Final

If you own 1 unit of stock and you are afraid of downside risk, you should buy a put option (protective put). Suppose you buy a put with strike $80 at a price of $10. What will be the value of your portfolio, net of the cost of the option, if the price end up at 95?

$85 The value of your one unit of stock is 95. However, you paid 10 for the put, which has now expired worthless. Hence, net of the cost of the option, the value of the portfolio is only 85.

Concept Questions #27

Go over on NYU Classes

An investor who can set up an arbitrage...

Is sometime earning positive cash flow but is never losing money We defined arbitrage as a trade in which you never lose money but in some cases you make a sure profit.

Bond Types:

Par bond - face value = price Discount bond - face value > price Premium bond - face value < price

Consider a company whose existing assets generate cash-flows worth $1B today. Suppose that the only new project the company is (or ever will be) considering costs $.1B to implement today and generates a payoff $.2B in 10 years. If the firm's required rate of return is 10%, what is the value of the firm? Suppose managers know the required rate of return and optimize shareholder value.

$1B because the firm will never undertake the new project as it has negative NPV. The project has negative NPV since .2/1.1^10<.1. Therefore, the firm will not undertake the project and it's current value equals the NPV of the cash flows of its current assets ($1B).

Suppose the price of a one-year zero coupon bond is $900, and that the price of a two-year zero coupon bond is $800. Both bonds have $1,000 face value. What is the price of a two-year coupon bond with a 10% coupon rate?

$970 We build a replicating portfolio that has the same cash flows as the coupon bond with a 10% coupon rate. These cash flows are $100 in period 1 and $1100 at maturity. Therefore, we need to buy 0.1 units of the one-year zero coupon bond which amounts to a cost of $90. Furthermore, to generate the same cashflow in period 2 as for the coupon bond, we need to buy 1.1 units of the two-year zero coupon bond which requires a payment of $880. Together, the cost for the portfolio amounts to $90+$880=$970.

According to the liquidity premium theory, the forward rate is...

an upward biased predictor of future expected short rates. The liquidity premium theory says that investors require extra compensation for risk for holding longer maturity bonds. Hence, forward rates incorporate a risk compensation. That is, forward rates are set by investors to be higher than their actual expectation of future short rates as compensation for risk. Thus, there is an upward bias in forward rates relative to future expected short rates.

If the current interest rate is 5% and your semi-annual coupon paying bond has a duration of 5.33 years, how much will the price of the bond change if the interest rate increases by 1 basis point?

change in price = -(5.33/1.05) * .0001 Use the formula that shows how the percentage change in the bond price is related to the Duration and the yield change. A basis point is 1/100 of a percent. Therefore, using the formula we have: ΔP/P= D/(1 + y) Δy = - 5.33/1.05 * 0.01%

If you own a European call option with strike price 100, 1 year to expiration, the riskless interest rate is 1% and the stock is trading at 112, what is the lowest price for which you should be willing to sell your option (minimum value or adjusted intrinsic value)?

112-100*e^(-.01*1) = $13 The adjusted intrinsic value is given by S - X exp(-rT) where T = 1 as the interest rate is given as an annual rate and there is one year to expiration. Substituting in gives 112 - 100 exp(-.01*1).

The Law of One Price states...

Any two securities with the same cash flows must carry the same price The idea behind the law of one price is that if two securities have the same cash flows and they do not carry the same price, there is an arbitrage opportunity (as long as there are no transaction cost).

Which one-year option strategy will give the highest expected net profit if a. you are very confident that the stock price will move VERY LITTLE over the next year, b. out-of-the-money options are much cheaper than at-the money options?

Short a straddle If you do not expect the stock to move much, then you should be selling straddles or strangles. This is because these positions pay off well if the stock moves around a lot from its current price. If you do not expect that the stock will move much, you should be selling these types of positions. For thinking about whether you should sell a straddle or strangle, let's consider straddles and strangles with strike prices centered around the current price. Then, since strangles involve selling out-of-the- money options, and you consider these to be cheap, you should instead sell straddles.

The yield on three-year zero-coupon bonds is 3% and the yield on one-year zero-coupon bonds is 2%. Investors expect that in two years, the yield on three-year zero-coupon bonds will still be 3% but the yield on one-year zero-coupon bonds will be 1%. What is the expected annualized holding period return (ann. HPR) of an investor who buys a three-year bond and sells it in two years?

