Interest Rate Scenario Reduction Algorithms

Transcription

1 University of California Los Angeles Interest Rate Scenario Reduction Algorithms A thesis submitted in partial satisfaction of the requirements for the degree Master of Science in Statstics by Ran Hao 2011

2 c Copyright by Ran Hao 2011

3 The thesis of Ran Hao is approved. Nicolas Christou Hongquan Xu Jan De Leeuw, Committee Chair University of California, Los Angeles 2011 ii

4 To my beloved parents, who support every decision I make; and my dear friends, who are always there for me. iii

5 Table of Contents List of Figures vi List of Tables vii 1 Background Introduction Actuarial Guideline 43 and Cash Flow Testing Purpose: Scenario Reduction Existing Algorithms Chueh s Pivot Method PAM CLARA Other Methods Assumptions Duration of Projection Simplify economic scenarios to interest rate scenarios Beginning Interest Rate Stochastically Generate interest rate scenarios Target Statistics: CTE(70) of investment income Investment Model Bond Return Stock Return iv

6 2.7 True value of CTE(70) of investment income Algorithms Chueh s pivot method Sample size = Sample size = K-means Clustering Algorithm Conclusion and Recommendation for Future Study Conclusion Chueh s pivot method Clustering method Recommendation for Future Study Default rate modeling Sensitivity to input Appendix: Codes in R Bibliography v

7 List of Figures 2.1 Histogram of generated interest rates Histogram of Investment Income Histogram of CT E(70)Chueh, sample size= Histogram of CT E(70)Chueh, sample size= vi

8 List of Tables 2.1 Summary of Linear Regression: r SP i Frequency Table of P stock Summary of Investment Income Summary of CT E(70)Chueh, sample size = Summary of CT E(70)Chueh, sample size = vii

9 Abstract of the Thesis Interest Rate Scenario Reduction Algorithms by Ran Hao Master of Science in Statstics University of California, Los Angeles, 2011 Professor Jan De Leeuw, Chair The Actuarial Guideline 43 (AG 43) C3 Phase II for variable annuities requires stochastic testing, which involves large number of interest rate scenarios to be tested. This paper presents a new method for reducing interest rate scenarios. Currently, the most well-known method of interest rate scenario reduction is Chueh s pivot method. In this paper, results show that Chueh s algorithm tends to select extreme scenarios, hence leads to biased cash flow projection. Instead, using k-means clustering algorithm, an efficient interest rate scenario subset could be selected, providing good estimation. viii

10 CHAPTER 1 Background Introduction 1.1 Actuarial Guideline 43 and Cash Flow Testing In actuarial profession, cash flow testing is defined by Actuarial Standard of Practice as a test performed under several sets of economic scenarios which require that consistency be maintained in the relationships between the economic scenarios and the other assumptions. Cash flow testing may be an element of several types of analyses, including pricing studies, evaluation of investment strategy, determination of non-guaranteed elements (e.g., current interest and mortality rates), financial projections or forecasts, reserve adequacy testing, and valuation of blocks of business or appraisal work. [1] In recent years, stochastic modeling is widely used in insurance, especially for products highly related to the investment market, e.g. variable annuities. Actuarial Guideline 43 (AG 43) C3 Phase II is a guideline for variable annuities to set up reserves. For AG 43, the American Actuarial Academy (AAA) has published a set of economic scenarios. Then, the companies would determine their reserves with the following process: Simulate cash flow under each of the scenarios for the next few years (usually the next 30 years); 1

