Investments.docx

Investments -- GRA6534 Innholdsfortegnelse {#innholdsfortegnelse.Overskriftforinnholdsfortegnelse} =================== [[Lecture 1] 2](#lecture-1) [[Overview of the Course] 2](#overview-of-the-course) [[Bonds] 2](#bonds) [[Equity & Housing] 2](#equity-housing) [[Funds] 2](#funds) [[Pricing di...

Investments -- GRA6534 Innholdsfortegnelse {#innholdsfortegnelse.Overskriftforinnholdsfortegnelse} =================== [[Lecture 1] 2](#lecture-1) [[Overview of the Course] 2](#overview-of-the-course) [[Bonds] 2](#bonds) [[Equity & Housing] 2](#equity-housing) [[Funds] 2](#funds) [[Pricing different Investments] 3](#pricing-different-investments) [[Risk-based explanation] 3](#risk-based-explanation) [[Behavioral explanation] 3](#behavioral-explanation) [[Risk-Expected Return relationship] 3](#risk-expected-return-relationship) [[Lecture 1 Videos] 5](#lecture-1-videos) [[Video -- Annuity, Geometric series] 5](#video-annuity-geometric-series) [[Video 2 -- Annuity, Final formula] 5](#video-2-annuity-final-formula) [[Video 3 -- Annuity due extension] 6](#video-3-annuity-due-extension) [[Video 4 -- Annuity generalization] 7](#video-4-annuity-generalization) [[Video 5 -- Perpetuity formula] 8](#video-5-perpetuity-formula) [[Video 6 -- Perpetuity due extension] 9](#video-6-perpetuity-due-extension) [[Lecture 2 Videos] 9](#lecture-2-videos) [[Consistency time-interest rate] 9](#consistency-time-interest-rate) [[Bond pricing] 10](#bond-pricing) [[Bond pricing exercise Holding Period Return (HPR)] 10](#bond-pricing-exercise-holding-period-return-hpr) [[Exercise Year-to-Maturity (YTM)] 11](#exercise-year-to-maturity-ytm) [[Lecture 2] 12](#lecture-2) [[Bonds] 12](#bonds-1) [[Bond Characteristics] 12](#bond-characteristics) [[Pricing] 12](#pricing) [[Accrued interest] 13](#accrued-interest) [[Yield to Maturity] 14](#yield-to-maturity) [[Lecture 3] 15](#lecture-3) [[What is the (expected) return on a bond?] 15](#what-is-the-expected-return-on-a-bond) [[No arbitrage bond pricing] 15](#no-arbitrage-bond-pricing) [[Relationship between inflation and bond pricing] 16](#relationship-between-inflation-and-bond-pricing) [[Interest rate risk] 17](#interest-rate-risk) [[Term structure of interest risk] 18](#term-structure-of-interest-risk) [[Expectations hypothesis vs Liquidity preference theory] 18](#expectations-hypothesis-vs-liquidity-preference-theory) [[Exam Question] 19](#exam-question) [[Videos lecture 3] 21](#videos-lecture-3) [[The term structure of interest rate] 21](#the-term-structure-of-interest-rate) [[Exercise forward rates] 21](#exercise-forward-rates) [[Exercise Expectation hypothesis] 22](#exercise-expectation-hypothesis) [[EH vs. LPT] 23](#eh-vs.