1. I nvestment A nalysis II Investment Analysis II - © 2012 Houman Younessi MGMT-6330 Investment Analysis II 1 Interest Rates, Forwards, Futures and Fixed Income Investments

2. I nvestment A nalysis II Investment Analysis II - © 2012 Houman Younessi 2 Interest and Interest Rates (perhaps a review) Interest is the fee paid by the borrower to the lender for the possession of an asset for a fixed period of time. Interest is usually calculated in terms of the percentage of monetary value of the asset for a specific period of time (e.g. a year). This is called an interest rate. In investment finance we define interest rate as the relative return of “risk-free” securities.

3. I nvestment A nalysis II Investment Analysis II - © 2012 Houman Younessi 3 Interest Rate Calculation Simple Time Independent interest (usually simply called interest) is the pure difference between the purchase price of the debt (the amount the borrower must pay at maturity) and the face value of the amount borrowed at contract time (principal) presented as a percentage with respect to the face value. Example: I borrowed $100 from my son and have agreed to pay him $110 “when he needs his money back”. What is the interest? Answer: Maturity time is “when he needs his money back”, it is fixed but not a constant. The interest is the pure difference the principal and purchase price: Note: Such interest is not time based. He may ask for his money back in ten years (his loss) or tomorrow (mine)

4. I nvestment A nalysis II Investment Analysis II - © 2012 Houman Younessi 4 Interest Rate Calculation Simple Time Normalized Interest (usually called simple interest RATE) is the difference between the purchase price of the debt (the amount to pay at maturity) and the face value of the amount borrowed at contract time (principal) when the time from contract to maturity is fixed and pre- determined; presented as a percentage of the face value. Example: I borrowed $100 from my son and have agreed to pay him $110 in exactly one year. What is the interest rate? Answer: Maturity time is exactly one year, it is fixed and constant. The interest rate still is the pure difference the principal and purchase price: Note: Such interest rate is now time based. He can now only ask for his money back in exactly one year per annum

5. I nvestment A nalysis II Interest Rate Calculation What if he wanted his money sooner, or I wanted to borrow the money for longer than one year ? There should be no problem. The interest rate may be calculated for any period of time. These periods usually range from a day (overnight) to multiple years (30 year loan). Each rate is calculated exactly the same. Note: the numerical value of the rate is the same, but the actual rate is not the same. This is 10% every three months Example: I borrowed $100 from my son and have agreed to pay him $110 in exactly 3 months. What is the interest rate? Per Quarter

6. I nvestment A nalysis II Interest rate calculation What if he offered to lend me the money at 2.5% a quarter or 10% per annum, which deal should I take? $2.5 Per Quarter x 4 Quarters = $10 Per Annum There is no difference. So: Note: I am not paying interest on the interest. I am borrowing $100, keeping it for 3 months, then return it along with the $2.50 interest; then borrow $100 again, keep it for 3 months, then return it along with the $2.5 interest………(a total of four times)

7. I nvestment A nalysis II What if he realized that if I kept the money for a whole year and return the principal and interest at the end of the year, at the end of the 3 rd month, he is actually lending me $102.5 and at therefore at the end of the 6 th month I have $102.5 (1+0.025)= $105.06 of his money and I should pay interest on that sum for the third quarter and so on. At what effective annual rate have I borrowed the $100? Note: Now I am paying interest on interest and as such end up paying $0.38 more This is called Compound Interest Interest Rate Calculation or in general:

8. I nvestment A nalysis II Investment Analysis II - © 2012 Houman Younessi 8 Time Value of Money (Present and Future Value) Because the notion of interest exists, money now, would be worth more than money in the future (if interest rates are assumed to be more than zero). One very good way of evaluating the value of various different payment patterns is to determine their present value. The better investment (ceteris paribus) is the one with a higher present value If we know the interest rate, in a simple situation, the present value is the future value divided by (1+r). So d=1/(1+r) is called the discount factor (at times discount rate, but this is confusing and is to be avoided) In the case of payments at a future different from a year, the rate is expressed in its equivalent annual rate (r). If the payment takes place in 1/n years, using simple interest rule the discount factor is: The compound version is:

