see table on next slide) ) </li></ul></ul></ul>
  • 16. Duration (in year) of a 12%, 12% yield 20 years bond
  • 17. 久期实用案例 1 <ul><li>债券免疫 Bond Immunization </li></ul><ul><ul><li>平衡再投资风险和价格风险 ( Balancing reinvestment risk and price risk is called immunization) </li></ul></ul><ul><ul><li>可通过匹配久期和投资期限来达到免疫 ( This can be done using duration matching: matching duration with the investor ’s horizon (See proof in class.)) </li></ul></ul>
  • 18. 久期实用案例 2 <ul><li>久期衡量债券价格的波动率(即对利率的敏感度) ( Duration as a measure of volatility to estimate price change due to change in interest rate): </li></ul>
  • 19. 案例 : 19.5 year 12% coupon bond priced at 12% yield. A 10 bp change in yield. <ul><li>Before coupon payment when duration = 7.48 </li></ul>
  • 20. 久期实用案例 3 <ul><li>当久期在债券付息日升高时,是否意味债券的风险增加了? When duration jumps on coupon payment day, does it mean that the bond suddenly begins to impose higher market risk on the investment? </li></ul><ul><ul><li>付息后,久期= 7.92 (duration = 7.92) </li></ul></ul><ul><ul><li>价格变化: </li></ul></ul><ul><ul><li>注意有现金流,而现金的久期等于零。投资经理必须决定如何投资收到的现金。这将影响其“组合”久期。 However, duration-matched manager has to decide how to reinvest the cash. His/her decision will change the “portfolio” duration. </li></ul></ul>
  • 21. 久期实用案例 4 <ul><li>随着期限的变短,久期也在下降 ( Duration of the bond change as time passes) </li></ul><ul><ul><li>举例: </li></ul></ul><ul><ul><ul><li>假设初始投资时: 20-year bond , 12 % coupon, 12% yield , duration = 8 years = 投资期限 (investment horizon) </li></ul></ul></ul><ul><ul><ul><li>4 年之后,投资期限= 4 years , 但是久期 = 7.5 years </li></ul></ul></ul><ul><ul><ul><li>所以必须重新平衡组合 rebalance the “portfolio”. </li></ul></ul></ul><ul><ul><ul><li>在什么时候最适合做重新平衡 When is the best time to do rebalancing? </li></ul></ul></ul><ul><ul><ul><li>在付息日。因此国债在 Feb 15, May 15, Aug 15, 及 Nov 15 的交易量特高。 </li></ul></ul></ul>
  • 22. 久期实用案例 5 <ul><li>债券市场用 DV01 来衡量久期 : </li></ul><ul><ul><li>That is, how much the price of bond will change for 1 bp change in yield. </li></ul></ul><ul><ul><li>DV01 = (Modified Duration x Full Price) x 0.0001 for $100 face value bond. </li></ul></ul>
  • 23. 久期实用案例 6 – 组合 <ul><li>一个债券组合的久期等于其加权平均久期。权重以债券市值计算。 ( A portfolio’s duration is simply the weighted average duration of the bonds in a portfolio. The weight is calculated by market value.) </li></ul>If yield curve shift up by 100 bp, what is the percentage value change of portfolio? Bond Market Value (mil $) Weight Mod Duration Weight x Mod Dur A 10 0.1 4 0.4 B 40 0.4 7 2.8 C 30 0.3 6 1.8 D 20 0.2 2 0.4 Total 100 1 5.4
  • 24. 凸性 Convexity <ul><li>凸性测量在利率和价格关系线上的弧线部分 Convexity measures the curvature (nonlinearity) in a bond-yield curve </li></ul><ul><li>所有没带期权的债券都存在正凸性 All non-callables exhibit some degree of positive convexity </li></ul><ul><li>正凸性意味在同量的利率下降和上升情况下, 债券价格的上升比下降来的大 Positive convexity indicates a bond ’s price rises more for a given decline in yield but fall less for a given increase in yield </li></ul><ul><li>凸性来自于债券的定价模式 Convexity exists because of bond price-yield relationship </li></ul><ul><li>凸性就像是在期权里的 Gamma Convexity is like Gamma in option </li></ul><ul><li>凸性越高越好 Convexity is desirable </li></ul>
  • 25. 从 Dp/Dy 到 D 2 p/Dy 2 <ul><li>From duration, we know: </li></ul><ul><li>The second derivative: </li></ul>
  • 26. 测量凸性 Measure Convexity 有些人用以下定义: 我们对凸性的定义 : <ul><li>注意: </li></ul><ul><li>我们测量凸性时收益是以小数表示。如果收益率是以%表示,那么当凸性= 100 时,凸性应写成 1 </li></ul>
  • 27. 凸性的特性 <ul><li>久期越长,凸性更大 Higher the duration, higher the convexity (see next slide). </li></ul><ul><li>现金流越分散,凸性越大 Higher the dispersion of cash flows, higher the convexity. </li></ul><ul><ul><li>比较两种债券:固定票息和零息,哪种债券的凸性较大? (no cash flow dispersion). </li></ul></ul><ul><ul><li>杠铃 ( Barbell) 组合的凸性比子弹的凸性来的高 Barbell has greater convexity than a bullet (draw a line between any two points on the duration-convexity curve to see the barbell has greater convexity.) </li></ul></ul>
  • 28. 凸性的特性 Coupon bond ’s convexity is sum of a duration-matched zero’s convexity.
