# Suppose a bond has a life of three periods. It offers $50 coupon payments in periods 1,2, and 3. It has a face value of $1000. If the relevant interest rate is 8% for period 1, 5% in period 2, and 10% in period 3, what is the present value of this bond? 2. Some bond has a life of 4 periods, a spot yield of 5%, pays a coupon of $35 on each of the years (including at maturity), a face value of $800, and currently has a price of $1000. a. What is the Macaulay duration of this bond?

**QUESTION**

1. Suppose a bond has a life of three periods. It offers $50 coupon payments in periods 1,2, and 3. It has a face value of $1000. If the relevant interest rate is 8% for period 1, 5% in period 2, and 10% in period 3, what is the present value of this bond?

2. Some bond has a life of 4 periods, a spot yield of 5%, pays a coupon of $35 on each of the years (including at maturity), a face value of $800, and currently has a price of $1000.

a. What is the Macaulay duration of this bond?

b. Now suppose you have a liability stream with a Macaulay duration of 2 periods. If you wanted to immunize the portfolio and the only other bond has a Macaulay duration of 1.5, in what proportions should you hold these 2 stocks?

3. Using your own words, briefly (1 paragraph max) explain why:

a. An investor may prefer a forward contract to a futures contract.

b. An investor may prefer a futures contract to a forward contract.

4. Construct a futures market in such a way that the number of rainy days for a given month in Des Moines, Iowa, is (hopefully) accurately predictable.

5. Suppose that for a group of European options, the strike price is $100, the asset is currently trading at $112, and the interest rate is 15%. What is the lower bound for:

a. The call option?

b. The put option?

**ANSWER**

**To calculate the present value of the bond, we need to discount the future cash flows back to their present value using the relevant interest rates for each period. Let’s calculate the present value of each coupon payment and the face value separately:**

Period 1:

Present Value (PV1) = Coupon Payment / (1 + Interest Rate) ^ Period

PV1 = $50 / (1 + 0.08) ^ 1 = $46.30

Period 2:

PV2 = $50 / (1 + 0.05) ^ 2 = $44.61

Period 3:

PV3 = ($50 + $1000) / (1 + 0.1) ^ 3 = $801.65

Now, we sum up the present values of all cash flows:

Present Value of the bond = PV1 + PV2 + PV3

Present Value of the bond = $46.30 + $44.61 + $801.65 = $892.56

Therefore, the present value of the bond is $892.56.

Macaulay duration measures the weighted average time until cash flows are received from a bond. To calculate the Macaulay duration, we need to find the present value of each cash flow, weight it by the period it is received, and then divide by the bond’s current price.

Year 1:

PV1 = $35 / (1 + 0.05) ^ 1 = $33.33

Year 2:

PV2 = $35 / (1 + 0.05) ^ 2 = $31.75

Year 3:

PV3 = $35 / (1 + 0.05) ^ 3 = $30.24

Year 4:

PV4 = ($35 + $800) / (1 + 0.05) ^ 4 = $770.23

Macaulay duration = (1 * PV1 + 2 * PV2 + 3 * PV3 + 4 * PV4) / Current Price

Macaulay duration = (1 * $33.33 + 2 * $31.75 + 3 * $30.24 + 4 * $770.23) / $1000

Macaulay duration = 3.84

Therefore, the Macaulay duration of the bond is 3.84.

**To immunize the portfolio with a liability stream duration of 2 periods, we need to match the Macaulay duration of the bond to the liability duration. Given that the other bond has a Macaulay duration of 1.5, we can calculate the proportion in which to hold these two stocks.**

Let’s assume the proportion of the bond with duration 1.5 is x, and the proportion of the bond with duration 3.84 is y.

To immunize the portfolio, we set up the following equation:

2 = (1.5 * x) + (3.84 * y)

Since the only other bond available has a duration of 1.5, we have x = 1, and substituting in the equation:

2 = (1.5 * 1) + (3.84 * y)

2 = 1.5 + 3.84y

0.5 = 3.84y

y = 0.5 / 3.84

y ≈ 0.1302

Therefore, to immunize the portfolio, you should hold the bond with a Macaulay duration of 1.5 fully (100%) and the bond with a Macaulay duration of 3.84 in approximately 13.02% proportion.

An investor may prefer a forward contract to a futures contract due to the flexibility it offers. In a forward contract, the terms and conditions can be customized to suit the specific needs of the parties involved. The contract can be tailored in terms of the underlying asset, quantity, price, and delivery date. This customization allows the investor to hedge against specific risks or achieve specific objectives. Additionally, forward contracts are usually traded over-the-counter (OTC), which means they are not subject to exchange regulations and can offer more privacy and confidentiality.

An investor may prefer a futures contract to a forward contract due to their standardized nature and the ease of trading on organized exchanges. Futures contracts are standardized contracts that are traded on regulated exchanges, making them highly liquid and easily tradable. Standardized terms, such as contract size, delivery date, and underlying asset quality, reduce negotiation and transaction costs. The exchange clearinghouse acts as a counterparty to all contracts, mitigating counterparty risk. Futures contracts also provide greater transparency in terms of pricing and market information. Additionally, the ability to take long or short positions and the ease of offsetting positions before expiry make futures contracts more flexible for active trading and speculation.

Unfortunately, it is not possible to construct a futures market that accurately predicts the number of rainy days for a given month in Des Moines, Iowa. The number of rainy days is influenced by various complex factors such as weather patterns, atmospheric conditions, and natural variability, which are inherently unpredictable. While weather derivatives exist to hedge against specific weather-related risks, they are typically based on objective weather parameters like temperature or rainfall index rather than directly predicting the number of rainy days. Predicting the exact number of rainy days accurately in advance is still a challenging task for meteorologists and climate scientists.

The lower bound for the call option can be determined using the concept of intrinsic value. Intrinsic value represents the immediate payoff that the option would provide if it were exercised immediately. For a call option, the intrinsic value is the difference between the current asset price ($112) and the strike price ($100). If the intrinsic value is negative, it is set to zero, as options cannot have a negative value. Therefore, the lower bound for the call option is $12 (i.e., $112 – $100).

Similarly, the lower bound for the put option can also be determined using intrinsic value. For a put option, the intrinsic value is the difference between the strike price ($100) and the current asset price ($112). If the intrinsic value is negative, it is set to zero. In this case, since the asset price is greater than the strike price, the put option has no intrinsic value, and thus the lower bound for the put option is zero.

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