11 In each of the cash flow projection, calculate the accumulated profit/ deficiency of each year: for an n year cash flow projection, there would be n accumulated profits/deficiencies calculated. Take present value of the most negative number, i.e. the biggest deficiency, the Present Value of Greatest Accumulated Deficiency (PVGAD) is obtained PVGAD s are calculated, take expectation of the largest 30% of these PVGAD s, denote this conditional tail expectation as CTE(70). CTE(70) is the reserve to be set up based on stochastic approach. Calculate the reserve to be set up based on prescribed non-stochastic approach (deterministic approach). If result from deterministic approach is greater than CTE(70), set it as the company s reserve, otherwise use CTE(70). 1.2 Purpose: Scenario Reduction Like most stochastic modeling, one big problem in AG 43 C3 Phase II is modeling efficiency. Insurance companies have large block of business: each company have several business lines, each business line involves hundreds of thousands of policies... As a result, there would be million times calculations in each simulation. Multiply the number of scenarios to be tested , the total number of calculation could be tens of billions. As a result, it is not unusual for a company to take weeks to implement all the tests. But the actuaries work might not just end here: it s possible that insurance companies submit their actuarial memorandums with such results, but regulators consider part of the company s asset-liability model as inappropriate (for example, overrated bond credibility, under-estimated lapse rate, etc). In such cases, the insurance company has to 2

12 fix their model and re-run all the tests, which would take another several weeks. Hence, improving modeling efficiency is very important. One efficient way to shorten the run-time of stochastic tests is to reduce the number of economic scenarios to be tested. By drawing a representative subset of the original scenarios, CTE(70) could be estimated with controllable estimation error. 1.3 Existing Algorithms Chueh s Pivot Method Chueh s pivot method is the first well-developed algorithm of interest rate scenario reduction. In 2002, Yvonne Chueh published a paper, introducing a Pivot Mehod as following[3]: Denote N as the total number of interest rate paths, n as the desired sample size, i P t as the interest rate in year t in pivot scenario. Define distance between a given 30-year interest rate path and a pivot interest rate path as D = 30 (i t t P t ) 2 v t t=1 Choose an arbitrary interest rate path out of the N simulated ones and call it Pivot 1. Calculate the distances from Pivot 1 to the remaining interest rate paths. Name the interest rate path with the largest distance to Pivot 1 as Pivot 2. Calculate the distances of the N 2 non-pivot interest rate paths to Pivot 1 and Pivot 2. Assign each of the remaining interest rate paths to the closest 3

13 of Pivot 1 or Pivot 2, thus forming 2 disjoint sets of interest rate paths. Flip a coin if the distances are equal. Each of the remaining interest rate paths now has a unique distance to its pivot scenario. Rank these N 2 distances in descending order. The interest rate path producing the top distance is called Pivot 3. Follow the above procedure to select the additional pivot scenarios, Pivot 4, Pivot 5,..., Pivot n. the n Pivot scenarios could be a representative sample of the original N scenarios PAM Partitioning Around Medoids (PAM) is an algorithm to find a set of k medoids (n-dimensional median). PAM is good for small data set, usually less than 200. Because of the large data set we are dealing with in actuarial industry, PAM is not suitable for interest rate scenario reduction CLARA Clustering Large Applications (CLARA) alleviates the small data set problem of PAM. It randomly select a subset from the large data set and uses PAM to choose from this subset. Repeat this process for several times, and choose the best from the subsets. 4

14 1.3.4 Other Methods Researchers are trying to apply stratified sampling, Metropolis-Hasting, and other algorithms to scenario reduction[15, 9]. Till now, these methods have not been applied in actuarial practice yet. 5

15 CHAPTER 2 Assumptions 2.1 Duration of Projection All cash flow projections are made for the next 30 years. 2.2 Simplify economic scenarios to interest rate scenarios Economic scenario is fundamental and essential to cash flow testing. An economic scenario includes the interest rate, inflation rate and other economic elements. To simplify the problem of scenario reduction, I decide to take the key element of economic scenario - interest rate, and ignore other economic factors. The reasons I choose interest rate are: First, interest rate is the most significant indicator of the economy. Both the principle of economics and actual facts show that bad economy comes with low interest rate and good economy comes with high interest rate. Now (2011) we are experiencing a hard time because of financial crisis, the risk-free treasury bond rate (10 year) is around 3% - 4%; back in 1990 s, when economy was good, the risk free treasury bond rate (10 year) was as high as 8%. 6

