Financial Economics Question - HOST

Question 1

St. Petersburg Game -Toss a fair coin until a head appears; if the head occurs on the k^th toss, the player gest a payoff of 2^k and the game ends?

(1) What is the fair value of the game? What is the expected payoff to a player?
(2) Give me bid-ask spread to play this game?

Answer 1

• (1/2)^k*(2)^k = 1
• 1+1+1+….= k
• The expected payoff to the game is Infinite
• “Value” is not the same thing as the “expected payoff”
• Value equals “utility” of expected payoff. Most people cannot distinguish between very large amount of money
• Ex: 2^k = 2⁵⁰ for all k>= 50, then the expected payoff of this game is:
  
  • 50 + 2⁵⁰* ( 1/2⁵¹ +1/2⁵² +..) = 51
• Spread can be; ($20, $200). How much would you pay to play this game, how much would you charge the customer?
• Another way to think about this problem:
  
  • Customer goes bankrupt after $1 million- that is about 20 tossses
  • Bank goes bankrupt after $1 billion - that is about 30 tosses (however you should charge more because your job and shareholder’s money is at stake)

Question 2

If the standard deviation of continuously compounded stock return is 10%. What is the standard deviation of continuously compunded four-year stock returns?

Answer 2

• Assuming continuously compounded returns follow an arithmetic Brownian motion
• Variance of returns grows linearly with the compounding return
  
  • Why? Consecutive returns in a random walk are independent. and the variance of a sum of independent returns in a random walk are independent.
  • Variance of the sum of indepdent random variables is just the sum of the variances
• This means that 4 year σ² equals four times one-year σ²
• It follows that 4 year σ equals two times one-year σ. Hence the answer is 20%

Question 3

Term-structure of the interest rate: You see that five year spot rate = 10% and 10 year spot rate = 15%, What is the implied forward rate from five year to year 10?

Answer 3

• (1+.10)⁵ x (1+x)⁵ = (1+.15)¹⁰
  
  • Think intutitively:
  • Since the 10 year rate is 15%, and first five year rate is only 10%
  • It must be that implied forward rate from 5 to 10 year must be 20% to average out the 10 year rate to 15%

Question 4

Explain carefully the difference between the “yield” on a bond and the “rate of return” on a bond?

Answer 4

• “Yield” on a bond is usually “internal rate of return” or “YTM” or “promised yield”
  
  • It is what you earn if you hold the bond till maturity
  • Assuming that we can re-invest at the constant rate.
  • In practice - reinvestment rate varies and our actual return might vary
• “Rate of return” on a bond is the internal rate of return of the “realized cash flows” to the bond-holder including the reinvestments (also capital PnL)

Question 5

What is “Chaos Theory”? Can you use it to predict stock returns? If so, how?

Answer 5

• Chaos Theory: Computer-simulated nonlinear mathematical equations describing the evoluitions of weather patterns are very sensitive to the starting values of the variables
• Sensitive depends on initial condition
• Non-linear systems describing chaotic systems are non-random
• Doesn’t work in finance

Question 6

Draw the bond price versus YTM. Why is the curve convex?

Answer 6

• Macaulay Duration: the slope of the price-yield curve: [- D/ (1+r)] P
  
  • r = yield, P = Price
• For the zero-coupon bond, the bond’s duration is ten years because it is a ten-year zero coupon bond

Question 7

CAPM says that plot of E(r) and Beta is an upward sloping line through (0,rf) and [1, E(rm)]. Reality is a bit different. Which of these scenarios is most likely?

(1) An upward sloping curve begin at (0,rf), wholly above the theoretical SML. Initially more steep but eventually paralle to sML
(2) fully below the theooretical SML

Answer 7

• If the empirical SML is wholly above the theoretical one, this means that stocks are under-priced relative to CAPM.
  
  • There are other factos than just the Market
  • The market is the only risk factor but market participants require higher compensation per unit of beta-risk than suggested by CAPM
• If the empirical SML is below the theoretical one. market participants do not require as much compensation per unit of beta-risk as theory suggests

Question 8

2 Year spot = 7.60%

1 Year Spot = 7.15%

What is the forward rate for the 2nd year?

Answer 8

• Average is 7.6 (which is two years spot)
• Difference between Average and 1 year spot is 45bps,
  
  • Multiply this by 2
  • add 90bps to 7.15
• 8.05% <<– Forward rate for the 2nd year

Question 9

Consider a six-month forward contract on a 10-year riskless ZCB. Is the bond selling at a forward premium or a forward discount?

(2) Does your answer change if the bond is riskless coupon bond?