(1.03^3/1.01)^0.5-1 = 4% The current price of the three year zero coupon bond is Facevalue/1.03^3. The expected price of the same bond two years from now is Facevalue/1.01. This is because the coupon bond has one year left until maturity and the expected yield on one year zero coupon bonds is 1%. The annualized holding period return is ann. HPR = (V(t)/V(0))^(1/t)-1 = [(Facevalue/1.01)/(Facevalue/1.03^3)]^(1/2)-1 = [1.03^3/1.01]^(.5)-1.

If you buy a European put with strike price $100 and expiration in 1 year, which costs $10, and the price on expiration is $70, what is the (annual) rate of return on your investment?

(100-70-10)/10 = 200% The payoff upon expiration is 100-70 = 30. To get the net payoff, subtract from this the cost of the put, which is 10, to get 20. Since expiration is in 1 year, the annual rate of return is just this net payoff divided by the initial investment of 10. Hence, the rate of return is 20/10 = 200%.

What is the yield to maturity on a 5 year zero coupon bond with face value 100 and price equal to 85?

(100/85)^.2 -1 The yield to maturity sets the NPV equal to the current price, hence 85 = 100/(1+YTM)^5. Solving this for YTM yields: YTM = (100/85)^.2-1. This is equivalent to YTM = (FV/PV)^(1/T) - 1 since the future value (FV) of the zero coupon bond is simply its facevalue at maturity and no other payments occur until maturity.

Suppose you buy a 2-year 4% coupon paying bond, with face value equal to $1000, and a YTM of 4%. The price of that coupon paying bond is

1000 The yield to maturity (YTM) equals the coupon rate only for bonds trading at par. Since YMT = 4% = coupon rate in the example, the bond needs to trade at par, i.e. its current price equals its face value, $1,000.

Treasure inflation protected bonds (TIPS) make payments based on the inflation adjusted facevalue of the bond, i.e. if inflation is 10%, the treasury makes payments as if the facevalue of the bond was: actual facevalue *(1+inflation rate). The inflation rate for the coming year is 2%. A treasury Inflation Protected Bond (TIPS) with one year maturity, 1 coupon at a coupon rate of 5% and face value of $100 will pay how much at maturity:

105*1.02 The difference between TIPS and a regular bond is that the TIPS have an adjustment factor that scales up the payments of the bond to compensate the investor for inflation. The adjustment factor is equal to the accumulated rate of inflation over the life of the bond. Hence, for this bond it is equal to 1.02 (since inflation is 2% for one year). As the face value paid by the bond is $100 and the coupon is $5, the answer is just $105*1.02 or $100*1.02 + ($100*1.02*.05).

What is true about American and European Options?

A European option can be exercised only at the day of maturity, whereas an American option can be exercised at or before the day of maturity. Since the option holder has higher flexibility with the American option, American options are more expensive, all else equal. A European option can be exercised only at the day of maturity, whereas an American option can be exercised at or before the day of maturity. Since the option holder has higher flexibility with the American option, and investors are willing to pay for flexibility, American options are more expensive, all else equal.

Why does an investor buy a call option?

Because he/she thinks the stock price will go up and he/she wants to limit the downside risk. The investor will benefit from the call option if the stock prices goes up above the strike price. However, the same is true if the investor buys the stock. The difference is that if he buys the call then his downside is limited to the price of the call. Of course, in return for this limited downside, he pays the price of the call option.

Suppose you are short a call option. To hedge your risk exposure, you would like to cover your position by buying the underlying stock (which does not pay dividends). What exactly do you do between now and expiration date?

Buy Delta = N(d1) shares of the stock initially and keep changing your portfolio until expiration, as Delta moves over time. The Delta of the option is the sensitivity of the option to a $1 move up in the stock price. If you buy Delta shares of stock, then you will be hedged against moves in the stock price because if the stock moves up $1 you will lose Delta dollars on the short call option position but make it back on your Delta shares of stock. Hence, you are hedged. Since the Delta of the call option moves around over time, you need to constantly adjust your shares of stock to remain hedged. This is called Delta hedging.

Suppose two companies, Zinc and Zonc, issue a one-year zero coupon bond. The yield to maturity (YTM) of Zinc bond is 3%, and the YTM of Zonc bond is 5%. Suppose there are no cost of shorting. How would you make money ?