16 All other economic factors are highly correlated with interest rate. Inflation rate, mortgage rate, industrial sales,..., change in other economic factors could be represented by change in interest rate. Hence, I believe interest rate can be the representative element of economic scenario. Algorithms to reduce interest rate scenario can be generalized into reducing economic scenarios. 2.3 Beginning Interest Rate Use annual treasury rate (10-year) on Feb 1, 2011 as the beginning interest rate: 3.43% Stochastically Generate interest rate scenarios Economic scenarios to be tested are determined in two ways: prescribed by regulators, or generated by insurance companies. For some type of cash flow testing, the company is allowed to use their own assumptions of future economy; in other cases such as AG 43, regulators make the rules and set up scenarios to be tested. To generalize the suitability of my research, I randomly generate interest rate scenarios. The interest rate scenarios are generated from a time series model ARM A(5, 0): the interest rate of a certain year is determined by that of the past 5 years and 1 Data Source: Unites States Department of Treasury, interest rate statistics. Website: 7

17 systematic noise. 5 X t = c + ɛ t + ϕ i X t i (2.1) i=1 ɛ t N(0, ˆσ) (2.2) ɛ is noise; I use ˆσ = standard error of 10-year treasury rate from 1982 to 2011, as an estimation of ɛ s standard error σ. For t = 1, ϕ 1 = 1; For t = 2, ϕ 1 = ϕ 2 = 1 2 ; For t = 3, ϕ 1 = 1 2, ϕ 2 = ϕ 3 = 1 4 ; For t = 4, ϕ 1 = 1 2, ϕ 2 = 1 4, ϕ 3 = ϕ 4 = 1 8 ; For t 5, ϕ 1 = 1 2, ϕ 2 = 1 4, ϕ 3 = 1 8, ϕ 4 = ϕ 5 = In (2.1), c is moving average. To decide c, a concept of New York 7 interest rate scenarios (NY7) is used. When doing cash flow testing, a certain minimum amount of analysis is required by the Standard Valuation Law. Such analysis includes cash flow testing under 7 scenarios, which represents typical trends of interest rate in the future. The 7 interest scenarios are[11]: NY1: Level. NY2: Uniformly increase 5% over 10 years and then level. NY3: Uniformly increase 5% over 5 years, then uniformly decrease 5% over next 5 years, and then level. NY4: Pop-up 3% in the first year and then level. NY5: Uniformly decrease 5% over 10 years and then level. 8

18 NY6: Uniformly decrease 5% over 5 years, then uniformly increase 5% over next 5 years and then level. NY7: Pop-down 3% in the first year and then level. Since NY7 represents typical moving trend of interest rate, I define 7 values of moving average c in (2.1). Each value of c represents one scenario in NY7. Then generate [10000/7] = 1429 scenarios from each ARM A(5, 0) model associating with a c value (as a result, I actually generated scenarios). NY1: c = 0 for t = 0,..., 30. NY2: 0.005, t = 1,..., 10 c = 0, t = 11,..., 30 NY3: c = 0.01, t = 1,..., , t = 6,..., 10 0, t = 11,..., 30 NY4: NY5: 0.03, t = 1 c = 0, t = 2,..., , t = 1,..., 10 c = 0, t = 11,..., 30 NY6: c = 0.01, t = 1,..., , t = 6,..., 10 0, t = 11,..., 30 9

19 NY7: 0.03, t = 1 c = 0, t = 2,..., 30 Currently, AG 43 working group set the floor of interest rate as 50% of beginning interest rate, which is 3.43% 50% = 1.715%. Based on the above assumptions, interest rate scenarios for the next 30 years are randomly generated. Figure 2.1 is a histogram of all the = interest rates, from figure 2.1, I see no extra-ordinary number. Frequency generated interest rates Figure 2.1: Histogram of generated interest rates 10