Answer 9

• Forward price F(t,T) is related to spot price
  
  • F(t,T) = S(t)*e^r(T-t) > = S(t)
  • The discount bond sells at a forward premium
• Coupon bond is different:
  
  • Continuous coupon: p
  • Forward price: F(t,T)
  • F(t,T) = S(t)^e(r-p)(T-t) < = S(t)
    
    • p is higher than r
  • The coupon bond sells at forward discount because of no-arbitrage

Question 10

I have a long position in $100M 30-year bond. what can I do to limit my exposure to only $50M?

Answer 10

• You can reduce the exposure by shorting the T-Bond futures contract
• CBOT - T-bond futures cover a face value of $100,000 of T-Bonds
• If the duration of my bond is the same as tthe duration of CTD (Cheapest to deliver) T-Bond, then we can short:
  
  • 50M/100K = 500 contracts.
• If the duration differs:
  
  • DB/DF * 500
    
    • DF = duration of the CTD T-Bond
    • DB = duration of my bond

Question 11

You hold 8% coupon, 30Yr, $1000 Par - Mexican Brady Bond. Interest rates in Mexico do not change. Rates in the US increase by 1%. What is the change in the price of your bond? make any necessary assumptions.

Answer 11

• Delta(P) = -D*P* [Delta(Y) / (1+y) []
  
  • Let’s say duration of this bond: 15
  • -15 * 1000 * [.0025/ 1.08]
  • -37.50/1.08 = -35
• With these assumptions - Brady bond prices go down by about 3 to 4% points
• About a quarter of Brady bonds are collaterized by the US Treasury

Question 12

Five year interest rate = 10%, 10 year interest rate = 15%, five year forward rate = 20%

In plain english, why the forward rate has to be higher than 20% approximation?

Answer 12

• Key argument lies upon the fact that interest on interest accumulates
• If you are offered 10% first five years and then 20% in second half. This won’t do well as receiving 15% every year
• To avoid this arbitrage, interest rate for the five year forward has to be higher than 20%

Question 13

Arithmetic, Geometric and Harmonic averages:

Answer 13

• Arithmetic = [x1 +x2 +…+xn] / n
• Geometric = (x1*x2*x3*…*xn)^(1/n)
• Harmonic = n / [1/x1 + 1/x2 + 1/x3 + ..+1/xn]
  
  • A > = G > = H

Question 14

If I get a head, I get $7 in 18 months. If I get a tail, I lose $2 instantly. One year rate is 12% and 2 year rate is 18%. How much should I pay for this game?

Answer 14

• 1 year forward 1 year from now: 24%
  
  • since for six months, we will earn about 12%
• E(V) = expected value of the game
  
  • E(V) = .5*(-2) + (.5)*(7 / (1.12)*(1.12) ]
    
    • = -1 + (3.5/1.24)
    • = 1.8

Question 15

Two stocks have the same expected return. One has standard deviation of 20% and 2nd one has standard deviation of 30%. Correlation of two stocks = .50. How do I allocate money s.t. I minimize my risk?

Answer 15

• σ1 = .20, σ2 = .30, p = .5
• w = weights in stock 1
• Portfolio Variance =
  
  • w²σ1² + (1-w)²σ2² + 2w(1-w)σ1σ2p
• Differentiate the above w.r.t to w
  
  • 2wσ1²-2(1-w)σ2² + 2(1-w)σ1σ2p -2wσ1σ2p
  • 2 [w(σ1² + σ2² - 2σ1σ2p) - σ2² + σ1σ2p]
• w = [σ2 (σ2 - σ1p) / (σ1² + σ2² - 2σ1σ2p]
• In our example w = .8571
• σ of portoflio = .1964

Question 16

Explore different correlations and vol of stock 1 and stock 2.

Answer 16

• if p = -1, then w = [σ2 / (σ1 + σ2)] and σ = 0
  
  • σ = vol of the portfolio (no shorting required)
• if -1 < p < [min(σ1, σ2) / max(σ1, σ2)]
  
  • w = [σ2 (σ2 - σ1p) / ( σ1² + σ2² - 2σ1σ2 p)]
  • No shorting requird
• if [min(σ1, σ2) / max(σ1, σ2)] < p < 1
  
  • w = same as above
  • Shorting required
• if p = [min(σ1, σ2) / max(σ1, σ2)],
  
  • all money goes to the lowest vol asset
• if p = 1
  
  • w = [σ2 / (σ2 - σ1)]
  • High vol asset is shorted to over-invest in the low vol asset. The optimum is the zero-risk portfolio