-lpt) [[Exercise HPR] 23](#exercise-hpr) Giovanni Pagliardi [[Giovanni.pagliardi\@bi.no]](mailto:[email protected]) Office hours: Tuesday 13-14 & Wednesday 13-14 Office B4-013 Lecture 1 ========= Overview of the Course ---------------------- Et bilde som inneholder tekst, skjermbilde, Font, nummer Automatisk generert beskrivelse No Arbitrage: Key to find the value of any investment ### ### Bonds - Pricing - Term structure of interest rate ### Equity & Housing - Are returns predictable? ### Funds - Hedge funds - Open-end funds - Closed-end funds Pricing different Investments ----------------------------- ### Risk-based explanation - Rationality (Friedman,1953) ### Behavioral explanation - Overreaction - Overoptimism - Herzfeld Caribbean Basin (CUBA fund)  1984, Shiller - Sophisticated investors wipe out mispricing **But Arbitrage can be [costly]:** - Interest rate - Liquidity costs (bid-ask spreads) - Possibility to replicate the position ### Risk-Expected Return relationship \ [\$\$E\\left\\lbrack r \\right\\rbrack = \\frac{E\_{t\\left\\lbrack \\text{Gai}n\_{t + 1} \\right\\rbrack}}{P\_{t}} = \\frac{E\\left\\lbrack P\_{t + 1} \\right\\rbrack - P\_{t}}{P\_{t}}\$\$]{.math.display}\ \ [\$\$= \\frac{E\\left\\lbrack P\_{t + 1} \\right\\rbrack}{P\_{t}} - 1 \$\$]{.math.display}\ \ [\$\$1 + E\\left( r \\right) = \\frac{\\left( E\\left\\lbrack P\_{t + 1} \\right\\rbrack \\right)}{\\left( P\_{t} \\right)}\$\$]{.math.display}\ \ [\$\$\\mathbf{P}\_{\\mathbf{t}}\\mathbf{=}\\frac{\\mathbf{E}\\left\\lbrack \\mathbf{P}\_{\\mathbf{t + 1}} \\right\\rbrack}{\\mathbf{1 +}\\mathbf{E}\\left\\lbrack \\mathbf{r} \\right\\rbrack}\$\$]{.math.display}\ **Expectation!** The Uncertainty around this **Must incorporate Risk** expectation is missing in the numerator Example +-----------------------+-----------------------+-----------------------+ | Lottery A -- Pays 100 | Lottery B -- | Lottery C | | for sure | [*P* = 0, 5]{.math | | | |.inline} | | +=======================+=======================+=======================+ | \ | \ | \ | | [\$\$P = | [*P* = 0, 5 = 0]{.mat | [*P* = 0, 5 = 100]{.m | | \\frac{100}{1 + | h | ath | | r\_{f}} = |.display}\ |.display}\ | | 38\$\$]{.math | | | |.display}\ | \ | \ | | | [1 − *P* = 0, 5 = 200 | [1 − *P* = 0, 5 = 300 | | | ]{.math | ]{.math | | |.display}\ |.display}\ | +-----------------------+-----------------------+-----------------------+ | | \ | \ | | | [*E*\[payoff\] = 100] | [*E*\[Payoff\] = 200] | | | {.math | {.math | | |.display}\ |.display}\ | +-----------------------+-----------------------+-----------------------+ | | Assume Investors are | | | | willing to pay 80 USD | | | | for Every 100 USD | | +-----------------------+-----------------------+-----------------------+ | | \ | \ | | | [\$\$80 = | [\$\$P = | | | \\frac{100}{1 + | \\frac{200}{1 + 0,25} | | | E\\left\\lbrack r | = 160\$\$]{.math | | | \\right\\rbrack} = |.display}\ | | | E\\left\\lbrack r | | | | \\right\\rbrack = | | | | 25\\%\$\$]{.math | | | |.display}\ | | +-----------------------+-----------------------+-----------------------+ | | | Lottery B and C is | | | | not equally risky. | | | | | | | | \ | | | | [StDe*v*~*B*~ = *StDe | | | | v*~*C*~]{.math | | | |.display}\ | +-----------------------+-----------------------+-----------------------+ **Realized returns** [**≠**]{.math.inline} **Expected returns** 1. A statement on expected returns is always a statement on prices - 2. Expected returns are **[only]** determined by risk! +-----------------------------------+-----------------------------------+ | 1. Incorporate risk in the | 2. Use the [*r*~*f*~]{.math | | denominator |.inline} as the discount rate | +===================================+===================================+ | \ | \ | | [\$\$\\mathbf{P}\_{\\mathbf{t}}\\ | [\$\$\\mathbf{P}\_{\\mathbf{t}}\\ | | mathbf{=}\\frac{\\mathbf{E}\_{\\m | mathbf{=}\\frac{\\mathbf{E}\_{\\m | | athbf{t}}\\left\\lbrack | athbf{t}}\^{\\mathbf{Q}}\\left\\l | | \\mathbf{C}\\mathbf{F}\_{\\mathbf | brack | | {t | \\mathbf{C}\\mathbf{F}\_{\\mathbf | | + 1}} \\right\\rbrack}{\\mathbf{1 | {t | | +}\\mathbf{r}\_{\\mathbf{f}}\\mat | + 1}} \\right\\rbrack}{\\mathbf{1 | | hbf{+ | +}\\mathbf{r}\_{\\mathbf{f}}}\$\$ | | E}\\left\\lbrack | ]{.math | | \\mathbf{r}\_{\\mathbf{M}} |.display}\ | | \\right\\rbrack\\mathbf{+ | | | RP}}\$\$]{.math.display}\ | | +-----------------------------------+-----------------------------------+ | | \ | | | [**Q** **=** **Risk** **−** **neu | | | tral** **probability**]{.math | | |.display}\ | +-----------------------------------+-----------------------------------+ Lecture 1 Videos ================ ### Video -- Annuity, Geometric series 1. For every annuity, as well as for every perpetuity, we compute the present value [(PV)]{.math.inline} of this annuity, one period before the first coupon [(*C*~1~)]{.math.inline} \ [\$\$PV = \\frac{c}{1 + r} + \\frac{c}{\\left( 1 + r \\right)\^{2}} + \\frac{c}{\\left( 1 + r \\right)\^{3}} + \\ldots + \\frac{c}{\\left( 1 + r \\right)\^{n}}\$\$]{.math.display}\ 2. We need to factor out the first term, [\$(\\frac{c}{1 + r})\$]{.math.inline} \ [\$\$PV = \\frac{c}{1 + r} + \\frac{c}{\\left( 1 + r \\right)\^{2}} + \\frac{c}{\\left( 1 + r \\right)\^{3}} + \\ldots + \\frac{c}{\\left( 1 + r \\right)\^{n}}\$\$]{.math.display}\ \ [\$\$\\frac{c}{\\left( 1 + r \\right)}\*\\left\\lbrack 1 + \\frac{1}{1 + r} + \\frac{1}{\\left( 1 + r \\right)\^{2}} + \\ldots + \\frac{1}{\\left( 1 + r \\right)\^{n - 