9. I nvestment A nalysis II Investment Analysis II - © 2012 Houman Younessi 9 Time Value of Money (Present and Future Value) Example: Today’s value of a payment of $105 in one year at 5% per annum simple interest rate is: Today’s value of a payment of $105 in three months at 5% per annum simple interest rate is: Today’s value of a payment of $105 in three months at 5% per annum compound interest rate is:

10. I nvestment A nalysis II 10 Investment Analysis II - © 2012 Houman Younessi Contracts Simple Contracts Derivatives Forward Contracts SwapsFutures Contracts

11. I nvestment A nalysis II Investment Analysis II - © 2012 Houman Younessi 11 Spot Rates A spot rate/price is the rate/price at which you can have an asset now A spot price/rate can be any price/rate, EXCEPT that it has to make sense in terms of other/future prices/rates Consider the following: Bank A lends out money at 5% a year today for customers with good credit. Bank B, a bank in good standing, pays out an interest of 6% per annum on savings accounts. What must any live investor with good credit do? This is called ARBITERAGE Making an investment profit without the potential of risk The theory of efficient markets, laws of supply and demand and profusion of information, tend to very quickly neutralize any arbitrage opportunity

12. I nvestment A nalysis II 12 Investment Analysis II - © 2012 Houman Younessi Forward Rates A forward rate/price is a rate/price in the future that reflects today’s no arbitrage spot rate Most simply, the forward price at an interest rate of 5% for an asset worth $100 today is $105 in one year. Alternatively the forward rate for an asset of $105 in one year, being sold today for $100 is 5% Forward rates do not need to necessarily connect a price or rate in the future to a price or rate now. They just have to reflect today’s spot rates/prices. That is, a forward rate may connect a price in three months to a price in six months (but both have to reflect the reality of today’s spot price which is the only actual fact available)

13. I nvestment A nalysis II 13 Investment Analysis II - © 2012 Houman Younessi Forward Rates A bit more formally: The forward rate f(t,u) is the interest rate for the money invested between dates t and u in the future but agreed upon today In reality, this is a theoretical rate often called implied forward rate, actual or market forward rates are often different as they are moderated by transaction costs, risk of incomplete knowledge of future behavior of rates, delays, etc. Forward rates are determined by the relationship between the spot rates (present value) of different maturities.

14. I nvestment A nalysis II 14 Investment Analysis II - © 2012 Houman Younessi Forward Rates Consider for example, the one year and two year present values of an asset. As the present value has to be the same, they cancel out so: So the forward rate a year from now for this asset maturing two year from now is

15. I nvestment A nalysis II 15 Investment Analysis II - © 2012 Houman Younessi Forward Rates In General: The forward rate f(t,u) for period (t,u) with t and u expressed in years is determined from the annualized forward rate formula with annual compounding: For compounding of m periods per year, the formula for the annualized forward rate f(i,j) between the ith and the jth period is: The formula for annualized forward rate with continuous compounding is:

16. I nvestment A nalysis II 16 Investment Analysis II - © 2012 Houman Younessi Forward Contract: An agreement between two parties in which the buyer agrees to buy from the seller an underlying asset at a future date at a price established at the start. (e.g. when you order a pizza for delivery or when you buy a 1000 tons of wheat in January for delivery in September). Rate-based Contracts Futures Contract: Similar in essence to a forward contracts with some important differences: A) a futures contract is not a private contract. Instead it is a public standardized transaction sold an exchange (called a futures exchange); B) a forward contract is subject to default, being essentially a private arrangement, whereas a futures contract is guaranteed by the selling exchange; C) the position in the futures contract is updated periodically (typically daily), that is it is marked to market. Swap Contract: A swap is a contract by which two parties agree to exchange two cash flows with different features.

