FDM - How to calculate stuff

This covers both CAPM, APT, AFP and corporate finance

CAPM: Find the minimum variance portfolio

1. What are the steps?

CAPM: Find the minimum variance portfolio

1. What are the steps?

1. Find the matrix z=A^-11

2. Find Z=sum(z)

3. Compute x=z/Z

CAPM: Find the tangent portfolio

1. What are the steps?

1. Find the matrix z = A^-1*(b-r_f1)

2. Find Z = sum(z)

3. Compute x = z/Z

CAPM: Find the efficient frontier:

What theorem do we use here?

1. What are the steps?

We use the two mutual fund theorem.

1. Set up the weights for each portfolio

2. Calculate the weights of each assets in the combined portfolio x=(w_mv*x_mv+w_tan*x_tan)^T

3. Calculate the variance of the portfolio: V(p)=xAx^T

4. Calcuate the expected return: E(p)=xb

CAPM: How to find the capital market line

How does it differ from finding the efficient frontier?

1. What are the steps?

The only difference is that we use the risk-free portfolio instead of the minimum variance portfolio.

1. Set up the weights for each portfolio

2. Calculate the weights of each assets in the combined portfolio x=(w_rf*x_rf+w_tan*x_tan)^T

3. Calculate the variance of the portfolio: V(p)=xAx^T

4. Calcuate the expected return: E(p)=xb

APT: Find the expected return of a factor equation:

Consider the factor equations:

r₁ = 0.04 + 3F₁ + 5F₂ +epsilon_1, V(epsilon_1) = 0.01

r₂ = 0.10 + 5F₁ + 7F₂ +epsilon_2, V(epsilon_2) = 0.02

r₃ = 0.15 + 4F₁ + 6F₂ +epsilon_3, V(epsilon_3) = 0.04

And we know that V(F1)=0.01 and V(F2) = 0.02

1. What assumptions do we make about factors and idiosyncratic risk?

2. What is the expected return of r₁?

3. What is the variance of r₁?

4. What is the covariance between r₁ and r₂?

1. We assume that F1, F2, epsilon have E(x) = 0, and that they are uncorrelated.

2. The expected return i E(r1)=0.04, since all else is mean 0.

3. The variance is: V(r2)= 5²V(R1) + 7²V(F2) + V(epsilon2) = 5²*0.01+7²*0.02+0.01=1.24

4. The covariance can be found as Cov(r1,r2), so just insert and use indepence assumptions.

APT: How do we find the minimum variance portfolio?

1. What are the steps?

Since we know the var-cov matrix, we can just use the same as CAPM:

1. Set up z = A^-11

2. Compute Z=sum(z)

3. Compute x=z/Z

APT: Construct a risk-fee portfolio.

1. What is the idea?

2. What are the steps?

1. Construct a portfolio where all risk-factors cancel out.

2. Setup the matrix equation: x_rf^T = (0^T|1) * (B_selected|1)^T

3. Solve the matrix system. Idea is to setup the matrix B with all the loadings, and then jsut solve it in Excel

APT: Construct a pure-factor portfolio

1. What is the idea?

2. What are the steps?

1. The idea is to 1-for-1 load on a single factor, and we can then calculate the risk-premium from it.

2. Setup the matrix equation: x_rf^T = (e^T_K|1) * (B_selected|1)^T_. Such that we load on the choosen factor K.

3. Solve the matrix system. Idea is to setup the matrix B with all the loadings, and then just solve it in Excel

APT: Find the expected return and risk premia of a pure factor portfolio.

1. How to calculate the expected return?

2. How do we calculate the risk premia?

1. Since we know the weights of the portfolio that yields a given pure-factor portfolio, we can just multiple them with the expected return of each asset.

2. The risk premia is the expected return of a pure factor portfolio minus the risk free rate.

APT: Introduction of a new assets and its impact on arbitrage?

If we consider a new asset being added, how do we determine if there is still no-arbitrage?