1}} \\right\\rbrack\$\$]{.math.display}\ A geometric series is one series according to which the ratio between two successive terms, is always constant. [\$\\frac{C\_{2}}{C\_{1}} = one\\ Constant = Lambda(\\lambda)\$]{.math.inline} \ [\$\$\\lambda = \\frac{1}{1 + r}\$\$]{.math.display}\ \ [1 + *λ* + *λ*^2^ + ... + *λ*^*n* − 1^]{.math.display}\ \ [*λ*^0^]{.math.display}\ \ [\$\$\\lambda\^{0} + \\lambda\^{1} + \\lambda\^{2} + \\ldots + \\lambda\^{n - 1} = \\sum\_{K = 0}\^{n - 1}\\lambda\^{K} = \\frac{1 - \\lambda\^{n}}{1 - \\lambda}\\ with\\ \\lambda = \\frac{1}{1 + r}\$\$]{.math.display}\ ### Video 2 -- Annuity, Final formula \ [\$\$PV = \\frac{c}{1 + r} + \\frac{c}{\\left( 1 + r \\right)\^{2}} + \\frac{c}{\\left( 1 + r \\right)\^{3}} + \\ldots + \\frac{c}{\\left( 1 + r \\right)\^{n}}\$\$]{.math.display}\ \ [\$\$\\frac{c}{\\left( 1 + r \\right)}\*\\left\\lbrack 1 + \\frac{1}{1 + r} + \\frac{1}{\\left( 1 + r \\right)\^{2}} + \\ldots + \\frac{1}{\\left( 1 + r \\right)\^{n - 1}} \\right\\rbrack\$\$]{.math.display}\ \ [\$\$PV = \\frac{C}{1 + r}\*\\frac{1 - \\lambda\^{n}}{1 - \\lambda}\\ with\\ \\lambda = \\frac{1}{1 + r}\$\$]{.math.display}\ \ [\$\$\\frac{C}{1 + r}\*\\frac{1 - \\left( \\frac{1}{\\left( 1 + r \\right)\^{n}} \\right)}{1 - \\left( \\frac{1}{1 + r} \\right)} = \\frac{C}{1 + r}\*\\frac{\\frac{\\left( 1 + r \\right)\^{n}}{\\left( 1 + r \\right)\^{n}} - \\left( \\frac{1}{\\left( 1 + r \\right)\^{n}} \\right)}{\\frac{1 + r}{1 + r} - \\left( \\frac{1}{1 + r} \\right)} = \\frac{C}{1 + r}\*\\frac{\\frac{\\left( 1 + r \\right)\^{n} - 1}{\\left( 1 + r \\right)\^{n}}}{\\frac{1 + r - 1}{1 + r}}\$\$]{.math.display}\ \ [\$\$= \\frac{C}{1 + r}\*\\frac{\\left( 1 + r \\right)\^{n} - 1}{\\left( 1 + r \\right)\^{n}}\*\\frac{1 + r}{r}\$\$]{.math.display}\ \ [\$\$\\frac{C\\left\\lbrack \\left( 1 + r \\right)\^{n} - 1 \\right\\rbrack}{r\*\\left( 1 + r \\right)\^{n}} = \\frac{C\*\\left( 1 + r \\right)\^{n}}{r\*\\left( 1 + r \\right)\^{n}} - \\frac{C}{r\*\\left( 1 + r \\right)\^{n}}\$\$]{.math.display}\ \ [\$\$C\\left\\lbrack \\frac{1}{r} - \\frac{1}{r\\left( 1 + r \\right)\^{n}} \\right\\rbrack = PV\_{0}\$\$]{.math.display}\ ### Video 3 -- Annuity due extension If the first payment is given at [*time* = 0]{.math.inline} Everything else stays the same, but remember, the [PV]{.math.inline} now goes to -1. We want the [PV]{.math.inline} to be presented in period {.math.inline}, not in [ − 1]{.math.inline}. \ [\$\$PV\_{- 1} = \\frac{c}{1 + r} + \\frac{c}{\\left( 1 + r \\right)\^{2}} + \\frac{c}{\\left( 1 + r \\right)\^{3}} + \\ldots + \\frac{c}{\\left( 1 + r \\right)\^{n}}\$\$]{.math.display}\ \ [\$\$\\frac{c}{\\left( 1 + r \\right)}\*\\left\\lbrack 1 + \\frac{1}{1 + r} + \\frac{1}{\\left( 1 + r \\right)\^{2}} + \\ldots + \\frac{1}{\\left( 1 + r \\right)\^{n - 1}} \\right\\rbrack\$\$]{.math.display}\ \ [\$\$PV = \\frac{C}{1 + r}\*\\frac{1 - \\lambda\^{n}}{1 - \\lambda}\\ with\\ \\lambda = \\frac{1}{1 + r}\$\$]{.math.display}\ \ [\$\$\\frac{C}{1 + r}\*\\frac{1 - \\left( \\frac{1}{\\left( 1 + r \\right)\^{n}} \\right)}{1 - \\left( \\frac{1}{1 + r} \\right)} = \\frac{C}{1 + r}\*\\frac{\\frac{\\left( 1 + r \\right)\^{n} - 1}{\\left( 1 + r \\right)\^{n}}}{\\frac{1 + r - 1}{1 + r}}\$\$]{.math.display}\ \ [\$\$= \\frac{C}{1 + r}\*\\frac{\\left( 1 + r \\right)\^{n} - 1}{\\left( 1 + r \\right)\^{n}}\*\\frac{1 + r}{r}\$\$]{.math.display}\ \ [\$\$\\frac{C\\left\\lbrack \\left( 1 + r \\right)\^{n} - 1 \\right\\rbrack}{r\*\\left( 1 + r \\right)\^{n}} = \\frac{C\*\\left( 1 + r \\right)\^{n}}{r\*\\left( 1 + r \\right)\^{n}} - \\frac{C}{r\*\\left( 1 + r \\right)\^{n}}\$\$]{.math.display}\ \ [\$\$C\\left\\lbrack \\frac{1}{r} - \\frac{1}{r\\left( 1 + r \\right)\^{n}} \\right\\rbrack\*\\left( 1 + r \\right) = PV\_{0}\$\$]{.math.display}\ ### Video 4 -- Annuity generalization \ [\$\$PV\_{1} = \\frac{c}{1 + r} + \\frac{c}{\\left( 1 + r \\right)\^{2}} + \\frac{c}{\\left( 1 + r \\right)\^{3}} + \\ldots + \\frac{c}{\\left( 1 + r \\right)\^{n}}\$\$]{.math.display}\ \ [\$\$\\frac{c}{\\left( 1 + r \\right)}\*\\left\\lbrack 1 + \\frac{1}{1 + r} + \\frac{1}{\\left( 1 + r \\right)\^{2}} + \\ldots + \\frac{1}{\\left( 1 + r \\right)\^{n - 1}} \\right\\rbrack\$\$]{.math.display}\ \ [\$\$PV = \\frac{C}{1 + r}\*\\frac{1 - \\lambda\^{n}}{1 - \\lambda}\\ with\\ \\lambda = \\frac{1}{1 + r}\$\$]{.math.display}\ \ [\$\$\\frac{C}{1 + r}\*\\frac{1 - \\left( \\frac{1}{\\left( 1 + r \\right)\^{n}} \\right)}{1 - \\left( \\frac{1}{1 + r} \\right)} = \\frac{C}{1 + r}\*\\frac{\\frac{\\left( 1 + r \\right)\^{n} - 1}{\\left( 1 + r \\right)\^{n}}}{\\frac{1 + r - 1}{1 + r}}\$\$]{.math.display}\ \ [\$\$= \\frac{C}{1 + r}\*\\frac{\\left( 1 + r \\right)\^{n} - 1}{\\left( 1 + r \\right)\^{n}}\*\\frac{1 + r}{r}\$\$]{.math.display}\ \ [\$\$\\frac{C\\left\\lbrack \\left( 1 + r \\right)\^{n} - 1 \\right\\rbrack}{r\*\\left( 1 + r \\right)\^{n}} = \\frac{C\*\\left( 1 + r \\right)\^{n}}{r\*\\left( 1 + r \\right)\^{n}} - \\frac{C}{r\*\\left( 1 + r \\right)\^{n}}\$\$]{.math.display}\ \ [\$\$C\\left\\lbrack \\frac{1}{r} - \\frac{1}{r\\left( 1 + r \\right)\^{n}} \\right\\rbrack\*\\left( 1 + r \\right)\^{- 1} = PV\_{0}\$\$]{.math.display}\ ### Video 5 -- Perpetuity formula \ [\$\$PV = \\frac{c}{1 + r} + \\frac{c}{\\left( 1 + r \\right)\^{2}} + \\frac{c}{\\left( 1 + r \\right)\^{3}} + \\ldots + \\frac{c}{\\left( 1 + r \\right)\^{n}} + \\ldots\$\$]{.math.display}\ \ [\$\$\\frac{c}{\\left( 1 + r \\right)}\*\\left\\lbrack 1 + \\frac{1}{1 + r} + \\frac{1}{\\left( 1 + r \\right)\^{2}} + \\ldots + \\frac{1}{\\left( 1 + r \\right)\^{n - 1}} + \\ldots \\right\\rbrack\$\$]{.math.display}\ \ [\$\$PV = \\frac{C}{1 + r}\*\\frac{1 - \\lambda\^{n}}{1 - \\lambda}\\ with\\ \\lambda = \\frac{1}{1 + r}\$\$]{.math.display}\ \ [\$\$\\lim\_{N \\rightarrow \\infty}\\frac{\\left( 1 - \\lambda \\right)\^{N}}{\\left( 1 - \\lambda \\right)} = \\frac{1}{1 - \\lambda}\\text{\\ if\\ }\\left\| \\lambda \\right\| \< 1\$\$]{.math.display}\ \ [\$\$\\frac{C}{1 + r}\*\\frac{1}{1 - \\left( \\frac{1}{1 + r} \\right)} = \\frac{C}{1 + r}\*\\frac{\\frac{1}{1 + r - 1}}{1 + r} = \\frac{C}{1 + r}\*\\frac{1 + r}{r}\$\$]{.math.display}\ \ [\$\$PV\_{0} = \\frac{C}{r}\$\$]{.math.display}\ ### Video 6 -- Perpetuity due extension \ [\$\$PV = \\frac{c}{1 + r} + \\frac{c}{\\left( 1 + r \\right)\^{2}} + \\frac{c}{\\left( 1 + r \\right)\^{3}} + \\ldots + \\frac{c}{\\left( 1 + r \\right)\^{n}} + \\ldots\$\$]{.math.display}\ \ [\$\$\\frac{c}{\\left( 1 + r \\right)}\*\\left\\lbrack 1 + \\frac{1}{1 + r} + \\frac{1}{\\left( 1 + r \\right)\^{2}} + \\ldots + \\frac{1}{\\left( 1 + r \\right)\^{n - 1}} + \\ldots \\right\\rbrack\$\$]{.math.display}\ \ [\$\$PV = \\frac{C}{1 + r}\*\\frac{1 - \\lambda\^{n}}{1 - \\lambda}\\ with\\ \\lambda = \\frac{1}{1 + r}\$\$]{.math.display}\ \ [\$\$\\lim\_{N \\rightarrow \\infty}\\frac{\\left( 1 - \\lambda \\right)\^{N}}{\\left( 1 - \\lambda \\right)} = \\frac{1}{1 - \\lambda}\\text{\\ if\\ }\\left\| \\lambda \\right\| \< 1\$\$]{.math.display}\ \ [\$\$\\frac{C}{1 + r}\*\\frac{1}{1 - \\left( \\frac{1}{1 + r} \\right)} = \\frac{C}{1 + r}\*\\frac{\\frac{1}{1 + r - 1}}{1 + r} = \\frac{C}{1 + r}\*\\frac{1 + r}{r}\$\$]{.math.display}\ \ [\$\$\\frac{C}{r}\*\\left( 1 + r \\right)\^{- 2} = PV\_{0}\$\$]{.math.display}\ Lecture 2 Videos ================ ### Consistency time-interest rate \ [\$\$\\left\\{ \\begin{matrix} \\text{Time\\ in\\ semesters} \\\\ i\_{\\text{annual}} = 3\\% \\\\ \\end{matrix} \\right.\\ \$\$]{.math.display}\ \ [1 \* (1+*i*~annual~)^1^ = 1 \* (1+*i*~6*m*~)^2^]{.math.display}\ \ [\$\${i\_{6m} = \\sqrt{1 + i\_{\\text{annual}}}\\ - 1 }{\\sqrt{1,03} - 1 = 0,0149 = 1,49\\%}\$\$]{.math.display}\ ### Bond pricing ---------------------------------------------------------------------------------------------------------------- T CR (annual) PAR \ Coupons paid [*R*~6*M*~]{.math.display}\ ---------- ----------------------- ------------------------------- ------------------------------ -------------- 10 years \ \ \ Every 6m [8%]{.math.display}\ [1 000*USD*]{.math.display}\ [3%]{.math.display}\ ---------------------------------------------------------------------------------------------------------------- \ [\$\$P = \\frac{40}{\\left( 1 + r \\right)\^{1}} + \\frac{40}{\\left( 1 + r \\right)\^{2}} + \\frac{40}{\\left( 1 + r \\right)\^{3}} + \\ldots + \\frac{40}{\\left( 1 + r \\right)\^{20}} + \\frac{1000}{\\left( 1 + r \\right)\^{20}} \$\$]{.math.display}\ \ [\$\$C\*\\left\\lbrack \\frac{1}{r} - \\frac{1}{\\left( r\*\\left( 1 + r \\right) \\right)\^{n}} \\right\\rbrack + \\frac{1000}{\\left( 1 + r \\right)\^{20}} =\$\$]{.math.display}\ \ [\$\$40\*\\left\\lbrack \\frac{1}{0,03} - \\frac{1}{0,03\*{\\left( 1 + 0,03 \\right))}\^{20}} + \\frac{1000}{\\left( 1 + 0,03 \\right)\^{20}} \\right\\rbrack = 1\\ 148,77\\ USD\$\$]{.math.display}\ ### Bond pricing exercise Holding Period Return (HPR) --------------------------------------------------------------------------------------------------------------- T CR (annual) PAR \ Coupons paid [*R*~6*M*~]{.math.display}\ --------- ----------------------- ------------------------------- ------------------------------ -------------- 3 years \ \ \ Every 6m [8%]{.math.display}\ [1 000*USD*]{.math.display}\ [4%]{.math.display}\ --------------------------------------------------------------------------------------------------------------- \ [*Cf*~0~ = (50,50,50,50,50,1050)]{.math.display}\ \ [\$\$HPR = \\frac{P\_{1} + C - P\_{0}}{P\_{0}}\$\$]{.math.display}\ \ [\$\$P\_{0} = \\frac{50}{\\left( 1 + r \\right)\^{1}} + \\frac{50}{\\left( 1 + r \\right)\^{2}} + \\frac{50}{\\left( 1 + r \\right)\^{3}} + \\frac{50}{\\left( 1 + r \\right)\^{4}} + \\frac{50}{\\left( 1 + r \\right)\^{5}} + \\frac{50}{\\left( 1 + r \\right)\^{6}} + \\frac{1000}{\\left( 1 + r \\right)\^{6}}\$\$]{.math.display}\ \ [\$\$P = C\*\\left\\lbrack \\frac{1}{r} - \\frac{1}{\\left( r\*\\left( 1 + r \\right) \\right)\^{n}} \\right\\rbrack + \\frac{1000}{\\left( r\*\\left( 1 + r \\right) \\right)\^{n}} \$\$]{.math.display}\ \ [\$\$P\_{0} = 50\*\\left\\lbrack \\frac{1}{0,04} - \\frac{1}{0,04\\left( 1 + 0,04 \\right)\^{6}} \\right\\rbrack + \\frac{1000}{\\left( 1 + 0,04 \\right)\^{6}}\$\$]{.math.display}\ \ [*P*~0~ = 1 052, 4 *USD*]{.math.display}\ \ [*Cf*~1~ = (50,50,50,50,1050)]{.math.display}\ \ [\$\$P\_{1} = \\frac{50}{\\left( 1 + r \\right)\^{1}} + \\frac{50}{\\left( 1 + r \\right)\^{2}} + \\frac{50}{\\left( 1 + r \\right)\^{3}} + \\frac{50}{\\left( 1 + r \\right)\^{4}} + \\frac{50}{\\left( 1 + r \\right)\^{5}} + \\frac{1000}{\\left( 1 + r \\right)\^{5}}\$\$]{.math.display}\ \ [\$\$P\_{1} = 50\*\\left\\lbrack \\frac{1}{0,04} - \\frac{1}{0,04\\left( 1 + 0,04 \\right)\^{5}} \\right\\rbrack + \\frac{1000}{\\left( 1 + 0,04 \\right)\^{5}}\$\$]{.math.display}\ \ [*P*~1~ = 1 044, 5 *USD*]{.math.display}\ \ [\$\$HPR = \\frac{P\_{1} + C - P\_{0}}{P\_{0}}\$\$]{.math.display}\ \ [\$\$HPR = \\frac{1044,5 + 50 - 1052,4}{1052,4} = 0,04 = 