{"text_format":"3","workbook":"xlsx","n_sheets":5,"sheets":["Basics","RNBondpr","RA World","2Factor","Exercises"],"not_rendered":{"charts":3,"images":10,"macros":1},"read":"whole","date_system":"1900"}

{"sheet":"Basics","index":0,"n_rows":3696,"n_cols":20,"rows_kept":3696}
["Let's assume that everyone in the economy is risk-neutral."]
["We need to assume that there exists a continuously evolving instantaneous rate of interest, r."]
["In this setting, the value of any T-period 0-coupon bond, with $1 Face-value is:"]
{"$rows_empty":9}
["Implication 1:"]
[null,"If the instantaneous interest rate is a constant (and known to be so) then the yield on any T-period bond is r."]
[]
["Of course to get more results, beyond Implication 1, we have to specify a law of motion for r--that describes probabilistically (and quantitatively)--how r can move around in the future."]
[]
["The most basic model of short-rate dynamics is Vasicek's:",{"$nulls":12},"The transition density in discrete time:",{"$nulls":3},"Normal with the following conditional mean and variance:"]
[{"$nulls":13},"    E[r(t) | r(t-1)] = theta + (r(t-1) - theta) * exp(-k * delta-t)"]
[{"$nulls":13},"  Example:  ",null,"delta-t","E(r(t+delta-t))",null,"Var(r+ delta-t)","sigma(r+ delta-t)"]
["Parameter Definitions:",{"$nulls":14},{"$f":"B44","v":2.7397260273972603E-3},{"$f":"$B$40+($B$39-$B$40)*EXP(-$B$41*P21)","v":0.030021893806242213},null,{"$f":"$B$42^2/(2*$B$41)*(1-EXP(-2*$B$41*P21))","v":2.7337299130782841E-7},{"$f":"SQRT(S21)","v":5.2285083083784844E-4}]
[null,"q","Long-term mean of the short-rate. (annualized)",{"$nulls":12},{"$f":"1/52.14","v":0.019179133103183737},{"$f":"$B$40+($B$39-$B$40)*EXP(-$B$41*P22)","v":0.03015226197666158},null,{"$f":"$B$42^2/(2*$B$41)*(1-EXP(-2*$B$41*P22))","v":1.8887848898092169E-6},{"$f":"SQRT(S22)","v":1.3743307061290658E-3}]
[null,"k","Pull toward the long-term mean.",{"$nulls":12},{"$f":"1/12","v":0.08333333333333333},0.03064493014968382,null,7.8016675598157825E-6,2.7931465338960973E-3]
[null,"s","standard deviation of short-rate (annualized)",{"$nulls":12},{"$f":"1/4","v":0.25},0.03181269246922018,null,2.0604997122772542E-5,0.004539272752630375]
[{"$nulls":15},{"$f":"1/2","v":0.5},0.03329679953964361,null,3.4416939742673656E-5,5.866595242785517E-3]
["Mathematical notation:",{"$nulls":14},1,0.035506710358827784,null,4.9881467625334043E-5,0.007062681333978905]
[null,"dr","The instantaneous change in the short rate, r.",{"$nulls":12},5,0.03981684361111266,null,6.24790335857561e-05,0.00790436800672616]
[null,"dt","The instantaneous passage of time (deterministic, or non-random)",{"$nulls":12},10,0.039996645373720975,null,6.2499992966551579E-5,0.0079056937055866]
[null,"dw","Standard Brownian Motion  (the way randomness is introduced).",{"$nulls":12},100,0.04,null,6.25e-05,0.007905694150420948]
[]
["Notes:",{"$nulls":12},"Steady state mean:",null,"theta",{"$f":"B40","v":0.04}]
[null,"If r equals theta, then there is no expected change in the short rate--the short rate is expected to be theta at any future date."]
[null,"If r exceeds theta, then the short rate is expected to fall (toward theta) in the future.",{"$nulls":11},"  Var[ r(t) | r(t-1) ] = sigma^2 / (2 * k) * (1 - exp(-2*k*delta-t))"]
[null,"Conversely, if r is less than theta, then the short-rate is expected to rise (toward theta) in the future."]
