Questions and Answers of Venture Capital And The Finance Of Innovation

In the 1-step binomial model shown in Figure 5.2, consider the portfolio consisting of one long position in the contingent claim and short\[Q_{0}=\left(C_{u}-C_{d}ight) /\left(A_{u}-A_{d}ight)\]of
For the random walk shown in Figure 5.6.(a) What is the expected movement during each period?(b) What is the standard deviation of the movement during each period?(c) What is the expected movement
A European cancelable swap allows the owner to cancel a swap at some point before the swap maturity. The cancellation option is economically equivalent to a European swaption into an offsetting
Using the formulas for ARA and RRA(a) Show that exponential utility: $U(x)=1-e^{-c x}$ for some $c>0$, has constant ARA (CARA).(b) For power utility: \(U(x)=\left(x^{1-\alpha}-1ight)
An investor has a base $10 \log$ utility function: $U(x)=\log _{10}(x)$. Investment 1 has payoff $(1,100)$ with probabilities $(0.8,0.2)$. Investment 2 has payoff $(10,1000)$ with
Assume you have $\$ 100,000$ to invest for one year, and decide to allocate $60 \%$ to an all equity fund trading at $\$ 400$ per share and $40 \%$ to a bond fund trading at $\$ 100$ per
Show the steps to get from Formula 3.4 to Formula 3.5.
Let $R_{1} \sim\left(\mu_{1}=4 \%, \sigma_{1}=10 \%ight)$ and $R_{2} \sim\left(\mu_{2}=6 \%, \sigma_{2}=6 \%ight), ho\left(R_{1}, R_{2}ight)=$ $25 \%$, where $R_{1}, R_{2}$ are the returns of
The canonical form of a shifted hyperbola in the $(\sigma, \mu)$ plane is $\sigma^{2} / c_{\sigma}^{2}-$ $\left(\mu-\mu_{0}ight)^{2} / c_{\mu}^{2}=1$.(a) Derive the expression for the two
Given three assets whose returns' means, variances, and correlations are\[\boldsymbol{\mu}=\left[\begin{array}{l}5 \% \\6 \% \\3 \%\end{array}ight] \quad \boldsymbol{\sigma}=\left[\begin{array}{r}6
Assume that the risky returns are jointly normal and use the method of Lagrange multipliers to find the optimal location on the MVF for an investor with exponential utility, that is, maximize \(\mu-c
To derive Formula 3.11, we observe that the slope of the Capital Market Line connecting R0R0 to MM is the maximum among all lines connecting R0R0 to any feasible portfolio. The weights for MP are
Let the risk-free rate be $3 \%$ and the market return and risk be $\left(\mu_{M}=ight.$ $\left.8 \%, \sigma_{M}=10 \%ight)$. Given an asset with $\beta=0.8$ and a return standard deviation
According to CAPM, $\mu_{X}$ is a linear function of $\beta_{X}$. The graph of $\mu_{X}$ versus $\beta_{X}$ in the $(\beta, \mu)$ plane is called the security market line (SML).(a) What is
For two assets, let their 1 -year simple returns, $R_{1}, R_{2}$, be independent and identically distributed as follows\[P\left[R_{i}=right]=p, \quad P\left[R_{i}=-right]=1-p\]for some $r>0$ and
It is argued that in the commonly used 60/4060/40 portfolio (60%(60% stocks, 40%40% bonds), since stocks are much riskier than bonds, the stocks' relative contribution to the portfolio risk is too
Starting with $N$ assets, construct two portfolios $P_{1}, P_{2}$ with corresponding allocations $\left\{w_{11}, \ldots, w_{1 N}ight\}$, and $\left\{w_{21}, \ldots, w_{2 N}ight\}$. Create a