  • 29. 凸性的特性 <ul><li>利率波动率增高时,凸性会越大 Higher the yield volatility, higher the value of convexity. </li></ul><ul><ul><li>波动率增高会对正凸性更有利 Higher volatility enhances the expected performance of positively convex positions. </li></ul></ul><ul><ul><li>那么,有谁会愿意买子弹式债券呢? Why would anybody want to buy bullet then? No free lunch. </li></ul></ul>
  • 30. 近似凸性 Approximate Convexity
  • 31. 应用久期和凸性测量价格变化 应用泰勒系列关系式 Taylor series expansion : http://en.wikipedia.org/wiki/Taylor_series
  • 32. Duration Approximation Duration and Convexity Approximation Yield Price
  • 33. 举例 <ul><li>假设修正久期= 4 年,凸性= 49 ,那么如果利率增加 100 基点,债券的价格会增加或降低多少? Suppose Modified Duration is 4 years and convexity is 49, what is the % change of price for 100 bp increase in yield? </li></ul><ul><li>-4x0.01+0.5x49x(.01) 2 =-0.03755 = -3.755% </li></ul>
  • 34. 近似久期 Approximate Duration <ul><li>举例 : 25-yr, 6% coupon, 9% yield, P 0 = 70.3570 </li></ul><ul><li>步骤: </li></ul><ul><ul><li>提高 YTM 10 bp (ie., Δ y=0.001), 用定价模式计算 P - = 69.6164 </li></ul></ul><ul><ul><li>降低 YTM 10 bp (ie., Δ y=0.001), 用定价模式计算 P + = 71.1105 </li></ul></ul><ul><ul><li>近似久期 Approximate duration = 10.62 </li></ul></ul>
  • 35. 久期实用案例 7 -相对久期 Matching $Duration <ul><li>假设你持有债券 X ,想置换债券 Y 。但是你期望保持其久期。 </li></ul><ul><li>Suppose you own Bond X and want to exchange for Bond Y. You want to ensure that after the swap, the $Duration is maintained. </li></ul>Bond Price Par Amount (PA) Market Value (MV) Modified Duration (MD) Dollar Duration ($D) X 80 10 mill 8 mill 5 8 mil x 5 / 100 = $400,000 (for 1% change in yield) Y 90 ? ? 4 $400,000 (imposed)
  • 36. 久期实用案例 7 -相对久期 Matching $Duration
  • 37. 债券免疫 Bond Immunization <ul><li>等配资产和负债的价值 </li></ul><ul><li>Matching assets and liabilities value. </li></ul><ul><ul><li>要等配资产和负债的现金流是很难的 </li></ul></ul><ul><ul><li>Cannot exactly match the cash flows. </li></ul></ul><ul><ul><li>但是我们确保资产的价值比负债高。当需要现金时,我们售出的资产大过负债的现金需求 </li></ul></ul><ul><ul><li>But can ensure the value of asset is greater than the value of liabilities. When fund is needed, can sell assets to cover cash needs. </li></ul></ul>
  • 38. 资金不足和资金过分充足 Under-funding and Over-funding <ul><li>考虑一退休基金有以下的现金需求: </li></ul><ul><li>Consider a pension fund has the following cash flow requirements: </li></ul>Year Pension Payment Present value (5% yield) 1 10,000 9,523.81 2 10,000 9,070.29 3 10,000 8,638.38 4 10,000 8,227.02 5 10,000 7,835.26 Total 50,000 43,294.77
  • 39. <ul><li>假设基金买了一只 5 %的平价债券( $43,294.77 ) </li></ul><ul><li>Suppose the fund invested in a bond with coupon 5%. Par amount = $43,294.77. </li></ul><ul><li>当利率上升后,基金的资产 < 负债 </li></ul>Modified Duration Value today If yield is 5% If yield is 6% Assets 4.33 $43,294.77 $43,294.77 $41,471.04 Liabilities 2.76 $43,294.77 $43,294.77 $42,123.64 Difference 0 0 -$652.6 (under-funded)
  • 40. 债券免疫理论 Bond Immunization Theorem <ul><li>Recall this: </li></ul>Where P 1 = Value after yield change (利率变化后的价值) P 0 = Value before yield change (利率变化前的价值) MD = Modified duration (修正久期) Δ y = change in yield (利率变化) C = Convexity (凸性)