Buy Zonc and sell Zinc. There are two ways you can see the right answer. First, you want to hold the bond that pays a higher yield to maturity. Hence you go long the Zonc bond and short the Zinc bond. The second line of thought first translates the yields into prices. High yields imply low prices and vice versa. Hence the Zonc bond is cheaper than the Zinc bond. By the same logic as in question 2, you want to buy the cheap bond and sell the expensive Zinc bond.

If the price of one year zero coupon bonds in New York is 80 and the price of 1 year zeros in San Francisco is 85, then how would you construct an arbitrage?

Buy the NY bond, sell short the SF bond now, unwind positions after 1 year. Buying the cheap bond while selling the expensive bond today creates an instantaenous profit. This profit comes from the sale that generates more income than the cheap bond costs. At the end of the year, the cash flows exactly cancel out.

Find the price of a European call with strike 75 and maturity 6 months. The underlying stock is currently trading at 60, it has a volatility of 0.6 per year (continuously compounded) and it pays no dividends. The riskless interest rate per year is 10 percent also continuously compounded. (you will need a calculator to compute the BS formula)

C0 = 6.25 Use the Black-Scholes formula. First find the value of d1, which is given by: d1 = [ln(60 / 75) + (0.1 + ½* 0.6^2) *0.5 ] / [ .6 * sqrt(.5)] = -.196 and d2 = d1 - 0.6*sqrt(.5) Using the standard-normal distribution table, N(d1) = 0.422 and N(d2) = 0.268. Then calculating the price of the call we get 60 * N(d1) - X exp(.1*.5) * N(d2) = 6.25.

Which of the following is true? A. For a par bond, coupon rate = current rate = yield to maturity. B. For a par discount bond, coupon rate< current rate <yield to maturity. C. For a par premium bond, coupon rate > current rate > yield to maturity. D. All of the above

D. All of the above Coupon rate, current rate and yield to maturity are only equal for par bonds. For discount bonds, we have coupon rate< current rate <yield to maturity and or premium bonds coupon rate > current rate > yield to maturity.

Which of the following 4 firms should have the highest price-earnings ratio?

Firm 4 with a low beta and a high dividend growth. We can see this using the valuation formula and by simply reasoning through it. First, a low beta means the required rate of return is relatively low. Hence, for a given rate of dividend or earnings growth, the price should be relatively high. This means the price-earnings ratio will be high. Secondly, holding fixed the beta and required rate of return, a high growth rate means intuitively that future earnings and dividends will be relatively large, so the price should be relatively high relative to current earnings or dividends. We can see this formally from the formula for the price-earnings ratio: P/E = (1 - b)(1 + g)/(R - g) We can see that, holding g fixed and increasing R lowers the price-earnings ratio. On the other hand, increasing g increases the numerator and lowers the denominator, both of which increase the price-earnings ratio.

Suppose you buy a 2-year 4% coupon paying bond, with face value equal to $1000, and a YTM of 4%. You hold the bond until maturity. From year 1 to year 2, you reinvest the coupon at 8% interest rate. The annualized holding period return on that investment is

Greater than the YTM of 4% In class we discussed how, if the reinvestment rate of return is equal to the YTM, then the YTM and annualized holding period return will be equal. If the reinvestment rate of return is higher than the YTM, as is the case here, then the annualized holding period return will be greater than the YTM. This is because we were able to reinvest funds at a rate of return that is even higher than the YTM, which raises the overall holding period return. Recall that the annualized holding period return is the return that one needs to earn to transform the initial investment into the total forward value of all the payouts from the investment. Here the initial investment is $1000 (par bond). The forward value of all payouts (forwarded to year 2) is: 40 * 1.08 + 1040. Hence, the annualized holding period return is given by: 1000 * (1+annual return)^2 = 40 * 1.08 + 1040. Solving this shows that annual return = 4.077%, which is greater than 4%.

If you are a bank with liabilities of duration 4 years, how will you choose the composition of your assets to eliminate the interest rate risk that you would be exposed to if you had a duration mismatch on your balance sheet? Assume there exist only 3 year and 25 year assets.

I would purchase a portfolio consisting for 95% of 3 year and for 5% of 25 year bonds with a total value equal to the value of my liabilities. This will equal the duration of assets and liabilities. You want your overall portfolio to have 0 duration (to be duration neutral). If you purchase a portfolio of assets that has total value equal to the value of your liabilities, then you need to match the duration of the asset portfolio with the duration of the liabilities. The duration of the liabilities is 4 years. The duration of the asset portfolio is given by the weighted average of the two assets that constitute it. If w is the weight on the 3 year assets then (1-w) is the weight on the 25 year assets and setting the asset duration equal to 4 means that: w * 3+(1-w) * 25 = 4 Solving for w gives w = 0.954 (rounded to full percentages, this is 95%).