20 Interval [0, 0.02] has the highest frequency, because the current interest rate is 3.43%, and the floor of generated interest rate is 1.715%, for ARMA model with decreasing trend (NY5, NY6, NY7), it s very likely to generate interest rate lower than floor, especially for the first 10 years. 2.5 Target Statistics: CTE(70) of investment income In AG 43 C3 Phase II, the target statistic is CTE(70) of PVGAD. To simplify the problem, I decide to study the investment income of a certain portfolio, and calculate CTE(70) of investment income. The reasons I choose investment income as a substitute of PVGAD are: Investment income is most sensitive to interest rate. Although other items like policy premium income and expenses are also influenced by interest rate, but the sensitivity is not as much as investment income. It is true that bad economy (low interest rate) would lead to a certain level of surrender (early withdrawal), or increasing expense on maintaining policies, but such changes are not very significant compared to change in investment income which totally depends on the economy; Investment income is a most important factor which decides the result of cash flow projection, especially for variable annuities. One of American Academy of Actuaries (AAA) s practice note points out: for variable products, one key consideration is usually the projection of variable fund performance. Thus, to perform cash flow testing on variable products, many actuaries try to specify how future fund performance correlates to fixed interest rate movements [16]. 11

21 2.6 Investment Model To simulate investment income under certain interest rate scenario, I assume the company s investment income comes from a portfolio with $1 starting asset, in which $a is invested in bonds and $b is invested in stocks, a + b = 1. Assume a = 0.6, b = Bond Return Equally divide $a into three parts ($ 1 a=$0.2) and invest into 10-year, 20-year, 3 30-year bonds. Make the following assumptions on bond return: Annual coupon rate: 5%; Default rate (possibility of the bond to default in year t): default rate is negatively co-related with interest rate. In this paper, I use a simple linear relationship: denote default rate as r t, interest rate as i t, assume r t = 0.15 it 5 ; All coupons will be accumulated with risk free interest rate; At the maturity date, reinvest the accumulated pay-offs of an m year bond into a 30 m year bond; If a bond defaults before maturity in year t, reinvest all pay-offs into a 30 t year bond. 12

22 2.6.2 Stock Return Use Standard&Poor 500 (S&P500) index to represent stock market performance. Use annual yield of S&P500 2 and 10-year treasury rate from 1981 to 2001, perform a regression model to study the relationship between S&P500 and interest rate. Denote S&P yield rate as r SP, a linear regression model can describe the relationship properly: Table 2.1: Summary of Linear Regression: r SP i Estimate Std. Error t value Pr(> t ) (Intercept) Interest Rate i t In this paper, annual yield rate of S&P500 is generated from formula (2.3) and (2.4): r SP t = i t + ɛ (2.3) ɛ N(0, σ ) (2.4) σ is standard error of S&P 500 yield rate from 1981 to Assume the beginning price of S&P 500 index is 1, then after 30 years, the final payoff of stock investment P stock is P stock = b (1 + r SP 1 )(1 + r SP 2 )... (1 + r SP 30 ) (2.5) One randomly generated result of P stock and associated average annual yield rate r stock 3 under all the scenarios is shown in Table 2.2: 2 Data Source: yahoo finance, adjusted close price of S&P500 from 1981 to r stock = (P stock /b)

23 Table 2.2: Frequency Table of P stock P stock [0,5) [5,10) [10,15) [15,80] r stock [0,8.8%) [8.8%,11.3%) [11.3%,12.8%) [12.8%,19.4%] Frequency True value of CTE(70) of investment income Run the investment model described in Section 2.6 through interest rate scenarios generated in Section 2.4, investment incomes are calculated. Summary of results: Table 2.3: Summary of Investment Income Minimum 1st Quarter. Median Mean 3rd Quarter Maximum Histogram of investment income: 14