  • 41. 债券免疫理论 Bond Immunization Theorem <ul><li>对负债来说 : </li></ul><ul><li>对资产来说 : </li></ul>
  • 42. 债券免疫理论 Bond Immunization Theorem <ul><li>我们的目的是: </li></ul><ul><ul><li>A 1  L 1 </li></ul></ul><ul><li>假设 : </li></ul><ul><ul><li>A 0 = L 0 和 </li></ul></ul><ul><ul><li>MD A = MD L </li></ul></ul><ul><li>那么 : </li></ul><ul><li>假如我们能保证 C A  C L , 那么 A 1  L 1 </li></ul>
  • 43. 债券免疫理论 Bond Immunization Theorem <ul><li>Back to the example. Suppose we invest in 2 bonds: </li></ul><ul><ul><li>2-year, value at par = 100, MD 2 = 1.85941 number of bonds = n 2 </li></ul></ul><ul><ul><li>5-year, value at par = 100, MD 5 = 4.329477, number of bonds = n 5 </li></ul></ul><ul><li>We can solve for the following equations: </li></ul>
  • 44. 债券免疫理论 Bond Immunization Theorem <ul><li>n 2 = 274.3406 </li></ul><ul><li>n 5 = 158.6071 </li></ul><ul><li>你可以证明其总凸性大过负债的凸性 </li></ul><ul><li>You can verify that the total convexity is greater than the convexity of the liabilities (=12.08251) </li></ul><ul><li>你也应该发觉到,如果要得到更大的凸性,你应该选择现金流分散的债券或债券组合 </li></ul><ul><li>You should also notice that in general, to get higher convexity, you should choose a bond portfolio with higher cash flow dispersion. </li></ul>
  • 45. 债券免疫理论应用时的问题 Bond Immunization in Practice <ul><li>实践中,可以供选择的债券很多 In practice, there are a large number of bonds to choose from, so there are many different combinations that may match the conditions: equal value, equal duration, and excess convexity. </li></ul><ul><li>要解决这个问题,我们需要多加其它的条件 To handle this problem, we need additional criteria </li></ul><ul><ul><li>收益最大化 Maximize yield (aggressive approach) </li></ul></ul>
  • 46. Max imize Yield (收益率最大化) <ul><li>要求收益最大化,但必须免疫(条件) </li></ul><ul><li>We can pick bond (n 1 ,n 2 …n N ) to maximize y A : </li></ul><ul><li>Subject to </li></ul>
  • 47. 债券免疫理论应用时的问题-重新组合 Bond Immunization in Practice - Rebalancing <ul><li>在一段时间内,负债的久期和资产的久期不会产生一样的变化 Change in duration of liabilities does not match change in duration of assets as time passes. </li></ul><ul><ul><li>举例 </li></ul></ul><ul><ul><ul><li>假设初开始投资期限为 8 年,我们购买了一只 20 年的债券( 12 %票息, 12 %收益率),久期为 8 年 Assume initial investment of a 20-year bond (12% coupon, 12% yield) with duration to match 8 years horizon. </li></ul></ul></ul><ul><ul><ul><li>4 年后, 20 年债券的久期掉到 7.5 年(假设) 4 years later, horizon reduces to 4 years but 20-year bond’s duration only reduced to say 7.5 years duration. </li></ul></ul></ul><ul><ul><ul><li>因此我们必须重新组合 Therefore he has to rebalance the “portfolio”. </li></ul></ul></ul>
  • 48. 为负债免疫的最佳方法是什么? What is the best way to immunize a liability? <ul><li>假设其它条件不变,买零息债券最好 Holding everything constant, buy zeros of same maturity period. </li></ul><ul><li>但是,零息债券的价格一般偏高(低收益),因此你需要更多的钱来达到目标收益 However, zeros provide lower yield, hence higher price. That is, you need more money to satisfy the yield target. </li></ul>

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