Continuously Compounded Rate of Return Cagr

Modeling (Optional)

Gary Smith , in Essential Statistics, Regression, and Econometrics (Second Edition), 2015

Continuous Compounding

As the examples have shown, more frequent compounding increases the effective return. Theoretically, we can even have continuous compounding, taking the limit of (1   + R/m) ^m as the frequency of compounding becomes infinitely large (and the time between compounding infinitesimally small):

(11.10) $1+S=\underset{m\to \infty }{\lim }{\left(1+\frac{R}{m}\right)}^{mn}={\text{e}}^{Rn}$

where e   =   2.718… is the base of natural logarithms. In our example, with R  =   12 percent, continuous compounding pushes the effective annual rate up to 12.7497   percent. Although advertisements trumpeting "continuous compounding" convey the feeling that the bank is doing something marvelous for you and your money, the improvement over daily compounding is slight.

On the other hand, continuous compounding is very useful for modeling growth because it simplifies many computations. If the continuously compounded rate of growth of X is 3   percent and the continuously compounded rate of growth of Y is 2   percent, then the continuously compounded rate of growth of the product XY is 3   +   2   =   5   percent, and the continuously compounded rate of growth of the ratio X/Y is 3   −   2   =   1   percent. It is as simple as that.

In Chapter 1, we looked at the distinction between nominal data (denominated in dollars, euros, or another currency) and real data (adjusted for changes in the cost of living). Using continuous compounding, if nominal income Y is increasing by 5   percent and prices P are increasing by 3   percent, how fast is real income Y/P increasing? By 5   −   3   =   2   percent. If real income Z is increasing by 2   percent and prices P are increasing by 1 percent, how fast is nominal income ZP increasing? By 2   +   1   =   3   percent.

Continuously compounded models are written like this:

(11.11) $Y_t=Y_0{\text{e}}^{gt}$

where Y _t is the value of Y at time t and g is the continuously compounded growth rate.

Suppose, for example, that Y is nominal income and that Y initially equals 3000 and grows at a continuously compounded rate of 5   percent:

$Y_t=3000{\text{e}}^{0.05t}$

We let the price level P initially equal 1 and grow at a continuously compounded rate of 3   percent:

$P_t=1.0{\text{e}}^{0.03t}$

Real income initially equals 3000 and grows at 2   percent:

$\begin{array}{c}\frac{Y}{P_t}=\frac{3.000{\text{e}}^{0.05t}}{1.0{\text{e}}^{0.03t}}\\ =3000{\text{e}}^{0.02t}\end{array}$

Figure 11.15 shows a graph of the growth of nominal income and real income over time.

Figure 11.15. Continuously compounded growth.

The continuously compounded growth rate can be estimated by taking the logarithm of both sides of Eqn (11.11):

$\ln \left[Y_t\right]=\ln \left[Y_0\right]+gt$

and using simple regression procedures with the logarithm of Y the dependent variable and time the explanatory variable.

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Brownian Motion and Stationary Processes

Sheldon M. Ross , in Introduction to Probability Models (Twelfth Edition), 2019

10.4.1 An Example in Options Pricing

In situations in which money is to be received or paid out in differing time periods, we must take into account the time value of money. That is, to be given the amount v a time t in the future is not worth as much as being given v immediately. The reason for this is that if we were immediately given v, then it could be loaned out with interest and so be worth more than v at time t. To take this into account, we will suppose that the time 0 value, also called the present value, of the amount v to be earned at time t is $v{e}^{-\alpha t}$ . The quantity α is often called the discount factor. In economic terms, the assumption of the discount function $e^{-\alpha t}$ is equivalent to the assumption that we can earn interest at a continuously compounded rate of 100 α percent per unit time.

We will now consider a simple model for pricing an option to purchase a stock at a future time at a fixed price.

Suppose the present price of a stock is $100 per unit share, and suppose we know that after one time period it will be, in present value dollars, either $200 or $50 (see Fig. 10.1). It should be noted that the prices at time 1 are the present value (or time 0) prices. That is, if the discount factor is α, then the actual possible prices at time 1 are either $200e^{\alpha }$ or $50e^{\alpha }$ . To keep the notation simple, we will suppose that all prices given are time 0 prices.

Figure 10.1.

Suppose that for any y, at a cost of cy, you can purchase at time 0 the option to buy y shares of the stock at time 1 at a (time 0) cost of $150 per share. Thus, for instance, if you do purchase this option and the stock rises to $200, then you would exercise the option at time 1 and realize a gain of $\mathrm{\$}200-150=\mathrm{\$}50$ for each of the y option units purchased. On the other hand, if the price at time 1 was $50, then the option would be worthless at time 1. In addition, at a cost of 100x you can purchase x units of the stock at time 0, and this will be worth either 200x or 50x at time 1.