24 Frequency Frequency investment income investment income under 10 Figure 2.2: Histogram of Investment Income Sort the investment incomes and calculate expectation of the tail 30%, true value of CTE(70) is obtained: CT E(70) = (2.6) 15

25 CHAPTER 3 Algorithms 3.1 Chueh s pivot method Use Chueh s pivot method described in Chapter 1, section 1.3.1, for sample size 100 and 500, run the algorithm for 100 times, results are shown section 3.1 and Sample size = 100 Summary of CT E(70): Table 3.1: Summary of CT E(70)Chueh, sample size = 100 Min 1st Quarter. Median Mean 3rd Quarter Max SD Histogram of CT E(70)Chueh, sample size = 100: 16

26 Frequency CTE(70): Chueh's Pivot Method, sample size=100 Figure 3.1: Histogram of CT E(70)Chueh, sample size=100 For sample size = 100, estimation of CTE(70) from Chueh s method is far away from the true value CT E(70) = In fact, if look deeper into the scenarios chosen by Chueh s pivot method, it could be found that all subsets contain a number of same scenarios (the number is around 60-70) Sample size = 500 Summary of CT E(70): Table 3.2: Summary of CT E(70)Chueh, sample size = 500 Min 1st Quarter. Median Mean 3rd Quarter Max SD

27 Histogram of CT E(70)Chueh, sample size = 500: Histogram of cte70ch1[1:100] Frequency cte70ch1[1:100] Figure 3.2: Histogram of CT E(70)Chueh, sample size=500 Chueh s pivot method performs much better when sample size = 500 than when sample size=100: estimation closer to true value of CT E(70), and estimation error becomes smaller. 3.2 K-means Clustering Algorithm With K-means clustering algorithm, a subset of the scenarios could be obtained: Use K-means clustering algorithm, find n cluster centers, which could minimize n k l=1 i l µ k 2 (3.1) 18

28 where i l = (i l 1,..., i l 30) is a interest rate scenario in cluster k, n k is number of scenarios in cluster k, µ k is the center (mean) of cluster k; Take the centers of each cluster as a sample of original scenarios. For k = 100, CT E(70)kmeans = ; For k = 500, CT E(70)kmeans = Compared to estimation of Chueh s method, k-means clustering algorithm has estimation CT E(70)kmeans much closer to the true value CT E(70) = Further conclusion is illustrated in Chapter 4. 19

29 CHAPTER 4 Conclusion and Recommendation for Future Study 4.1 Conclusion Chueh s pivot method According to results in section 3.1, Chueh s method is very stable: the estimation variance is small. But all estimation of of the true value: true CT E(70) is , while CT E(70)Chueh are much larger than CT E(70)Chueh > In each step of Chueh s method, a furthest scenario from all other pivot scenarios is chosen. As a result, Chueh s pivot method tends to choose extreme scenarios, which leads to extreme outcomes. In this paper, interest rates are generated on a floor of 1.715%, so extreme scenarios are those with extraordinarily high interest rates. Consequently, investment incomes of the chosen subsets are higher, hence the estimation of CT E(70) becomes larger than true value. On the other hand, if the original economic scenarios are relatively uniformly distributed, i.e., there are not many lonely points which are far from other scenarios, Chueh s method will well better. 20

30 4.1.2 Clustering method According to results in section 3.2: K-means clustering method gives much better estimation of CT E(70) than Chueh s pivot method: the difference of estimated CT E(70) from true value is within 3% of the true value. The result of sampling size 100 is very close to that of sampling size 500. So if using k-means clustering, small sample size can almost perform as well as large sample size. Limit of application of k-means clustering: it may not fit more complex asset-liability models. The scenarios selected are not actually scenarios from the original scenarios. Instead, the chosen subset are centers of each cluster, i.e., average of all scenarios in each cluster. Average is an linear transformation of interest rate, while investment income is not. It might work well in this paper, but in reality, insurance companies have much more complex asset-liability models which involve various financial instruments such as derivatives. Hence, investment income calculated from interest rate scenario cluster average may be biased. 4.2 Recommendation for Future Study Default rate modeling In Chapter 2, Section 2.6.1, I assumed default rate of a bond as a function of interest rate: r t = 0.15 i t /5. This is because I could not find a better way to describe the negative correlation between default rate and interest rate. In 21

