Arbitrage Pricing Theory KELOMPOK 2 : Dwi Ariestiyanti ( 0906612384 ) Ira Purdiningtyas ( 0906612466 ) Mulya Kurniawan ( 0906612 )

Pengertian Arbitrage Arbitrage (Arbitrase) merupakan pembelian dan penjualan berkesinambungan dari sekuritas pada dua harga yang berbeda di dua pasar yang berbeda Arbitrase bertujuan mengambil keuntungan dengan memanfaatkan perbedaan harga pada aset atau sekuritas yang sama.

Arbitrage Pricing Theory Model Model APT yang diciptakan oleh Stephen A. Ross adalah alternatif dari Capital Asset Pricing Model yang digunakan untuk memperhitungkan risiko dan imbal hasil dari sebuah aset tertentu. Prinsip dasar yang muncul pada model mean varians bahwa pada setiap asset berlaku sebuah persamaan: (1)  Di mana adalah suku bunga bebas risiko λ adalah lebihan pemasukan yang diharapkan dalam pasar () adalah koefisien beta yang bisa digambarkan sebagai berikut   Di mana adalah kovarians antara pemasukan pada asset ke-i dan portfolio pasar adalah varians dari portfolio pasar

Continued Dari persamaan (1), Ross mengajukan sebuah teori alternatif yang dapat dijabarkan sebagai berikut: (2) Untuk mendapatkan rumus akhir APT perlu dilakukan langkah-langkah berikut: membuat sebuah portfolio arbitrase, 𝜂, yang terdiri dari asset-aset n. dengan hukum angka-angka besar, akan menjadi (3) Dengan kata lain, diabaikan.

Continued Risiko sistematis dihilangkan, Sehingga dari persamaan (3) didapatkan bahwa E dijabarkan dengan e dan β, atau (4) pada ρ dan λ yang konstan. ρ adalah risk of return pada semua portofolio zero-beta, sehingga persamaan (4) akan menjadi persamaan (5) (5)

Continued Rumus APT single factor Rumus APT multi factor

Factor Models: Announcements, Surprises, and Expected Returns The return on any security consists of two parts. First the expected returns Second is the unexpected or risky returns. A way to write the return on a stock in the coming month is: return the of part unexpected is expected where U R + =

Risk: Systematic and Unsystematic A systematic risk is any risk that affects a large number of assets, each to a greater or lesser degree. An unsystematic risk is a risk that specifically affects a single asset or small group of assets. Examples of systematic risk include uncertainty about general economic conditions, such as GNP, interest rates or inflation. On the other hand, announcements specific to a company, such as a gold mining company striking gold, are examples of unsystematic risk.

Risk: Systematic and Unsystematic We can break down the risk, U, of holding a stock into two components: systematic risk and unsystematic risk:  risk unsystematic the is systematic where becomes ε m R U + = Total risk; U  Nonsystematic Risk;  Systematic Risk; m n

Systematic Risk and Betas The beta coefficient, b, tells us the response of the stock’s return to a systematic risk. In the CAPM, b measured the responsiveness of a security’s return to a specific risk factor, the return on the market portfolio. We shall now consider many types of systematic risk. ) ( 2 , M i R Cov s b =

Systematic Risk and Betas For example, suppose we have identified three systematic risks on which we want to focus: Inflation GNP growth The dollar-euro spot exchange rate, S($,€) Our model is: risk ic unsystemat the is beta rate exchange spot GNP inflation ε β F R m S I + =

Systematic Risk and Betas: Example Suppose we have made the following estimates: bI = -2.30 bGNP = 1.50 bS = 0.50. Finally, the firm was able to attract a “superstar” CEO and this unanticipated development contributes 1% to the return. ε F β R S GNP I + = % 1 = ε % 1 50 . 30 2 + ´ - = S GNP I F R

Systematic Risk and Betas: Example We must decide what surprises took place in the systematic factors. If it was the case that the inflation rate was expected to be by 3%, but in fact was 8% during the time period, then FI = Surprise in the inflation rate = actual – expected = 8% – 3% = 5% % 1 50 . 30 2 + ´ - = S GNP I F R % 1 50 . 5 30 2 + ´ - = S GNP F R