Which of the following is true about a security's price and expected holding period return?

If a security is underpriced, its expected holding period return is higher than the required rate of return. The expected holding period return and the required rate of return are equal to each other in equilibrium, when the security is correctly priced. An underpriced security has a higher expected return than its required rate of return, i.e. it is an attractive investment. Such underpricing cannot persist in the long run since investors would want to buy underpriced securities and by doing so, would drive up their prices until expected holding period return and the required rate of return are the same, i.e. the security is fairly priced.

What is true about the predictions of the expectations hypothesis (EH) and liquidity premium (LP) theories of the yield curve about the average slope of the yield curve?

In EH, the average yield curve is flat and in LP it is upward sloping. In EH, forward rates are equal to expected short rates so the average forward rate is equal to the average short rate. Hence, the average for all forward rates is the same. It is just equal to the average short rate. Hence, the average slope of the yield curve is at (sometimes the yield curve is upward sloping, sometimes downward sloping, but taking the average across all times gives a at yield curve). On the other hand, the LP says that as compensation for holding long-term bonds, investors demand a liquidity premium. This means yields are higher for longer maturity bonds and the yield curve tends to be upward sloping. Equivalently, forward rates are higher than expected short rates and on average the yield curve is upward sloping.

If the YTM on a 20 year T-bond is lower than the YTM on a 3 month T-bill, then, according to the expectations hypothesis theory...

Investors expect future short rates to be lower than the current 3 month interest rate. According to the expectations hypothesis, forward rates are equal to investors' expectations of future short interest rates. Since the 20 year YTM is lower than the 3 month interest rate, the forward rates must be lower than the 3 month interest rates. According to the expectations hypothesis, that forward rates are below the current 3 month interest rate means that investors expect future short rates to be lower than the current 3 month interest rate.

If the YTM on a 2 year zero coupon bond that starts today is 5% and the YTM on a 1 year zero coupon bond that starts today is 3%. What does the no-arbitrage condition tell you about the interest rate on a one year bond that starts next year?

It's ((1.05)^2 / 1.03) -1 = 7% To figure this out, we need to equate the returns one earn on two alternative, but equivalent, investments: (1) Invest in a two-year zero coupon bond or (2) Invest in the 1 year bond and the one year bond that starts in one year. Investment (2) will return 3% over the first year and f over the second year (the yield on the one year bond that starts in one year). Investment (1) returns 5% per year. Hence, we are equating 1:05^2 = 1.03 (1 + f). Solving for f gives f = 1.05^2/1:03 - 1.

What is the value of an option on a risk-free asset? Specifically, consider a European put option with one year to expiration (T=1) and strike price of $110 (X=110) written on a one-year zero coupon bond with face value $100 (F=100) and yield-to-maturity 2% (YTM=2%). What will be the option premium today (P_0)? (After you answer this question, consider whether an option on a risky asset will have a higher or lower premium, all else equal.)

It's P_0 = (110-100)/1.02 = 9.8. You know for sure that the price of the risk-free bond at maturity will equal its facevalue, $100. Since the put option gives you the right to sell the bond for $110, you will exercise for sure. The value of the option at expiration is therefore $10. You have to discount the value of the option at that point (a year from now) by the current YTM, hence the current value of the option is 10/(1.02). Since there is not uncertainty about what the price will be and, hence, whether you exercise the option, there is no value in being able to decide later. For risky assets, the value of being able to decide later (time value of option) is positive. Therefore, the premium is higher.

Which one-year option strategy will give the highest expected net profit if a.) you are very confident that the stock price will move A LOT over the next year, b.) out-of-the-money options are much cheaper than at-the money options?

Long a call and long a put Since you believe the stock price will move a lot over the next year, but do not know which direction, you should buy a call and a put. Since you believe that the stock will move a lot, you will make money whether it goes up or down. If the market does not realize that the stock is likely to move a lot, then calls and puts will be underpriced (cheap) and you should be buying them.

Yahoo stock pays the same expected cash flows as a portfolio of stocks containing Amazon, IBM and Nextel. What do you know about the price of Yahoo stock. Transacting stocks involves no transaction costs.