31 further study, regression could be used to seek a better expression of default rate based on interest rate Sensitivity to input Since the original scenarios are randomly generated, it is possible that the scenarios I generated are not meaningful in practice. For example, from results of Chueh s method, it is clear that there are several extreme scenarios which are very far from other scenarios. In reality, especially when regulators prescribe scenarios, extreme scenarios are rare. In further study, I suggest changing the input of sampling frame, i.e., the original scenarios 1, to see whether Chueh s method and k-means clustering still perform the same as they do in this paper. 1 in this paper, the original scenarios. 22

32 CHAPTER 5 Appendix: Codes in R ###################################################### # Base 10,000 Interest Rate Scenarios ###################################################### # 30 year projection n <- 30 # 1-year treasury rate # -management/interest-rate/yield.shtml r <- 3.43/100 # 10-yr bond rate for the past years TNR <- read.csv("tnr_10.csv") yr <- seq(1,1+12*29,by=12) tnr <- TNR[yr,7]/100 sigma <- sd(tnr)/5 # 10,000 scenarios # interest rate has a floor: 50% of current rate floor <- 0.5*r ## int.f is a function to draw a interest scenario for the next n years 23

33 ## int.f_input: n - number of years to be simulated ## int.f_input: c - moving average parameter, based on NY7 ## int.f_output: int - a vector with the length n, representing interest rate for the next n years int.f <- function(n,c) { int <- rep(0,n) int[1] <- max(c[1] + r + rnorm(1,0,sigma),floor) int[2] <- max(c[2] + int[1] + rnorm(1,0,sigma), floor) int[3] <- max(c[3] + 1/2*int[2] +1/2*int[1] + rnorm(1,0,sigma), floor) int[4] <- max(c[4] + 1/2*int[3] + 1/4*int[2] + 1/4*int[1] + rnorm(1,0,sigma), floor) int[5] <- max(c[5] + 1/2*int[4] + 1/4*int[3] + 1/8*int[2] + 1/8*int[1] + rnorm(1,0,sigma), floor) for (i in 6:n) { int[i] <- max(c[i]+1/2*int[i-1]+1/4*int[i-2] + 1/8*int[i-3] +1/16*int[i-4] +1/16*int[i-5] +rnorm(1,0,sigma), floor) } return(int) } c1 <- rep(0,30) c2 <- c(rep(0.005,10),rep(0,20)) c3 <- c(rep(0.01,5),rep(-0.01,5),rep(0,20)) c4 <-c(0.03,rep(0,29)) c5 <- c(rep(-0.005,10),rep(0,20)) 24

34 c6 <- c(rep(-0.01,5),rep(0.01,5),rep(0,20)) c7 <- c(-0.03,rep(0,29)) c <- rbind(c1,c2,c3,c4,c5,c6,c7) ## int.f1 is a function to draw 7 scenarios at one time, based on NY7 directions ## int.f1_input: n - number of years ## int.f1_input: c - moving average parameter matrix ## int.f1_output: sce.1 - matrix with 7 rows and n cols int.f1 <- function(n,c) { sce.1 <- matrix(,nrow=7,ncol=n) for (i in 1:7) { sce.1[i,] <- int.f(n,c[i,]) } return(sce.1) } ## int.sce is a function to simulate a number of scenarios for the next n years ## int.sce_input: N - number of simulations needed ## int.sce_output: sce - a data frame with N rows and n cols. ## each row is an interest scenario int.sce <- function(n) { N1 <- round(n/7) sce <- NULL for (i in 1:N1) { sce <- rbind(sce,int.f1(n,c)) } return(sce) 25