[{"$nulls":13},"Steady state variance: sigma^2 / (2*k)",{"$nulls":4},{"$f":"B42^2/(2*B41)","v":6.25e-05}]
[]
["To develop intuition about how this process works, and what the implications for different parameter values affect possible short-rate paths, we can simulate."]
[null,"I wrote the macro OnePdemo attached to this spreadsheet to take a single path drawn from this process."]
["r",0.03]
["q",0.04]
["k",0.8]
["s",0.01]
["T",10,"  (in years)"]
["Delta-t",{"$f":"1/365","v":2.7397260273972603E-3},"  (in years)"]
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{"sheet":"RNBondpr","index":1,"n_rows":54,"n_cols":11,"rows_kept":54}
["Under the Vasicek model, the assumption is that the instantaneous interest rate follows the following mean-reverting stochastic process:"]
[]
[]
[]
[]
["If the economy is risk averse, then the only thing that affects the price of a long-term, 0-coupon bond is the interest rate path over that bond's term."]
[]
["Here is the value of a $1 0-coupon boind that matures at time T:"]
[]
[]
[]
[]
[]
["In order to implement this in discrete time (Monte Carlo), we need the transition density: the density of r(t+1) conditional on r(t)."]
[null,"The reason that this is not jjust the continuous-time analog is that we have to consider the discrete path."]
[null,"Consider for example the case where k = 2, r = .05, theta = .04, and sigma = .015.  If delta-t is one year and we simply took the drift in the diffusion at"]
[null,"  face value, then we would have an expectation that the expected r in one year is 6%.  This is obviously wrong as we deon't expect"]
[null,"   the interest rate to overshoot the long-run mean (theta)."]
[]
["The density of r(t) conditional on r(0) is Normal with mean "]
[]
["So the correct expectation of the instantaneous interest rate after one year in the above example is:  ",{"$nulls":8},{"$f":"0.04+0.05*EXP(-2*1)","v":0.046766764161830635}]
[null,null,"similarly, the variance of the rate at time t + tau conditional on the rate at time t is: ",{"$nulls":7},{"$f":"((0.015^2)/(2*2))*(1-EXP(-2*2*1))","v":5.52197453125087e-05}]
[{"$nulls":3},"and the conditional standard deviation is:",{"$nulls":6},{"$f":"SQRT(K23)","v":0.007430998944456169}]
["Term",{"$f":"4","v":4},"  (in years)"]
["r",0.04]
["q",0.04]
["k",0.8]
["s",0.03]
["Delta-t",{"$f":"1/365","v":2.7397260273972603E-3},"  (in years)"]
[]
["nsims",10000]
[]
[]
["The VB module Bondprice uses the above inputs to compute the zero-coupon bond price."]
[]
[]
["Price:",{"v":0.8535683825515629,"d":"0.85356838255"},"  (Price per $1 face value of 0-coupon default-free bond.)"]
[null,{"$f":"VALUE(B38)","v":0.8535683825515629},"  (I don't fully understand why VBA writes the price out as a text value, but this makes it a number)"]
["YTM:",{"$f":"LN(B39)/-B25","v":0.039582404950736186}]
[]
[]
["Exercises:"]
[null,"You should develop intuition about the Vasicek model in a risk-neutral setting."]
[null,"   This should include: "]
[{"$nulls":3},"plotting out yield curves for given r(0), theta, k, and sigma."]
[{"$nulls":3},"evaluate the effect on the base-case yield curve of a change in each of the parameters in turn (sigma, k, and theta)."]
[{"$nulls":3},"evaluate the effect on the base-case yield curve of changes in r."]
[{"$nulls":4},"Look at r less than theta."]
[{"$nulls":4},"Look at r greater than theta."]
[]
[null,"In particular, there are 2 effects that you should understand:"]
[null,null,"Convexity  (This effect is larger for a bigger sigma, cet par., and for longer term, cet. par.)"]
[null,null,"Expectations (Theta is the long-run mean of the short-rate.)"]

{"sheet":"RA World","index":2,"n_rows":87,"n_cols":8,"rows_kept":87}
[]
["We now extend the model from the previous worksheet to allow for a risk premium associated with term."]