1. How can we use APT to replicate the risk factors?

2. How do we determine if there is arbitrage?

1. Replicate the risk factors of the new assets, with the exisiting asset. So set up the Factor model pricing: E(r_i) = rf + beta*lamda_F1 beta2 * lambda_F2

2. If the replicated portfolio has a different payoff than E(R_new), then there is a arbitrage oppurtinity, othervise: No-Arbitrage.

APT: Finding arbitrage-free risk premia

MANGLER

AFP: How to find state prices

1. What is the interpretation of state prices?

2. What are the steps to find the state prices?

3. What must be check?

1. State prices is the value of one unit of CF in state j.

2. To solve for State prices we use the FoAP: p = Vd → d=V^-1p

3. Check that all state price are stricly positive.

AFP: State prices and probablities.

1. How can state prices be interpreted as probabilities?

2. How do we find r?

3. How do we find the probablities?

1. MANGLER

2. We find r as: r= 1/sum(d) -1

3. We then find the probablities as prob_i = d_i / sum

AFP: Replicate a new assets payoff.

1. How can we replicate a new asset?

2. What does possible price descriptiencies do to possible Arbitrage?

1. We try to do a combination of existing assets that replicate the value of the new asset in all future states. We then calculate the price of the replicated portfolio and compare it to the price of the new asset.

2. If the price differs there is an arbitrage opportunity

   1. Type 1: p > replication: Sell asset i and buy portfolio

   2. Type 2 : p < replication: Buy asset and sell portfolio

MM Theorem: What are debt and equity payoffs

Assuming perfect capital markets:

1. What is the payoff for debt holders?

2. What is the payoff for equity holders?

1. Succes = P, Failure: min{C_F,P}

2. Succes = C_S - P, Failure: max{C_F-P,0}

MM theorem: Calculatig value of firm and its claims

1. How is any claim valued?

2. What is the value of debt?

3. What is the value of equity?

4. What is the value of the unlevered firm?

1. V_i = d_FV_if + d_S*V_iS

2. D = d_F min{C_F,P) +d_s P

3. E = d_F*max(C_F-P,0) + d_S * (C_S - P)

4. V^U = d_F * C_F +d_S+C_f

MM with taxes: Adjusted firm value with tax shield.

1. How do you find the interst paid?

2. What is the formular for the tax shield?

3. What is the adjusted firm value?

1. D = (df + ds)P → Interest = P - D

2. tau_C * (P - D)

3. V^L = V^U + tau_C *(P-D)

MM with taxes: Effective tax advantage

1. What is the difference here?

2. What is the formula for effective tax advantage

1. Now investors are also taxed.

2. r* = 1- [equity tax] / [interest tax]

MM with distress costs:

1. What is the value of debt and equity

2. What is the value of the firm?

3. What is the financial distress costs?

1. D = d_F (rho*C_F) + d_S * P and E = d_S (C_S - P)

2. V^L = D + E = d_F (rho*C_F) + d_S * C_S

3. d_F*(1-rho)C_F

Financial Options:

1. What is the formular for the Put-Call parity?

2. What is the formular for the Put-call Parity with dividend?

1. C₀ - S₀ = P₀ + PV(K)

2. C₀ - S₀ = P₀ + Dv₀ PV(K).

Binomial Case: AFP

1. What formula must any asset satisfy?

2. What does the payoff matrix look like?

3. What is the formula for replicating a binomial case?

4. What is the formula for option premium in AF markets?

1. p = dv

2. Like this.
   [ S(U) S(D))]

   [. 1. 1. ]

3. Replicating by portfolio x, meants that x^TV is the options payoff vector v. Thus: x^T = vV^-1

4. C = x₁*S + x₂ * B. Where S is the current stock price and B is the current bond price.

Price call option using Risk-neutral pricing.