Prove Jensen's inequality for discrete random variables via induction as follows. For a convex function $f$ :(a) Let a random variable $X$ take on two values \(x_{1}
Arithmetic versus geometric average. (a) Let a1,a2>0a1,a2>0 and show that a1+a22≥√a1a2a1+a22≥a1a2 (b) Use Jensen's inequality applied to the log function to show that the arithmetic
Let $T_{0}=0, T_{1}, T_{2}, \ldots$ denote dates where $T_{n}$ is $n$ days from today $\left(T_{0}ight)$. An asset's price is $\$ 1,000$ today, $A\left(T_{0}ight)=1000$, and the
Given a discount curve, $D(T)$, one can extract continuously compounded zero-coupon rates for any date via\[D(T)=e^{-T \times Z(T)} \Leftrightarrow Z(T)=-\frac{1}{T} \ln D(T)\]when interest rates
Using the same no-arbitrage argument to derive Formula 4.3(a) Provide an expression for the forward exchange rate, $F_{X}(0, T)$, for any $T$ using simple (add-on) domestic and foreign interest
A 2 -year bond with a semiannual coupon rate of $4 \%$ per annum is trading at par $(100 \%)$.(a) What is its spot semiannual yield?(b) Assume one can borrow at $3 \%$ p.a. simple interest rate
The nn-year inflation rate, InIn, is the growth rate in the price of a basket of goods. In the United States, the Consumer Price Index (CPI) serves as the price index and is related to the inflation
A stock trading at $\$ 100$ per share can be financed at the continuously compounded interest rate of $5 \%$ per annum.(a) What is the 1-year forward price of the stock if it pays quarterly
Assume that the spot JPY/USD exchange rate is 120 Yen per 1 USD, and that the continuously compounded interest rate in the United States and Japan are $1 \%$ and $2 \%$, respectively.(a) What is
Let the noncompounding (simple) 3-month and 6-month interest rates be $2 \%$ and $3 \%$, respectively.(a) What is the $[3 \mathrm{~m}, 6 \mathrm{~m}]$ noncompounding forward rate?(b) What is
If the continuously compounded interest rate is a constant $r$, what is the instantaneous forward interest rate (see Formula 4.5)?
Assume the risk-free continuously compounded interest rate is $4 \%$ per annum. For an asset with today's price $A(0)=\$ 100$, you are told that its expected return is $10 \%$ per annum and
For each step of the risk-neutral binomial model, what is the expected continuously compounded yield\[\frac{1}{t_{i+1}-t_{i}} E_{t_{i}}\left[\ln
Let $r$ be the risk-free continuously compounded rate and $A(0)$ today's value of an asset. For a given horizon $T$, assume the asset can take on two values $A_{u}, A_{d}$ with risk-neutral
In a 1-step binomial model, compute the risk-neutral probabilities for some $\Delta A$ when(a) $A_{u, d}=A_{0} \pm \Delta A$(b) $A_{u, d}=F_{A}(0, T) \pm \Delta A$
Using the same numerical values as in the 2-period binomial model in Example 2(a) Calculate today's price, $P_{0}$, of a 6-month European put option with strike $K=\$ 100$.(b) Calculate today's
Replication via Forward Contracts. In the 1-step binomial model, replicate the option payoff at expiration via $Q_{0}$ amount of a forward contract with delivery price of $K_{0}$.(a) Solve for
Given two independent random variables $X_{1}, X_{2}$(a) Provide an expression for $E\left[X_{1}+X_{2} \mid X_{1}ight]$.(b) Evaluate the above when $E\left[X_{1}ight]=E\left[X_{2}ight]=0$.