We will suppose that both x or y can be either positive or negative (or zero). That is, you can either buy or sell both the stock and the option. For instance, if x were negative then you would be selling −x shares of the stock, yielding you a return of $-100x$ , and you would then be responsible for buying −x shares of the stock at time 1 at a cost of either $200 or $50 per share.

We are interested in determining the appropriate value of c, the unit cost of an option. Specifically, we will show that unless $c=50/3$ there will be a combination of purchases that will always result in a positive gain.

To show this, suppose that at time 0 we

$\begin{array}{c}\mathrm{buy}x\text{units of stock, and}\hfill \\ \mathrm{buy}y\text{units of options}\hfill \end{array}$

where x and y (which can be either positive or negative) are to be determined. The value of our holding at time 1 depends on the price of the stock at that time; and it is given by the following

$\mathrm{value}=\{\begin{array}{cc}200x+50y,\hfill & \text{if price is}200\hfill \\ 50x,\hfill & \text{if price is}50\hfill \end{array}$

The preceding formula follows by noting that if the price is 200 then the x units of the stock are worth 200x, and the y units of the option to buy the stock at a unit price of 150 are worth $(200-150)y$ . On the other hand, if the stock price is 50, then the x units are worth 50x and the y units of the option are worthless. Now, suppose we choose y to be such that the preceding value is the same no matter what the price at time 1. That is, we choose y so that

$200x+50y=50x$

or

$y=-3x$

(Note that y has the opposite sign of x, and so if x is positive and as a result x units of the stock are purchased at time 0, then 3x units of stock options are also sold at that time. Similarly, if x is negative, then −x units of stock are sold and $-3x$ units of stock options are purchased at time 0.)

Thus, with $y=-3x$ , the value of our holding at time 1 is

$\mathrm{value}=50x$

Since the original cost of purchasing x units of the stock and $-3x$ units of options is

$\mathrm{original cost}=100x-3xc,$

we see that our gain on the transaction is

$\mathrm{gain}=50x-(100x-3xc)=x(3c-50)$

Thus, if $3c=50$ , then the gain is 0; on the other hand if $3c\ne 50$ , we can guarantee a positive gain (no matter what the price of the stock at time 1) by letting x be positive when $3c>50$ and letting it be negative when $3c<50$ .

For instance, if the unit cost per option is $c=20$ , then purchasing 1 unit of the stock ( $x=1$ ) and simultaneously selling 3 units of the option ( $y=-3$ ) initially costs us $100-60=40$ . However, the value of this holding at time 1 is 50 whether the stock goes up to 200 or down to 50. Thus, a guaranteed profit of 10 is attained. Similarly, if the unit cost per option is $c=15$ , then selling 1 unit of the stock ( $x=-1$ ) and buying 3 units of the option ( $y=3$ ) leads to an initial gain of $100-45=55$ . On the other hand, the value of this holding at time 1 is −50. Thus, a guaranteed profit of 5 is attained.

A sure win betting scheme is called an arbitrage. Thus, as we have just seen, the only option cost c that does not result in an arbitrage is $c=50/3$ .

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Some Applications of the Fundamental Theorem

Robert L. Kosowski , Salih N. Neftci , in Principles of Financial Engineering (Third Edition), 2015

Exercises

1.

(Black–Derman–Toy model.) You observe the following default-free discount bond prices B(t, T _i), where time is measured in years:

(13.150) $B(0,1)=95,B(0,2)=93,B(0,3)=91,B(0,4)=89$

These prices are assumed to be arbitrage-free. In addition, you are given the following cap-floor volatilities:

(13.151) $\sigma (0,1)=0.20,\sigma (0,2)=0.25,\sigma (0,3)=0.20,\sigma (0,4)=0.18$

where σ(t, T _i ) is the (constant) volatility of the LIBOR rate $L_{T_i}$ that will be observed at T _i with tenor of 1 year.

a.

Using the Black–Derman–Toy model, calibrate a binomial tree to these data.

b.

Suppose you are given a bond call option with the following characteristics. The underlying, B(2, 4), is a two-period bond, expiration T=2, strike K _B =93. You know that the BDT tree is a good approximation to arbitrage-free LIBOR dynamics. What is the forward price of B (2, 4)?

c.

Calculate the arbitrage-free value of this call option using the BDT approach.

2.