[null,"This would be the case if investors were not indifferent to buying a 5-Year zero-coupon bond, or buying a 1-year bond, five years in a row--for the same expected holding period return."]
[]
["Recall the assumed interest rate process in the Vasicek model:"]
[]
[]
["In the past worksheet we looked at bond prices in this model, under the assumption that all investors are risk-neutral."]
[]
["In general, investors are risk-averse, so we have to adapt the model to handle this."]
[]
["The modeling structure we find convenient is the same as we use in Black & Scholes: Expectations in the equivalent risk-neutral world."]
["  In order to use this powerful method, we have to find the unique equivalent risk-neutral world."]
[]
["We adjust the drift of the short-rate to reflect the effect of a risk premium in the market."]
[]
[]
["This r* process is the risk-neutralized short-rate process, such that when we plug it into the fundamental pricing equation, we will get the correct bond prices (under the assumptions of the Vasicek model)."]
{"$rows_empty":7}
["We can use Monte Carlo, as we did in the previous worksheet, but use r* instead of r."]
[null,"Note that all this involves is changing the theta to theta-hat, and then everything else is the same as in the risk-neutral case."]
[{"$nulls":3},"The process with theta-hat as the long-run mean is the answer to the question:  What would the interest rate process be"]
[{"$nulls":3},"    so that when we take expectations we get a yield curve that has the same risk premium as the actual yierld curve."]
[]
[{"$nulls":3},"A couple of points to note about the risk-poremium parameter, lambda:"]
[{"$nulls":3},"    1. If lambda is 0, then there is no risk premium associated with the term of interest rates."]
[{"$nulls":3},"    2. lambda can be positive or negative."]
[{"$nulls":3},"    ","If lambda is positive, this means that there is a liquidity premium.  Investors require--and borrowers are willing to pay--"]
[{"$nulls":4},"   a premium above expected future rates to lend and borrow for a longer term."]
[{"$nulls":4},"If lamba is negative, this means that there is a solidity premium.   Investors require--and borrowers are willing to pay--"]
[{"$nulls":4},"   a premium above expected future rates to lend and borrow for a shorter term."]
[{"$nulls":3},"For the most part we expect lambda to be positive."]
[null,"We can obtain the value of a 0-coupon bond using Monte Carlo, as before but we use theta-hat instead of theta."]
[]
[]
["However, since the transition densities are normal, the solution to this expectation is available in closed form."]
[]
["This is a one-factor model, since there is only one source of randomness -- the instantaneous short rate of interest, r."]
["It is a member of the affine class of term structure models."]
["To see why, let's look at a closed-form expression for the value of a T-period 0-coupon bond at time t:"]
["Let: "]
[{"$nulls":4},"and,"]
[]
[]
[]
["Now we have that:"]
[{"$nulls":6},"and we know:"]
[]
[]
[]
["So this means that we can write the T-period spot rate:"]
[]
["I have coded the expression for the value of a zero-coupon bond in the VBA function VasZero1F()."]
[]
["Example:","r(0)",0.04,"  This input to the model is a \"state variable\""]
[null,"k",1,"  "]
[null,"sigma",0.05,"     These inputs to the model are \"parameters\""]
[null,"theta",0.055,null,"              Half life = ln(2) / k:",null,{"$f":"LN(2)/C62","v":0.6931471805599453},"     -- it takes this many years for the short rate to cover half the distance from its"]
[null,"lambda",0.01,{"$nulls":4},"        current level to the long run mean on average."]
[null,"theta-hat",{"$f":"C64+C65/C62","v":0.065},"  In terms of bond prices, this is the relevant long-run mean of the short-rate* process."]