17. I nvestment A nalysis II 17 Investment Analysis II - © 2012 Houman Younessi Rate-based Contracts and Derivatives Please note that: A)Rate-based contracts are not derivatives in and of themselves. However, due to market and economic uncertainty impacting prices and rates, they may be traded at variable prices. When they are thus traded, only, are they derivatives. B)Futures are not securities in the strict sense and cannot therefore be sold to a third party before maturity. However, because futures are marked to market, for all economic intents and purposes, they become equivalent to securities.

18. I nvestment A nalysis II 18 Investment Analysis II - © 2012 Houman Younessi Financial Assets Securities Cash Contract EquitiesFixed Income Bonds Money-Market Accounts Savings Accounts … Fixed income: instruments that pay a fixed amount of money to their owners Fixed-Income Investments

19. I nvestment A nalysis II 19 Investment Analysis II - © 2012 Houman Younessi Fixed-Income Investments Savings and Money Market Accounts Although there may be differences from a legal and banking standpoint, as investments, all accounts of this type follow the same rule. They are all different types of cash flow series. A cash flow series determines the present (or future) value of a series (may be only one) of cash flows. A cash flow is a deposit or withdrawal of a specific amount of funds made at a specific time. Cash flows are subject to interest/inflation rates that may be constant or variable. Examples of such deposit oriented instruments are: commercial bank time deposits, personal savings accounts, commercial papers, certificates of deposits, bankers’ acceptances, Eurodollar CDs, repurchase agreements

20. I nvestment A nalysis II 20 Investment Analysis II - © 2012 Houman Younessi V 0 Current deposit figure V 1 Deposit next term V 0 Value of current deposit now Value of next term’s deposit now Value now of deposit made in two term’s time, Value now of deposits made in two consecutive terms V 2 Deposit the term after next So, Represents the present value of all deposits made over n terms Fixed-Income Investments Savings and Money Market Accounts Example: Regular deposits made at constant interest over n terms

21. I nvestment A nalysis II 21 Investment Analysis II - © 2012 Houman Younessi Fixed-Income Investments Savings and Money Market Accounts In general, we can write: Of course if the deposit values are a constant fixed figure, we can write Or if the yield rate remains constant, we have:

22. I nvestment A nalysis II 22 Investment Analysis II - © 2012 Houman Younessi Fixed-Income Investments Bonds and Bond Pricing Bond: A security purchased by its owner called the creditor; at an agreed amount called the bond price that gives the owner the right to a fixed, pre-determined payment called the nominal value (or face value, par value or principal); at a future, predetermined date called the maturity date (or simply maturity). There may also be periodically paid sums, called coupon payment in the interim. Examples of issuers of bonds include the US Federal, State and municipal governments, government sponsored agencies, foreign governments, and corporations

23. I nvestment A nalysis II 23 Investment Analysis II - © 2012 Houman Younessi Fixed-Income Investments Bonds and Bond Pricing Examples of government issued bonds in the US include: US Treasury bills (<1 yr) US Treasury notes (1<N<10 yr) US Treasury bonds (>10 yr) US savings bonds (nonmarketable) Housing bonds (e.g. Freddie Mac) Education Bonds (e.g. Sallie Mae) Municipal bonds: General obligation bonds Revenue bonds Industrial development bonds Tax anticipation notes Examples of Corporate bonds issued in the US include: Mortgage bonds Collateral trust bonds Equipment obligations Debentures Asset-backed securities Guaranteed bonds Bond-like instruments: Income bonds Convertible bonds Callable bonds Putable bonds

24. I nvestment A nalysis II 24 Investment Analysis II - © 2012 Houman Younessi Fixed-Income Investments Bonds and Bond Pricing Bond Pricing We learned that the present value of a series of cash flows may be determined by: Now, let P be the current price of a bond with a remaining life of n years, and promising cash flows to the investor of C 1 in year 1, C 2 in year 2 and onwards. The yield to maturity (actually the promised yield to maturity) of the bond is the value y that solves the value equation below. This concept is also called internal rate of return (IRR)

25. I nvestment A nalysis II 25 Investment Analysis II - © 2012 Houman Younessi Fixed-Income Investments Bonds and Bond Pricing Bond Pricing Example: Consider a bond that is currently selling for $900 and has a remaining life of three years. Assume also that the bond makes annual coupon payments of $60 per year and has a par value of $1000. The market is paying 9.00% for investments of similar risk. Would you buy this bond? Answer: This bonds is yielding over a full percentage above the market and is a good investment.