(Exchange rates and LIBOR rates.) You know that the euro/dollar exchange rate e _t follows the real-world dynamics:

(13.152) $\mathrm{d}{e}_t=\mu \mathrm{d}t+0.15e_t\mathrm{d}{W}_t$

The current value of the exchange rate is e _o =1.1015. You also know that the price of a 1-year USD discount bond is given by

(13.153) $B{(t,t+1)}^{\mathrm{US}}=98.93$

while the corresponding euro-denominated bond is priced as

(13.154) $B{(t,t+1)}^{\mathrm{EU}}=98.73$

Both of these prices are arbitrage-free and there is no credit risk.

a.

What are the 1-year LIBOR rates in these two currencies at time t?

b.

What are the continuously compounded interest rates $r_t^{\mathrm{US}},r_t^{\mathrm{EUR}}$ ?

c.

Obtain the arbitrage-free dynamics of the e _t . In particular, state clearly whether we need to use continuously compounded rates or LIBOR rates to do this.

d.

Is there a continuous time dynamic that can be written using the LIBOR rates?

3.

(European option.) Consider again the data given in the previous question.

a.

Use Δ=1 year to discretize the system.

b.

Generate five sets of standard normal random numbers with five random numbers in each set. How do you know that these five trajectories are arbitrage-free?

c.

Calculate the value of the following option using these trajectories. The strike is 0.95, the expiration is 3 years, and the European style applies.

4.

(European FX option.) Suppose you know that the current value of the peso–dollar exchange rate is 3.75 pesos per dollar. The yearly volatility of the Mexican peso is 20%.

The Mexican interest rate is 8%, whereas the US rate is 3%. You will price a dollar option written on the Mexican peso. The option is of European style and has a maturity of 270 days. All processes under consideration are known to be geometric.

a.

Price this option using a standard Monte Carlo model. You will select the number of series, the size of the approximating time intervals, and other parameters of the Monte Carlo exercise.

b.

Now assume that Mexico's foreign currency reserves follow a geometric SDE with a volatility of 10% and a drift coefficient of 5% a year. The current value of reserves is USD7 billion. If reserves fall below USD6 billion, there will be a one-shot devaluation of 100%. Is this information important for pricing the option? Explain.

c.

Use importance sampling to reprice the option. Your pricing is supposed to incorporate the risk of devaluation.

EXCEL Exercises

5.

(European Option.)

Write a VBA program to simulate M=100 stock prices using a Monte Carlo technique to calculate the prices of European Call and Put options based on the following data:

•

S(0)=100; K=105; T=1; r=8%; σ=50%

Gradually increase the value of M and report the observed resulting price of the options.

6.

(Digital Currency Options.) Write a VBA program to simulate M=100 stock prices using a Monte Carlo technique to calculate the prices of digital call and put options FX options as discussed in the text. Use the following parameters:

•

S(0)=$1.54; K=$1.58; T=1; r=8%; r _f =6%; σ=30%; payoff R=$10

Gradually increase the value of M and report the observed price of the options.

7.

(Barrier Option.) Write a VBA program to simulate M=100 stock prices using a Monte Carlo technique to calculate the price of Barrier down-and-out and down-and-in call options based on the following data:

•

S(0)=100; K=110; T=1; r=8%; σ=50%; H=90

Gradually increase the value of M and report the observed price of the options.

MATLAB Exercises

8.

(European Options.) Write a MATLAB program to document the efficiency of a Monte Carlo approach to the estimation of European Call and Put option prices based on the following data:

•

S(0)=100; K=105; T=1; r=8%; σ=30%

Plot a graph of estimated prices as a function of the number of stock price simulations.

9.

(Barrier Option.) Write a MATLAB program to document the efficiency of a Monte Carlo approach to the estimation of the price of Down-and-Out and Down-and-In Call options based on the following data:

•

S(0)=100; K=110; T=1; r=8%; σ=30%; H=90

•

Plot a graph of estimated prices as a function of the number of stock price simulations.

10.

(Digital Currency Option.) Write a MATLAB program to observe the efficiency of Monte Carlo technique to estimate the price of Digital Call and Put price with the following data:

•

S(0)=$1.54; K=$1.58; T=1; r=8%; r _f =6%; σ=30%; R=$10

Plot the graph of estimated price v/s the no. of stock price simulation.

11.

(BDT Model Calibration.) Write a MATLAB program to calibrate the BDT model based on the following data on bond prices and implied volatilities

•

B(t ₀, t ₁)=0.95; B(t ₀, t ₂)=0.93; B(t ₀, t ₃)=0.91; B(t ₀, t ₄)=0.89

•

σ(0,1)=20%; σ(0,2)=25%; σ(0,3)=20%; σ(0,4)=18%

Draw the LIBOR tree based on the output results.