Systematic Risk and Betas: Example If it was the case that the rate of GNP growth was expected to be 4%, but in fact was 1%, then FGNP = Surprise in the rate of GNP growth = actual – expected = 1% – 4% = – 3% % 1 50 . 5 30 2 + ´ - = S GNP F R % 1 50 . %) 3 ( 5 30 2 + ´ - = S F R

Systematic Risk and Betas: Example If it was the case that dollar-euro spot exchange rate, S($,€), was expected to increase by 10%, but in fact remained stable during the time period, then FS = Surprise in the exchange rate = actual – expected = 0% – 10% = – 10% % 1 50 . %) 3 ( 5 30 2 + ´ - = S F R % 1 %) 10 ( 50 . 3 5 30 2 + - ´ = R

Systematic Risk and Betas: Example Finally, if it was the case that the expected return on the stock was 8%, then % 1 50 . %) 3 ( 5 30 2 + ´ - = S F R % 8 = R

Portfolios and Factor Models Now let us consider what happens to portfolios of stocks when each of the stocks follows a one-factor model. We will create portfolios from a list of N stocks and will capture the systematic risk with a 1-factor model. The ith stock in the list have returns: i ε F β R + =

Portfolios and Diversification We know that the portfolio return is the weighted average of the returns on the individual assets in the portfolio: N i P R X + = L 2 1 i ε F β R + = ) ( 2 1 N P ε F β R X + = L N P ε X F β R + = L 2 1

Portfolios and Diversification The return on any portfolio is determined by three sets of parameters: N P R X + = L 2 1 The weighed average of expected returns. F β X N ) ( 2 1 + L The weighted average of the betas times the factor. N ε X + L 2 1 The weighted average of the unsystematic risks. The unsystematic risk is diversified away because the variance of row three tends to 0 as N goes to infinity. In a large portfolio, the third row of this equation disappears as the unsystematic risk is diversified away.

Portfolios and Diversification So the return on a diversified portfolio is determined by two sets of parameters: The weighed average of expected returns. The weighted average of the betas times the factor F. F β X R N P ) ( 2 1 + = L In a large portfolio, the only source of uncertainty is the portfolio’s sensitivity to the factor.

A Critical Reexamination of The Empirical Evidence on The Arbitrage Pricing Theory (By P.J. Dhrymes, I. Friend, and N.B. Gultekin) The basic methodology of analyzing small groups of securities in order to gather confirmatory or contrary evidence relative to the APT model is seriously flawed It’s not possible to test directly whether a given “factor” is priced. → how many factors there are and whether they are priced? There are 3 to 5 factors don’t appear to be robust. → Result: how many factors one ‘discovers’ depends on the size of the group of securities one deals with.

A Critical Reexamination of The Empirical Evidence on The Arbitrage Pricing Theory : A Reply (By Richard Roll and Stephen A. Ross) Reply 2 Uji pada faktor harga tunggal tetap memiliki arti. Pertanyaanya adalah seberapa jauh perbedaan nilai yang didapatkan dari estimasi yang menggunakan model APT terhadap keadaan sebenarnya.

A Critical Reexamination of The Empirical Evidence on The Arbitrage Pricing Theory : A Reply (By Richard Roll and Stephen A. Ross) Reply 3 Semakin banyak jumlah sampel dalam suatu kelompok, faktor yang berpengaruh akan semakin banyak. Tidak semua faktor yang muncul akan signifikan dalam penghitungan APT.

Summary and Conclusions Prinsip dari APT adalah sekuritas yang mempunyai karakteristik yang sama, tidak akan bisa dihargai dengan harga yang berbeda. APT mengatakan bahwa tingkat keuntungan suatu saham dipengaruhi oleh faktor-faktor tertentu yang jumlahnya bisa lebih dari satu. APT tidak dapat menjelaskan faktor-faktor apa saja yang mempengaruhi pembentukan harga sekuritas.

Summary and Conclusions The APT assumes that stock returns are generated according to factor models such as: As securities are added to the portfolio, the unsystematic risks of the individual securities offset each other. A fully diversified portfolio has no unsystematic risk. The CAPM can be viewed as a special case of the APT. ε F β R S GNP I + =

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