26. I nvestment A nalysis II 26 Investment Analysis II - © 2012 Houman Younessi Fixed-Income Investments Bonds and Bond Pricing Bond Pricing part 2: What is the intrinsic value of this bond? part 3: What is the net present value (NPV) of this bond? part 4: What is the coupon rate of this bond. The coupon rate is the ratio of the annual coupon payment and the par value. In this case:

27. I nvestment A nalysis II 27 Investment Analysis II - © 2012 Houman Younessi Fixed-Income Investments Bonds and Bond Pricing Bond Pricing Part 5: What is the current yield of this bond. The current yield is the ratio of the annual coupon payment and the current price. In this case:

28. I nvestment A nalysis II 28 Investment Analysis II - © 2012 Houman Younessi Fixed-Income Investments Bonds and Bond Pricing Bond Pricing Bonds have six primary attributes important in their valuation: Time to maturity Coupon rate Call and put provisions Tax status Liquidity Risk (likelihood of default) The overall interplay of these factors leading to a specific yield-to-maturity is called YIELD STRUCTURE. The set of yields of bonds of different maturities otherwise identical bonds is called TERM STRUCTURE The set of yields of bonds of different default risk is called RISK STRUCTURE

29. I nvestment A nalysis II 29 Investment Analysis II - © 2012 Houman Younessi Fixed-Income Investments Bonds and Bond Pricing Bond Pricing Call and Put Many bonds have a call provision. A call provision allows the issuer to redeem the bond before maturity. Generally, when market conditions move so that the issuer finds access to cheaper source of financing, they often call in their bonds. Such provision of course creates a risk to the creditor. Such risk, like all other risk must therefore be compensated for. Such bonds must therefore have a higher yield to maturity compared to non callable bonds of otherwise identical structure. The value declared at which the issuer would call the bond back in is called the call price and is generally a bit higher than the par value. That difference between the par value and the call price is called the call premium

30. I nvestment A nalysis II 30 Investment Analysis II - © 2012 Houman Younessi Fixed-Income Investments Bonds and Bond Pricing Bond Pricing Call and Put The converse of the call provision, that is the ability for the investor to return the bonds to the issuer before maturity and receive the par value upon return is called the put provision and such bonds are called putable bonds. A putable bond provides an advantage to the investor and as such putable bonds have lower yields than non-putable bonds with otherwise identical structure.

31. I nvestment A nalysis II 31 Investment Analysis II - © 2012 Houman Younessi Fixed-Income Investments Bonds and Bond Pricing Bond Pricing Tax Provisions Investors are mostly interested in net gain. So, ceteris paribus, any yield that attracts less tax would be more attractive to the investor. Some bonds have nil or favorable tax structures. For instance municipal bonds traditionally have been tax exempt and as such have had a historical 20% to 40% lower yield to maturity compared to non-tax- exempt bond of otherwise similar structure.

32. I nvestment A nalysis II 32 Investment Analysis II - © 2012 Houman Younessi Fixed-Income Investments Bonds and Bond Pricing Bond Pricing Liquidity Liquidity refers to the speed and ease at which an investor may liquidate a position. Most bonds are bought and sold in dealer markets. Dealers charge a premium for dealing in bonds that are not very liquid. As such, the bid- ask spread (cost to acquire and to liquidate) a bond is a good measure of the liquidity of a bond.

33. I nvestment A nalysis II 33 Investment Analysis II - © 2012 Houman Younessi Fixed-Income Investments Bonds and Bond Pricing Bond Pricing Default Risk Not all bonds are created equal. Depending on the creditworthiness, business prospects and the debt management practices of the bond issuer, bonds may range in their “rating”. Standard & Poor’s and Moody’s are two firms that provide such ratings internationally. In general, there are three classes of bonds: Investment grade, Speculative grade, and junk