4\\%\$\$]{.math.display}\ ### Exercise Year-to-Maturity (YTM) ------------------------------------------------------------------------------------------------------------------ T CR (annual) PAR \ Coupons paid [*P*~0~]{.math.display}\ --------- ----------------------- -------------------------------- -------------------------------- -------------- 2 years \ \ \ Every year [8%]{.math.display}\ [1 000 *USD*]{.math.display}\ [1 000 *USD*]{.math.display}\ ------------------------------------------------------------------------------------------------------------------ \ [*CF* = (40, 1040)]{.math.display}\ \ [\$\$P\_{0} = \\frac{40}{\\left( 1 + YTM \\right)\^{1}} + \\frac{40}{\\left( 1 + YTM \\right)\^{2}}\$\$]{.math.display}\ \ [*Define* *x* = 1 + *YTM*]{.math.display}\ \ [\$\$1\\ 000 = \\frac{40}{\\left( x \\right)\^{1}} + \\frac{40}{\\left( x \\right)\^{2}}\$\$]{.math.display}\ \ [1000*x*^2^ = 40*x* + 1040]{.math.display}\ \ [1000*x*^2^ − 40*x* − 1040 = 0]{.math.display}\ \ [\$\$X\_{\\frac{1}{2}} = \\frac{40 \\pm \\sqrt{1600 + 4\*1000\*1040}}{2\*1000}\$\$]{.math.display}\ \ [\$\$\\frac{40 \\pm \\sqrt{2040}}{2000}\$\$]{.math.display}\ \ [*x*~1~ =  − 1  ∨ *x*~2~ = 1, 04 ]{.math.display}\ \ [*x*~1~ =  − 1 = 1 + *YTM* =  \> *YTM* =  − 2]{.math.display}\ \ [*x*~2~ = 1, 04 = 1 + *YTM* =  \> *YTM* = 0, 04 = 4%]{.math.display}\ Lecture 2 ========= Recap Lecture 1 \ [\$\$P\_{t} = \\frac{\\sum\_{K}\^{}\\mathbb{E}\\left\\lbrack CF\_{t + k} \\right\\rbrack}{\\left( 1 + r \\right)\^{K}}\$\$]{.math.display}\ +-----------------------+-----------------------+-----------------------+ | Shock | Realized Return | Expected Return | +=======================+=======================+=======================+ | 1. Cash Flow Shock | \ | Does not (necessary) | | (+) | [ \> 0]{.math | change | | |.display}\ | | | | | | | | (Lottery B to C) | | +-----------------------+-----------------------+-----------------------+ | 2- Discount Rate | \ | Increase | | shock | [ \ 0]{.math.inline}) Long term interest rate is ONLY a function of future expected short- term interest rates [↓]{.math.inline} [(1+*r*~0, 2~)^2^ = (1+*r*~1, 2~)^1^ \* (1\**E*\[*r*~1, 2~\])^1^]{.math.inline} [ ]{.math.inline}No Risk Risk [Risk neutrality] Exam Question ------------- +-----------------------+-----------------------+-----------------------+ | | **EH** | **LPT** | +=======================+=======================+=======================+ | **Upward** | \ | !I don't know! | | | [*E*\[*r*~1, 2~\] \>  | | | | *r*~1~]{.math | (Unless you observe | | |.display}\ | very steep term | | | | structure) | +-----------------------+-----------------------+-----------------------+ | **Flat** | \ | \ | | | [*E*\[*r*~1, 2~\] = * | [*E*\[*r*~1, 2~\] \