35 } # generate (actually 10003) generated scenarios N < set.seed(0626) sce <- int.sce(n) hist(sce,xlab="interest rates",main="histogram of generated interest rates") ################################################## # Investment Model ################################################## # Assume the initial capital is $1 # Assume $a is in bonds, $b in stocks. # # Bonds # # Assume coupon rate c: 5% # Assume principle K=1 # so the price at year t is P=sum(C*(1+r_t)^(-k))+K*(1+r_t)^(-n) ## bond.price is a function to calculate bond price ## Input: c - coupon rate; m - duration; it - interest rate at the starting year bond.price <- function(c,m,it) { price <- 0 for (i in 1:m) { price <- price + c/(1+it)^i 26

36 } price <- price + 1/(1+it)^m return(price) } # Each year, the bond default at a default rate, which is related to the interest rate of that year ## default.rate is a function to draw default rate based on interest rate default.rate <- function(it) { return(abs(0.15-it)/5) } # assume 1/3 a: 30yr bond; 1/3 a: 20yr; 1/3 a: 10 a <- 0.6 b <- 0.4 # assume once a bond is default in year t, re-invest all the paid coupons into a 30-t year bond ## bond.m is a function to simulate a m-year cash flow of one share of bond ## Input: i _ a vector of 30 year interest rate projection ## Output: the final payouts of this m-year bond bond.m <- function(c,i, mm) { ret <- 0 #return for (k in 1:mm) { dr <- default.rate(i[k]) seed <- sample(c(0,1),1, prob=c(dr,1-dr)) ret <- ret*(1+i[k])+c*seed+1*seed*as. numeric(k == mm) 27

37 if (seed==0) { break() } } return(list(ret=ret,year = k)) } ########### # a function to run a 30 year bond return simulation: first a m-year bond, if default, re-invest into a 30-m year bond. assume 1/3*a assets are invested in such bond. # input: m; i - interest scenario # output: bond.m.return bond.mto30 <- function(c,i,m) { bond.m30 <- bond.m(c,i,m) judgeyear <- bond.m30$year bond.return <- bond.m30$ret bond.m.return <- 1/3 * a /bond.price(c,m,i[1])*bond.return repeat { if (judgeyear ==30 ) {break()} rep <- bond.m(c,i[judgeyear:30],30-judgeyear) judgeyear <- judgeyear + rep$year bond.m.return <- bond.m.return/bond.price(c,30-judgeyear, i[judgeyear])*rep$ret } return(bond.m.return) } 28

38 # # Stock # ## use lognormal distribution. mean/sd of historical S&P 500 index sp500 <- read.csv("sp500.csv") yrs <- seq(1,1+12*29,12) sp500 <- sp500[yrs,] n.row <- nrow(sp500) sp500.return <- (sp500$adj.close[1:n.row-1] -sp500$adj.close[2:n.row]) /sp500$adj.close[2:n.row] # a linear regression of sp500.return, to each year s interest rate, use data (the last few years data are weird): sptoint <- lm(sp500.return[9:29]~tnr[10:30]) res.mu <- mean(sptoint$res) res.sd <- sd(sptoint$res) ## assume stock price at time 0 is 1 ## stock is a function to simulate n year stock price serie ## Input: n - number of years to be simulated ## Input: int - interest rate of that year ## Output: st.pr - a vector containing stock prices of the following n years set.seed(0626) stock <- function (n,int) { st.pr <- numeric(n+1) st.pr[1] <- 1 r <- numeric(n) 29