Nothing, because the stock and the portfolio pay the same expected cash flows, but cash flows are risky here. As we know from the section on the Capital Asset Pricing Model, expected values do not determine the price. If there is risk involved with the payouts of cash flows, we need to take systematic risk into account. Since we do not have any information on it, we cannot make a statement.

If the risk-free rate is 5%, the expected market return is 10%, the beta of the firm is 2, the current dividend is 1 and dividends are expected to grow at a rate of 10% per year, what price should the firm's stock be trading at?

P = 1.1 /.05 = $22 We simply use the discounted-dividend formula (aka the Gordon growth formula) that was shown in class to value the rm. The required return is calculated using the CAPM and is given by R = 0.05+2(0.1-0.05) = 0.15. Then the price of the stock should be given by P = D(1 + g)/(R - g) = 1 (1 + 0.1)/(0.15 - 0.10) = 1.1/(0.05) = 22

A company is about to go public. It announces that it plans to pay a $1 per share dividend in its first year of existence and 2$ in its second year. From year 3 onwards dividends are expected to grow at a constant rate of 10% per year. The risk free rate is 5%, the company's beta is 2 and the expected market return is 20%. What should be the IPO stock price?

P = 1/(1+.35) + 2/(1+.35)^2 + 2(1+.1)/(.35-.1)(1+.35)^2 To value the company we use the two-stage dividend discount model that we discussed in class. In the first stage, we value the dividends paid by the company in the first two years. The required rate of return is 0.05+2*(0.2-0.05) = 0.35. Hence, the value of the first two cash flows is: 1/(1+0.35) + 2/(1+0.35)^2. We can then value the cash-flows paid out from that point on. We do this in two parts. First, we compute the value of the stream of dividends as if we were standing at the end of year 2, and then we discount this value back two years to today. At the end of year 2, the company just looks like a firm that has $2 dividend, growing at 10% per year. The value of this firm as of the end of year 2 is D(1+g)/(R-g) = (2(1+0.1))/(0.35-0.1). We then discount this back two years to get the value of this today, by dividing it by (1+0.35)^2. Adding it to the value of the first two years of dividends, we get P = 1/(1+0.35) + 2/(1+0.35)^2 + 2(1+0.1)/((0.35-0.1)(1+0.35)^2)

Suppose the Black-Scholes-Merton formula gives the price of a one-year European call as $15.36. The interest rate is 6%, there are no dividends. The stock price is $101 and the strike price is $100. What is the value of a one-year European put with the same strike price?

P = C-S+X*e^(-r*T) = $8.54 You can use the put call parity to determine the price of the put. The put call parity gives the price of a put as a function of the price of a call with the same strike price and maturity, the underlying asset and the risk free rate. The price of the put is P = C-S+X exp(-rT), where C is the value of the call option, S is the current stock price, X is the strike price, T the time to maturity and r is the risk-free rate.

What is the duration D on the following annual pay bond: Face value = 100, maturity, 3 years, coupon rate = 10%, yield to maturity 10%? (you do not need a calculator to solve this question)

P= 100, D = (10/(1.1*P))*1+ (10/(1.1^2*P))*2+(110/(1.1^3*P))*3 Note, first, that the price of the bond is 100 (equal to the face value) since the coupon rate and YTM are equal. Then, just substitute into the formula for Duration. From that formula, the weight wt on the payment in year t is wt = 1 / 1.1^t* P The payments are just 10 in the first and second year and 110 in the third year. Substituting in the payments and weights into the duration formula, we get the answer.

If the call option on stock A is more expensive than the same call option on stock B, the stocks are currently trading at the same price and they pay the same dividend, then...

Stock A must be more volatile than stock B Since the call option on A is more valuable than on B, and everything else about the stocks is equal (same strike, no dividends), stock A must be more volatile than stock B as volatility increases the price of a call. Note that, the risk-free rate is not a stock-specic quantity - it is the same rate for stock A as stock B.

Company ABC has a return on equity (ROE) that is equal to the required rate of return (given by the CAPM). It is considering retaining more earnings (i.e. increasing the plowback ratio b). What will the effect be on ABC's price-earnings ratio?

The P-E ratio will be unaffected. It doesn't matter how much earnings the company retains because the earnings that are kept in the firm and reinvested internally provide the same rate of return as earnings that are paid out. As the ROE is exactly equal to the required rate of return, the return on extra retained returns is exactly what investors demand. Hence, retaining extra earnings is a 0 NPV decision, it neither creates nor destroys value for shareholders. Thus, the price of the stock per earnings will be unchanged.