1. What are the steps?

1. Find the value of the states:

   1. S(D) = max{S(D) - K ,0 }

   2. S(U) = max{S(U) - K, 0 }

2. Find the risk-neutral probabilities

   1. p(U) = [(1+rf)S - S(D)] / [S(U) - S(D)]

   2. p(D) = 1- p(U)

3. Find discounted stock value and bond value.

   1. S = [p(U)*S(U) + (1-p(U)) * S(D)] / (1+ rf)

   2. B = 1/[1+rf]

4. Find the option premium:

   1. C = x₁*S + x₂ * B

Price put option using Risk-netural pricing.

1. How does the method differ from call options?

1. Inly the first step. So they payoff is:

   1. S(D) = max{K - S(D), 0}

   2. S(U) = max{K - S(U), 0}

Price American call option using Risk-netural pricing.

1. What are the steps?

1. Find the value of the states:

   1. S(D) = max{S(D) - K ,0 }

   2. S(U) = max{S(U) - K, 0 }

2. Find the risk-neutral probabilities

   1. p(U) = [(1+rf)S - S(D)] / [S(U) - S(D)]

   2. p(D) = 1- p(U)

3. Find discounted stock value and bond value.

   1. S = [p(U)*S(U) + (1-p(U)) * S(D)] / (1+ rf)

   2. B = 1/[1+rf]

4. Find the option premium:

   1. C = x₁*S + x₂ * B

Price American call option using Risk - neutral pricing.

1. What are the steps?

1. Exactly the same as European

Multi Binomial Option Pricing: European Call option

1. What are the steps to price the option?

1. Setup the asset value in each state. Setup payoff in each end node.

2. Calculate risk-neutral prob with formular:

p(U) = [(1+rf)S_t-1 + S(D)] / (S(U) - S(D)]

3. Calculate option premium in t =1 with formular:

   V = [p(U)*S(U) + p(D) * S(D)] / (1+rf)

4. Calculate the option premium at t = 0, with same formular

Multi Binomial Option Pricing: European Put option

1. What are the steps to price the option?

The exact same to price the call option, (just copied from that):

1. Setup the asset value in each state. Setup payoff in each end node.

2. Calculate risk-neutral prob with formular:

p(U) = [(1+rf)S_t-1 + S(D)] / (S(U) - S(D)]

3. Calculate option premium in t =1 with formular:

   V = [p(U)*S(U) + p(D) * S(D)] / (1+rf)

4. Calculate the option premium at t = 0, with same formular

Multi Binomial Option Pricing: American Call option

1. What are the steps?

The exact same to price the EU call option, (just copied from that):

1. Setup the asset value in each state. Setup payoff in each end node.

2. Calculate risk-neutral prob with formular:

p(U) = [(1+rf)S_t-1 + S(D)] / (S(U) - S(D)]

3. Calculate option premium in t =1 with formular:

   V = [p(U)*S(U) + p(D) * S(D)] / (1+rf)

4. Calculate the option premium at t = 0, with same formular

Multi Binomial Option Pricing: American Put option

1. How can we calculate the price of the US put option?

1. Trick question, this is not possible

Choose optimal time to exercise Call.

1. What steps are different compared to the EU option?

3. What always hold for the American Call options value?

1. Calculate the intrinsic value at each middel note: IV = max{S - K, 0}

2. Then check which is larger

3. Intrinsic value <= Premium before expiration.

American put option: Check if it is optimal to exercise before time.

1. What are the steps that are different from the EU put option?

2. What always holds for American Put options?

Do the exact same as European Put, but also:

1. Calculate the intrinsic value at each middel note: IV = max{K - S , 0} .

2. Then check which is larger.

3. Intrinsic value >= Premium before expiration.

Price European options with dividend

1. How does the steps differ from without dividend?

The exact same, the only thing that changes is:

1. Calculate premium based on the value efter dividend is payed.

2. Risk-neutral prob is still calculated with old stock value.

Price American option With dividend

1. How does the steps differ from without dividend?

Exact same, but also:

1. Calculate premium based on the value efter dividend is payed. But risk-neutral prob is still calculated with old stock value.

2. Calculate intrinsiv value after divided is payed

3. Calculate intrinsiv value before divided is payed

4. Choose the metric with the highest value