Let $X_{1}, X_{2}, \ldots$ be independent and identically distributed random variables with $E\left[X_{i}ight]=0$, and let\[S_{n}=\sum_{i=1}^{n} X_{i}\]Show that $S_{1}, S_{2}, \ldots$ form a
Let r=4%,A(0)=100r=4%,A(0)=100, and let the price of a 3 -month (T=0.25)(T=0.25) ATMF ( K=100e0.04/4)K=100e0.04/4) call option be 2.50 . (a) Using your favorite solver, calculate the implied
In the CRR model, let A0=100,r=4%,σ=12%,T=1A0=100,r=4%,σ=12%,T=1, and N=12N=12. (a) Calculate the price of a 1-year call option with K=100K=100. (b) Calculate the price of the same option using
Normal and lognormal random variables.(a) Let $X \sim N\left(\mu, \sigma^{2}ight)$. Express CDF of X in terms of $N(\cdot)$, the CDF of a standard normal random variable, $N(0,1)$.(b) Let \(X
Derive the BSM call formula by completing the following steps. Let $F=F_{A}(0, T)$ for notation ease. We have\[\begin{align*}C(0) e^{r T} & =E[\max (0, A(T)-K)] \\& =F \times E\left[\max \left(0,
As seen in the proof of the BSM call formula, for a call\[N\left(d_{2}ight)=P[A(T) / F \geq K / F]=P[A(T)>K]\]that is, $N\left(d_{2}ight)$ is the probability that the call option finishes in the
The BSM formula for a call is\[\begin{aligned}C(0) & =e^{-r T}\left[F_{A}(0, T) N\left(d_{1}ight)-K N\left(d_{2}ight)ight] \\& =A(0) N\left(d_{1}ight)-K e^{-r T} N\left(d_{2}ight)\end{aligned}\]One
In addition to futures, options on futures contracts are actively traded on exchanges. The expiration date $T_{e}$ of the option need not coincide with the forward date $T$ of the futures
Normal and Lognormal Diffusions. Let $A(t, \omega)$ be a diffusion, and let $\mu, \sigma$ be two constants.(a) Let\[d A(t, \omega)=\mu d t+\sigma d B(t, \omega)\]and show that\[A(t, \omega) \sim
Bermudan Options. Using the CRR model with $A_{0}=100, r=4 \%$, $\sigma=10 \%, T=0.5, N=6$.(a) Using backward induction, calculate the price of a 6-month ( $T=$ $0.5)$ Bermudan put option
Confirm that the call Formula 6.6 is a solution to the BSM PDE.
Show that the European call and put formulas are convex functions of the underlying asset price.
Let the volatility $\sigma$ be nonconstant and a function of the underlying asset, $\sigma(A(t))$. Use the chain rule to compute the smile-adjusted delta of a call option.
A chooser option allows the owner to decide on $T_{1}$ whether to own a European-style $K$-strike call or put option with expiry $T_{2}>T_{1}$.(a) At $T_{1}$, the payoff of the chooser option
Using Table 7.3 and linear interpolation in discount factors(a) Calculate the spot and quarterly forward 3-month simple (add-on) rates, $f\left(\left[T_{i}, T_{i+1}ight]ight)$ for \(T_{0}=0,
Using Table 7.3 and linear interpolation in discount factors, for a $\$ 1 \mathrm{M}$ 1 -year swap with semiannual fixed rate of $4 \%$ per annum and quarterly floating leg based on 3-month
Using the same setup as Example 2(a) Compute today's value of each of the four floorlets for a \$1M 1-year forward start 1 -year $K=4 \%$ quarterly floor on 3 -month rates.(b) Compute today's value
Using the data in Table 7.3(a) Compute the semiannual forward swap rate for a 1-year swap, 2 -year forward, $F_{2,1}$.(b) Using Black's normal formula with $\sigma_{N}=0.80 \%$, compute the value
Let $R, P$ be the value of a European receiver, payer swaption with the same expiry, swap term, and strike $K$.(a) Let $S$ be the value to the receiver of the underlying swap with fixed rate
Using the Black's normal call formula(a) Show that the delta of a call option, $\partial C(0) / \partial A(0)$, equals $N(d)$.(b) Using put-call parity, compute the delta of a put option.(c) What
Consider an ATMF straddle's price under lognormal and normal dynamics, and use the first central difference approximation\[N^{\prime}(x) \times x \approx N(x / 2)-N(-x / 2)\]to relate the normalized
Using the example in Figure 7.7 and the results in Table 7.5(a) Show the computations for the $T_{1}$ discount factor curves in each of the $(u, d)$ states starting from the $T_{2}$ discount
A Bermudan cancelable swap allows the owner to cancel a swap at some point before the swap maturity. The cancellation option is economically equivalent to a Bermudan swaption into an offsetting swap.