The book value of a company's equity is equal to...

The current value on the balance sheet of the assets minus the liabilities. This is the denition of book value. Answer (a) is the replacement value and (b) is the liquidation value. Book value uses the numbers from the accounting statements to subtract balance sheet liabilities from balance sheet assets.

If the current one year zero coupon bond yield is 4% and the current two year zero coupon bond yield is 3%, then what do investors expect about the one year zero coupon bond yield next year if the expectations hypothesis is true?

They expect next year's one year yield to be roughly 2%. By the EH, the forward rate is investors' expectation for next year's one-year interest rate, which is just next year's zero coupon bond yield. This forward rate is the rate one needs to earn next year to get the same return from holding two one-year zero coupon bonds in succession as from holding the two-year zero coupon bond. Hence, f is given by: (1+0.04)(1+f) = (1.03)^2. Solving for f gives that is f is approximately 2%.

What is true about the seller of a put option (short a put)?

This person has the obligation to buy the underlying asset at the strike price. The seller of the put has the obligation to buy the underlying at the strike price from the buyer of the put. For taking on this obligation, the put seller is compensated by receiving the price of the put. If the price of the underlying ends up above the strike price, then the seller of the put will be happy. On the other hand, if the price of the underlying plummets, then the put seller will be unhappy that he must buy the underlying for the strike price.

What is true about the relationship of bond yields and prices?

When the yield to maturity increases, bond prices fall. When yields increase, bond prices fall, since there is a negative relationship between bond prices and yields.

How do you compute the effective annual yield (EAY) on a semi-annual coupon paying bond?

You find the semi-annual IRR using six-month periods to discount the bond's cash-flows (if the bond has a T-year maturity, then it has 2*T semi-years to maturity). The EAY is (1+IRR)^2 -1. The return per period is given by the IRR per period. The EAR properly annualizes this per period return, i.e. it tells you what is the annual rate of return if you compound the per period rate. Since the bond is semi-annual paying, coupons are paid every six months. Thus each period is six months. Hence, the computed IRR is per six months and to annualize the rate we do: (1+IRR)^2 -1 since, of course, there are two six month periods in each year.

Which of the following factors unambiguously cause an increase in the duration of a coupon paying bond?

an increase in maturity, a decrease in coupon rate and a decrease in the YTM. Increasing the maturity makes the principal payment occur further in the future and also adds coupon payments that are made further away. A decrease in the coupon rate increases the relative weight on the principal payment, which is the furthest away of all the payments. Hence, this also increases the duration - the weighted average time to payment. Finally, a decrease in the YTM increases the relative weight on payments further out in time because the decrease in YTM cause the greatest increase in their present-value discount factors.

Tobin s q is the ratio of market value to replacement cost. In the long run Tobin's q ratio, which is the ratio of market value to replacement cost must be one because:

entry and exit of firms in the industry will cause an adjustment of the market value. Competition means that firms will enter into industries here the cost of setting up operations is less than the value the market puts on those operations, since that will be protable. In the long run, this increased entry will increase competition and push the market value towards the replacement cost. Conversely, if market value is below replacement cost, new firms will not enter and existing firms will exit. Reduced competition will allow market values in the long run to move up towards replacement cost.

A coupon bond with face value $100 that is has a price of $105...

is a premium bond

The fundamental value of a firm:

is the sum of expected future dividends discounted by the required rate of return given by the CAPM. This is how we dened fundamental value. It is the value of the cash paid to investors by the rm (dividends), discounted at the rate of return required to invest in the rm given its riskiness. The CAPM gives this required rate of return.

The yield to maturity and the annualized holding period return of an investor are the same if

the investor holds the bond until maturity and if he can re-invest coupons at the yield to maturity. For yield to maturity and the annualized HPR to be the same, the investor has to hold the bond until maturity (otherwise, the sale price may have changed) and he has to be able to invest any coupons at the yield to maturity. If either condition is not true, yield to maturity and annualized HPR are not the same.

An option is out of the money when...

the option holder would not want to exercise the option given current prices. For call options this is the case when the price of the underlying asset is below the strike price. For put options this is the case when the price is above the strike price. An option is out of the money if the option holder would not want to exercise given the current prices. Call options are exercised whenever the underlying price is higher than the strike price, so they are out of the money when the price is lower than the strike price. Put options are exercised whenever the underlying price is lower than the strike price, so they are out of the money when the price is higher than the strike price.