[]
[]
[]
["T","r","V(t,T)"]
[{"$f":"1/12","v":0.08333333333333333},{"$f":"LN(C71)/-A71","v":0.04101060488659687},{"$f":"VasZero1F($C$62,$C$66,$C$63,$C$61,A71)","v":0.996588282771224}]
[{"$f":"3/12","v":0.25},0.04285839712570798,{"$f":"VasZero1F($C$62,$C$66,$C$63,$C$61,A72)","v":0.989342597574721}]
[{"$f":"6/12","v":0.5},0.04525372898853277,0.9776272031450363,0.045300323546676145]
[1,0.04898687197838031,0.9521936300350136]
[2,0.05371574557356462,0.8981380507507384]
[3,0.05641558608612912,0.8443005395486483,0.05684180868896845]
[5,0.05915532643648711,0.7439535847510507]
[7,0.060449359694847195,0.654983314403865]
[9,0.06118086408112341,0.5765877283803066]
[10,0.06143760214997078,0.5409783488753044]
[15,0.06220833379218682,0.3933226518460796,0.06292833382481638]
[20,0.06259375000231879,0.2859681036599124]
[30,0.06297916666666674,0.15116625822925403]
[40,0.063171875,0.07990834408983491]
[50,0.0632875,0.042240533899407816]
[100,0.06351875000000001,1.7434750539164442E-3]
[10000,0.0637476875,1.4038181288415253E-277]

{"sheet":"2Factor","index":3,"n_rows":33,"n_cols":63,"rows_kept":33}
["If the term structure did follow a 1-factor Vasicek model, then all rates would be perfectly correlated, and long-term rates would be very stable."]
["To get more realistic yield curve dynamics we need multiple factors."]
["The idea in a 3-factor Vasicek model is that the instantaneous short rate is the sum of factor 1, factor 2, and factor 3."]
["Where each factor follows its own mean reverting process:"]
[]
[{"$nulls":10},"   etc."]
{"$rows_empty":7}
["Let's look at a 2-factor model.  Both factors evolve according to the Vasicek law of motion.  So we need 4 parameters for each factor, as well as the correlation between the factors, for  a total of 9 parameters."]
["  In particular, one of the factors is a ``short-term'' factor in that it has a high k, and the other is a long-term factor in that it has a low k."]
["Parameter","Factor 1","Factor 2"]
["k",1.6,0.2]
["theta",0.02,0.008]
["lambda",0.0075,0.003]
["sigma",0.01,0.02]
["theta-hat",{"$f":"B18+B19/B17","v":0.0246875},{"$f":"C18+C19/C17","v":0.023}]
["factor value",0.008,0.005,"  According to the model, it is these 2 factors whose movement through time causes the yield curve to change."]
["Also, the factors can be correlated, so we need a ninth parameter: rho."]
["rho:",-0.4755,"  This is the instantaneous correlation between shocks to the 2 factors."]
["delta-t",0.5,"(expressed in terms of number of years)"]
[]
["The negative correlation between the factors seems to be required by the data.  The reason for this is that the long end of the yield curve has a lot of volatility -- relative to the short-end."]
[]
["There is a closed-form expression for a bond price in the multi-factor Vasicek model.  I have programmed the 2-factor model into a User-defined function, VasZero"]
["  which is linked to this spreadsheet.",null,null,"You should change the parameters and state variables to see the effect of each on the yield curve, and its dynamics."]
["Term (Years):",null,0.25,0.5,1,1.5,2,2.5,3,3.5,4,4.5,5,5.5,6,6.5,7,7.5,8,8.5,9,9.5,10,10.5,11,11.5,12,12.5,13,13.5,14,14.5,15,15.5,16,16.5,17,17.5,18,18.5,19,19.5,20,20.5,21,21.5,22,22.5,23,23.5,24,24.5,25,25.5,26,26.5,27,27.5,28,28.5,29,29.5,30]
[null,"df:",{"$f":"VasZero(C31,$B$22,$B$18,$B$17,$B$20,$C$19,$C$22,$C$18,$C$17,$C$20,$C$19,$B$24)","v":0.99603819193731},{"$f":"VasZero(D31,$B$22,$B$18,$B$17,$B$20,$C$19,$C$22,$C$18,$C$17,$C$20,$C$19,$B$24)","v":0.9909497313844889},0.9786377944701256,0.9646858133444306,0.9498737482283549,0.9345945871348189,0.9190611019601411,0.9033996106812864,0.88769276745755,0.8719994224693853,0.8563641378116886,0.8408219942645084,0.8254012138152087,0.8101247346646618,0.7950112559321241,0.7800759921734857,0.7653312531705588,0.7507869077110344,0.7364507636887663,0.7223288841128431,0.7084258520901511,0.6947449942217427,0.6812885696360