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The concept of comonotonicity in actuarial science and finance: applications

J. Dhaene , ... D. Vyncke , in Insurance: Mathematics and Economics, 2002

Assume that we are currently at time 0. Consider a risky asset (a non-dividend paying stock) with prices described by the stochastic process {A(t),t ≥0}, and a risk-free continuously compounded rate δ that is constant through time. In this section, all probabilities and expectations have to be considered as conditional on the information available at time 0, i.e. the prices of the risky asset up to time 0. Note that in general, the conditional expectation (with respect to the physical probability measure) of e^−δt A(t), given the information available at time 0, will differ from the current price A(0). However, we will assume that there exists a unique "equivalent probability measure Q" such that the discounted price process {e^−δt A(t),t≥0} is a martingale under this equivalent probability measure. This implies that for any t≥0, the conditional expectation (with respect to the equivalent martingale measure) of e^−δt A(t), given the information available at time 0, will be equal to the current price A(0). Denoting this conditional expectation under the equivalent martingale measure by E ^Q [e^−δt A(t)], we have that

(52) $\text{E}{}^{\mathrm{Q}}\text{[}\text{e}{}^{\mathrm{-\delta t}}\text{A(t)]=A(0),}\text{t≥0.}$

The notation F _A(t)(x) will be used for the conditional probability that A(t) is smaller than or equal to x, under the equivalent martingale measure Q, and given the information available at time 0. Its inverse will be denoted by F _A(t) ⁻¹(p).

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Recent advances in robust optimization: An overview

Virginie Gabrel , ... Aurélie Thiele , in European Journal of Operational Research, 2014

3.2.3 Other optimization models

Derivatives. Zymler, Rustem, and Kuhn (2011) propose a robust optimization model for portfolios that may include European-style options, leading to tractable, convex second-order cone programming problems, and derive guarantees on the worst-case portfolio return, both weak (when the return falls within the specified uncertainty set) and strong (when it falls outside). The presence of derivatives in the portfolio creates nonlinearities, in the uncertainty, of the portfolio return, and makes problems involving VaR difficult to solve. Zymler, Kuhn, and Rustem (2013) address this issue by developing two tractable approximations to the problem of computing the VaR of a portfolio with derivatives when the first two moments of the underlying asset prices are known.

Uncertain continuously compounded rates of return. Kawas and Thiele (2011a) consider the case where the worst-case value of a portfolio must be maximized in a static framework, when the objective is linear in the decision variables (asset allocation) and nonlinear but convex in the uncertain parameters, in line with the Log-Normal model of stock prices. The work shows that tractable and insightful robust optimization models can still be derived for this specific class of nonlinear robust problems. This is extended in Kawas and Thiele (2011b) to the case of nonlinear nonconvex portfolio management problems, which arise when short sales are allowed, i.e., decision variables can be negative. Further extensions are provided in Kawas (2010).

Additional topics. Bienstock (2007) develops a framework for robust portfolio management that is based on empirical histograms and uses a cutting-plane approach that proves to be effective for large, real-life data sets. Leibfritz and Maruhn (2009) consider the question of designing a hedging portfolio, which produces a payoff greater than or equal to that of another portfolio (financial instrument) in all states of the economy, in the presence of model or implementation errors. The resulting optimization problem is solved by a sequence of linear and nonlinear semidefinite programming (SDP–NSDP) problems. The NSDP problem is related to an eigenvalue minimization problem. The authors show convergence of the iterates of the sequential approach. Florez-Lopez (2010) provides a comparative analysis of methods to achieve robustness in credit scoring in the presence of missing data.

Gregory, Darby-Dowman, and Mitra (2011) evaluate the cost of robustness in portfolio optimization by considering the maximum return portfolio optimization problem. The authors derive the robust model under polyhedral uncertainty sets from a min-regret perspective and examine the properties of robust models with respect to portfolio composition. Fonseca, Zymler, Wiesemann, and Rustem (2011) consider the robust optimization of currency portfolios, where the uncertainty is on the foreign exchange rates. A key issue that does not affect stock-only portfolios is that exchange rates must satisfy the non-arbitrage property, to avoid "free" money-making opportunities generated by repeated conversion through a set of currencies. Robust decision times rather than asset allocations (e.g., when to sell an underperforming stock) are analyzed in Dziecichowicz (2011). Selecting assets to maximize similarity compared to an index when the similarity coefficients are uncertain is studied in Chen and Kwon (2012).

Delage and Ye (2010) and Ben-Tal et al. (2010) also make theoretical contributions to the robust optimization literature that are illustrated on financial management examples.

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