Use the Binomial Formula\[(a+b)^{n}=\sum_{k=0}^{n}\left(\begin{array}{l}n \\k\end{array}ight) a^{k} b^{n-k}\]to show(a)\[e=\lim _{n ightarrow \infty}\left(1+\frac{1}{n}ight)^{n}=\sum_{k=0}^{\infty}
Using an annual interest rate $r=4 \%$, recompute Table 2.1 for $F V(T)$ for a horizon date of 6 months, $T=0.5$. Repeat for a horizon date of $1 \mathrm{y}$ and 6 months, $T=1.5$.Table 2.1
How long does it to take to double your money at a given rate $r$ ? Begin with\[F V=P V e^{r t}=2 P V\]and approximate $\ln (2) \approx 0.72$ to come up with the Rule of 72 .(a) At $3 \%$,
Let $r_{m}$ be the interest rate with $m$ compoundings per year, for example, $m=12$ means monthly compounding.(a) Derive the formula to convert $r_{m}$ to $r_{n}$ for general \(m,
A 2 -year semiannual coupon bond is issued on $1 / 1 / 2020$ and matures on $1 / 1 / 2022$ with a semiannual coupon rate of $4 \%$ per annum and coupon dates 7/1/2020, 1/1/2021, 7/1/2021,
Geometric Series (a) In a Fractional Reserve system of banking, each bank need only keep a fraction αα, say 5%5%, of its deposits and can lend out 1−α1−α. The loan recipient will then deposit
Using the bisection method, compute the semiannual yield of a 5-year bond paying a semiannual coupon of $2 \%$ per annum and trading at $98 \%$. Start with bracketing levels of $0 \%, 10 \%$,
Using a semiannually compounded yield of $4 \%$ p.a., numerically or via evaluating the formula, compute the PV01 of(a) $\$ 1$ million 5 -year zero-coupon bond(b) $\$ 1$ million 5 -year \(4
For a $T$-year zero-coupon bond with continuously compounded yield $y$, compute(a) Its modified duration, $1 / P \times d P / d y$(b) Its convexity, $d^{2} P / d y^{2}$(c) The price, modified
Limits at $y=0$(a) Evaluate $P(C, y, N, m)$ (Formula 2.5) at $y=0$.(b) Evaluate $d P / d y$ (Formula 2.9) at $y=0$.(c) Evaluate $d P / d C$ (Formula 2.10) at $y=0$.
Convexity(a) If $f^{\prime \prime}$ exists, use the definition 2.11 and\[f^{\prime \prime}(x)=\lim _{h ightarrow 0} \frac{f(x+h)+f(x-b)-2 f(x)}{h^{2}}\]to show that $f^{\prime \prime}(x) \geq 0$
For a 30 -year monthly level-pay home mortgage loan of $\$ 500,000$ with a $3 \%$ interest rate(a) Compute the monthly level payment.(b) What is the total amount of interest you pay during the
Derivation of $S(x, m, N)$(a) Derive the formula for $S(x, m, N)=\sum_{n=m}^{N} n x^{n}$ in Formula 2.16. Hint: Begin by evaluating $S(x, m, N)-x S(x, m, N)$.(b) Alternatively, derive the
Let a 2 -year bond with a semiannual coupon rate of $2.25 \%$ p.a. have a semiannual yield of $2 \%$ p.a., and a 10 -year bond with a semiannual coupon rate of $2.5 \%$ p.a. have a semiannual
Given instruments in the following table(a) Extract the semiannual discount factors for 10 years using the bootstrap method with linear and log-linear interpolation in discount factors.(b) On the
Using the price-yield Formula 2.5 for a bond with a periodic coupon of $C / m$ with $N$ remaining coupons, show that $P(N, m, C, y) \times(1+y / m)^{N}$ equals the sum of all coupon payments
U.S. Treasury Bills are quoted based on an Act /360/360 discount yield, PB=PB= 1−yBN/3601−yBN/360, where NN is the number of days from settlement date to maturity date. (a) Provide a formula for
Compute price $P$, sensitivity $d P / d y$, convexity $d^{2} P / d y^{2}$ of $2 \mathrm{y}, 5 \mathrm{y}$, $10 \mathrm{y}, 30 \mathrm{y}, 100 \mathrm{y}$ bonds, all with a semiannual coupon
Compute total interest for a level pay loan, $I=\sum_{n=0}^{N-1} I_{n}$, to show that $I=A L \times C / m$. Does this make sense?