39 for (i in 1:n) { err <- rnorm(1, res.mu, res.sd) r[i] <- sptoint$coef %*% c(1,int[i]) + err st.pr[i+1] <- st.pr[i] * (1+r[i]) } return(list(price=st.pr[2:(n+1)],r=r)) } ############################################### #CTE(70) of Investment Income of the original 10,000 scenarios ############################################## N <- nrow(sce) #total number of scenarios: stock.income <- bond.return <- NULL set.seed(0626) for (i in 1:N) { stock.income <- c(stock.income, b * stock(30,sce[i,])$price[30]) bond.return <- c(bond.return, bond.mto30(0.05,sce[i,],10) +bond.mto30(0.05,sce[i,],20)+bond.mto30(0.05,sce[i,],30)) inv.inc <- stock.income + bond.return } inv.inc.sort <- sort(inv.inc) qt <- round(n * 0.7) cte70 <- mean(inv.inc.sort[qt:n]) cte70 # true value of CTE(70): ###################################################### 30

40 #Chueh s Pivot Alg ###################################################### # Chueh is a function to implement Chueh s Pivot Alg # Input: n, target sample size Chueh <- function(n) { p1 <- sample(seq(1:n),1) indices <- vector(length=n) indices[1] <- p1 distances <- rep( ,n) for (i in 2:n) { for (k in 1:N) { distances[k] <- min(sum((sce[k,]-sce[indices[i-1],])^2),distances[k]) } indices[i] <- which.max(distances) } return(list(index=indices,distance=distances)) } n <- 100 # Chueh s method would like to choose extreme points. # Variance of Chueh s method cte70ch <- NULL for (i in 1:100) { x <- Chueh(n) index <- x$index inv.inc.chueh <- sort(inv.inc[index]) cte70.chueh <- mean(inv.inc.chueh[71:n]) 31

41 cte70ch <- c(cte70ch,cte70.chueh) } mean(cte70ch) # sd(cte70ch) # hist(cte70ch,18,xlab="cte(70): Chueh s Pivot Method, sample size=100",main="") ##################### ## sample size = 500 ##################### n1 <- 500 cte70ch1 <- NULL for (i in 1:100) { x1 <- Chueh(n1) index1 <- x1$index inv.inc.chueh1 <- sort(inv.inc[index1]) cte70.chueh1 <- mean(inv.inc.chueh[355:n1]) cte70ch1 <- c(cte70ch1,cte70.chueh1) } mean(cte70ch1) sd(cte70ch1) hist(cte70ch1,18,xlab="cte(70): Chueh s Pivot Method, sample size=500",main="") #################################### # clustering method ######################################### # simple k-means # sample size =

42 sce.kmeans <- kmeans(sce,100)$centers bond.return <- stock.inc <- NULL for (i in 1:100) { stock.inc <- c(stock.inc, b * stock(30,sce.kmeans[i,])$price[30]) bond.return <- c(bond.return, bond.mto30(0.05, sce.kmeans[i,],10)+bond.mto30(0.05,sce.kmeans[i,],20) +bond.mto30(0.05,sce.kmeans[i,],30)) inv.inc.kmeans <- stock.inc + bond.return } inv.inc.kmeans <- sort(inv.inc.kmeans) cte70.kmeans <- mean(inv.inc.kmeans[71:100]) cte70.kmeans # sample size = 500 sce.kmeans1 <- kmeans(sce,500)$centers bond.return1 <- stock.inc1 <- NULL for (i in 1:500) { stock.inc1 <- c(stock.inc1, b * stock(30,sce.kmeans1[i,])$price[30]) bond.return1 <- c(bond.return1, bond.mto30(0.05, sce.kmeans1[i,],10)+bond.mto30(0.05,sce.kmeans1[i,],20) +bond.mto30(0.05,sce.kmeans1[i,],30)) inv.inc.kmeans1 <- stock.inc1 + bond.return1 } inv.inc.kmeans1 <- sort(inv.inc.kmeans1) cte70.kmeans1 <- mean(inv.inc.kmeans1[355:500]) cte70.kmeans1 33

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