862,0.668057930391707,0.6550536578995216,0.6422756791858856,0.6297233661599628,0.6173956205172838,0.6052909464763075,0.593407513186438,0.5817432083495507,0.5702956843512488,0.5590623979937263,0.5480406447519447,0.537227588332843,0.5266202861986242,0.5162157116157698,0.5060107727080277,0.49600232892148566,0.48618720525073766,0.47656220452524384,0.4671241180127297,0.45786973456063923,0.44879584846618,0.43989926623953673,0.43117681240265116,0.4226253344469872,0.4142417070574206,0.4060228356953957,0.3979656596224443,0.3900671544347538,0.38232433417048745,0.37473425304376295,0.3672940068524448,0.36000073410102984,0.3528516168747955,0.3458438814969293,0.33897479899646615,0.33224168541146615,0.32564190194889636,0.3191728550200737]
[null,"spot rate:",{"$f":"LN(1/C32)/C31","v":0.015878707256469306},{"$f":"LN(1/D32)/D31","v":0.018182942160999336},0.021593679914243394,0.02396854178865539,0.02571309991040904,0.02705697614836424,0.028134223797871966,0.029025796202870758,0.02978239461215051,0.03043700386202833,0.03101191973104398,0.0315227820047975,0.0319809485144304,0.03239493073203976,0.03277128657469241,0.033115191757419685,0.033430816024650346,0.033721577906683134,0.03399032189561594,0.03423944480467165,0.03447098801055814,0.034686706241514806,0.0348881198752483,0.035076555396727396,0.0352531771870748,0.035419012850344614,0.03557497364332091,0.0357218711383461,0.035860430948684886,0.035991304134786836,0.03611507675901339,0.03623227794703673,0.03634338673366452,0.03644883791085765,0.036549027050420485,0.03664431483925367,0.03673503083835361,0.03682147675591812,0.03690392930852694,0.03698264273135386,0.03705785098795507,0.03712976972178496,0.03719859798477832,0.03726451977277245,0.03732770539297115,0.037388312684874046,0.03744648811295769,0.03750236774677648,0.03755607814195685,0.03760773713371035,0.03765745455292985,0.03770533287360767,0.03775146779918668,0.03779594879449098,0.03783885956905678,0.03788027851697361,0.03792027911773279,0.0379589303020508,0.03799629678617526,0.03803243937778141,0.03806741525621917]

{"sheet":"Exercises","index":4,"n_rows":18,"n_cols":6,"rows_kept":18}
["1.  Modify the Monte Carlo Visual Basic program, BondPrice to account for risk-aversion."]
[]
["2.  Use the program you wrote to \"verify\" the closed-form solution in the previous worksheet."]
[]
["3.  Write a user-defined function that asks for the parameters and short-rate as inputs, and provides the yield-to-maturity as output."]
[]
["4. Develop intuition about the Vasicek model, by evaluating the effect on the yield curve of changing each of the inputs in turn (relative to a base case)."]
[null,"  ---- In particular, the yield curve depends on three things:"]
[null,null,"1. Expectations (How does the current state variable compare to its (risk-adjusted) long-run mean?"]
[null,null,"2. Convexity (We are averaging over a non-linear function).  The more non-linear (bigger sigma) the bigger this effect will be."]
[null,null,"3. Risk."]
[{"$nulls":3},"Zero lambda: This is the risk-neutral case.  Here the yield on a T-period bond only depends on expected average bond prices between now and T."]
[{"$nulls":3},"Positive lambda: Liquidity premium:  Yields on a T-period bond are higher than they would be in the risk-neutral case to compensate lenders for"]
[{"$nulls":5},"the risk inherent in tying up their loans (relative to the flexibility inherent in a sequence of short-term loans)."]
[{"$nulls":3},"Negative lambda: Solidity premium.  Here, yields on a T-period bond are lower than they would be in the risk-neutral case to compensate lenders for"]
[{"$nulls":5},"the risk of having to roll over their loans at uncertain future rates over the T period."]
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[{"$nulls":3},"The model is general enough to deal with all three cases.  In the data (US interest rates most periods), we tend to find a liquidity premium."]