Evaluate Formula 2.18 when $s=1$, and explain the result.
For a given time series $x_{1}, x_{2}, \ldots$, the $N$-period arithmetic Moving Average (MA), $A(n, N)$, is defined as\[A(n, N)= \begin{cases}\left(x_{1}+\ldots+x_{n}ight) / n & \text { if } n
Milton Friedman famously said that changes in money growth affect the economy with “long and variable lags.” That means that if the government increases growth in the monetary base this month,
One of the reasons it’s difficult to be a monetary policymaker is because it’s so hard to tell what’s actually going on in the economy. It’s a lot like being a doctor in a world before
Complete the following sentences: With a real shock, when real growth is worse than usual, inflation is _______________ than usual. With an aggregate demand shock, when real growth is worse
Define the following:a. The monetary base, MBb. M1c. M2
Does the House of Representatives get to vote on who becomes the chairperson of the Federal Reserve Board? If not, who does get to vote?
Using the FRED economic database (https://fred.stlouisfed.org/), graph the Effective Federal Funds rate from 2007 to the present. Now click Edit Graph and then Add Line. Add the Unemployment rate.
Central banks and voters a like usually want higher real growth and lower inflation. What kind of shock makes that happen?
Let’s reenact a simplified version of the 1981–1982 Volcker disinflation. Expected inflation and actual inflation are both 10%, real growth is 3%, and to keep it simple, velocity growth is zero.
a. Suppose that the central bank follows a fixed 3% annual monetary growth rule, as Milton Friedman sometimes recommended. In the short run, what will the velocity shock do to real growth and to
Practice with the best case: You are the central banker, and you have to decide how fast the money supply should grow. Your economy gets hit by the following AD shocks and your job is simply to
Consider the following figure. Suppose that there’s a rise in →v due to business optimism—what Keynes called the “animal spirits” of investors. This pushes us to AD(2). If the
a. Do most federal government transfers of cash go to the elderly or to the poor?b. Do most federal government purchases of health care go to the elderly or to the poor?
Using the FRED economic database (https://fred.stlouisfed.org/), create a series showing federal expenditures as a share of GDP. First find Federal Government current expenditures. Then Edit Graph
Let’s take a look at one aspect of the U.S. tax system over the past 100 years. Using the FRED economic database (https://fred.stlouisfed.org/), look for “U.S Individual Income Tax: Tax Rates for
There are many ways to help poor people in foreign countries. You can find charities, for example, that will buy a cow or shoes for people in poverty or help to dig a well in a low-income village.
Assume that the economy is growing steadily and hasn’t had a recent recession. A new administration increases spending on the military by $200 billion, a significant amount.a. What is your estimate
What shifts AD to the left: A rise in taxes or a cut in taxes? Does this push →v up or push it down?
Let’s see what the “three difficulties with using fiscal policy” look like in real life. Categorize each of the following three stories as either (1) crowding out, (2) magnitude, or (3) a
When people “buy government bonds,” are they borrowing money or saving money?
Imagine you live in the land of Ricardia, where every citizen is a Ricardian and thus “Ricardian equivalence” is 100% true. Government spending never changes in Ricardia: It’s a fixed amount
If the U.S.-debt-to-GDP ratio were 100% and if the interest rate on the debt were 5% (not far from the truth at present), then what fraction of U.S. GDP would go toward paying interest on the debt?

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