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2

Your calculation for 1 looks good. If you use the rule of $72$, the bond should double in $20$ periods if the interest is $3.6\%$ per period. It doesn't quite double in $20$ periods and you have an interest rate a little less. Bingo. For 2, we need to know what she does with the cash that comes in during the period. Naively, you would be expected to ...

2

You may be missing a sign. I need to plug in =RATE(60,94.36,-5300) to get 0.00218984 This is essentially a special case of an IRR calculation, where there is a large amount at time $0$, followed by a number of a constant smaller amounts of the opposite sign. In this particular case it gives the solution to $\displaystyle -5300 + \sum_{n=1}^{60} ... 2 The amount of money after$x$year ($x$a real number) is not multiplied by$(1+r)^x$. Because the compound is not continuous. The amount invested after$k\in \Bbb N$years is multiplied by$(1+r)^x$and this fixed amount is invested until the end of the year, and only then the interests are reinvested. This leads to the multiplier (with$E(x)$the integer ... 2 The objective of the investor is to receive a return. The expected return is indicated by the yield rate. If the bond were not callable, the investor will receive back the face value of$100$in$15$years along with$5\%$coupon payments on earlier dates. If the bond is callable for$ 10 \leqslant t \leqslant 15$at$80$, then the issuer has the option ... 2 Myszzzzz had the right idea, but I'll outline this with an example. Suppose in an account of$5\%$interest I have two payments scheduled to be made: one of$100$now, and$100$one year later. What is the interest from now to one year later? Well, if I put in$100$now, I get$105$one year later, so that's interest of$105-100 = 5$. But, do I include ... 2 Perhaps you misunderstood the question. In particular, note that the payments are flat in the first decade and last decade but vary in the middle decade. Apart from tiny rounding issues,$97.44$appears to be correct for payments in the third decase. Year Start Interest Payment End 1 1000.00 100.00 -100.00 1000.00 2 1000.00 100.00 -100.00 1000.00 ... 1 Let: $$C:Capital=32000\\$$ $$I:Interest=10\\$$ $$N:Number of months = 36\\$$ $$P:Payment\\$$ To find P, plug the numbers in the formula: $$P=\frac{C\times I\div 1200}{1-(1+(I\div1200))^{-N}}\\$$ And you'll find P = 1032.55 1 My interpretation of the question is as follows: the accumulated value of the reinvested interest plus the principal is equal to the accumulated value of a corresponding level payment annuity at 8%. Your last equation clearly cannot be what was intended, since the right-hand side of that equation merely corresponds to the accumulated value of a single ... 1 Covariance matrices, SVD decomposition. etc... 1 I always write out the cash flow explicitly, then express it in terms of actuarial notation, for complex situations with non-level payments. We have $$77.1 = (v^2 + 2v^3 + 3v^4 + \cdots + nv^{n+1}) + (nv^{n+2} + nv^{n+3} + \cdots),$$ where$v = (1+i)^{-1} = (1.105)^{-1}$is the present value discount factor. Now that we can see the cash flow written as a ... 1 Aha! I think I see where your error is.$i=0.10$in the equation involving the annuity symbols. 1 You can rearrange your equation by multiplying through to get$a(n-t) a(t) = a(n)$. Let$n = s+t$and you get$a(s) a(t) = a(s+t)$. So you want to find a function satisfying$a(s) a(t) = a(s+t)$for all choices of$s$and$t$. Now, you can rewrite this to get$a(s) a(s) = a(2s)$, and then$a(s) a(2s) = a(3s)$so$a(3s) = a(s)^3$, and in general$a(ms) = ...

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Hint: The effective annual rate may be computed as $\left(1+\frac{r}{n}\right)^n-1$ where $r$ is the stated annual rate, and $n$ is the number of compoundings per year.

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Using the formula for annual effective rate: $\displaystyle i_{effect}= \bigg(1+ \frac{i}{n}\bigg)^n-1$, we may compute the following. Smart Visa: \begin{align}i_{eff} &= \bigg(1 + \frac{0.24}{365} \bigg)^{365}-1 \\ &=0.271149 \\ &\approx 0.27 \end{align} Principle Card: \begin{align}i_{eff} &= \bigg(1 + \frac{0.25}{52} \bigg)^{52}-1 \\ ...

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Hint for B): The discount is $\$220 \cdot 0.3$. And the new price is$\text{original price - discount}$. 1 Hint Say that the cost for the retailer is$C$and that he makes a profit of$x$%; that is to say that to$1$he pays, he adds$x$percent. So, the sale price looks to be $$P=C \times (1+\frac{x}{100})$$ I am sure that you can take from here. 1 I like the book Brownian Motion - An Introduction to Stochastic Processes by René Schilling & Lothar Partzsch pretty much. It does not only cover stochastic differential equations (in particular, several possibilites are presented how to solve SDEs, e.g. by transforming them into linear SDEs), but also contains a very interesting and detailed exposition ... 1 I like "Brownian Motion Calculus" by Ubbo F. Wiersema (2008). I found it more approachable than other books I've seen. But it depends on your level of mathematical sophistication (I'm a mathematical hick). 1 In general, every security's price is equal to the present value of its cash flows. You have interpreted redeemable at par correctly. c can be seen to have a greater value than a by the reason you gave and similarly d can be seen to have a greater value than b. This can be seen by examining the present value equation and seeing that it increases with ... 1 Bonds have an inverse relationship with yield. So if Smith purchases a$10\%$bond and now the interest rate is$12\%$, the face value of the bound has gone down. Therefore, the bond is trading at a discount at present. If at some time later, the interest rate was$8\%$, the bond would be worth more and trading at a premium. Assuming there is still some ... 1 By the premium/discount formula, which is$ P = C + (Fr - iC)a_{n|i}$you can see that, if$ Fr -iC >0 $, the bond is selling at premium so the earliest redemption date is the most favorable for the issuer. (Because he would like to stop repaying the premium via the coupon payments as soon as possible.) For$ i^{(2)} = 0.08 $, the bond is ... 1 Hint: under the risk neutral probability, the prices of the securities having an$L^2$payoff are martingales. So you probably need to compute the Ito-differential of $$(t,\omega)\to S^3_t e^{(2r+3σ^2)(T−t)}$$ 1 It may not be possible to determine VaR analytically and you will need to run a Monte Carlo simulation. As you indicate, for a univariate distribution$F$, VaR is specified as the$\alpha$-percentile: $$P(X \leq \text{VaR}) = \alpha \implies \text{VaR}=F^{-1}(\alpha).$$ In the multivariate case, with$X = X_1 + X_2$, you need the distribution ... 1 In the Black-Scholes model, the underlying price$S_0$is positive. Then for$T>0\$, "vega", the partial derivative of the option price with respect to volatility is positive $$\frac{\partial X}{\partial \sigma} = S_0N'(d_1)\sqrt{T}=\frac1{\sqrt{2\pi}}\exp(-d_1^2/2)S_0\sqrt{T}>0$$

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