MAT 3788 Lecture 3, Feb

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1 The Tie Value of Money MAT 3788 Lecture 3, Feb 010 The Tie Value of Money and Interest Rates Prof. Boyan Kostadinov, City Tech of CUNY Everyone is failiar with the saying "tie is oney" and in finance there is a precise atheatical eaning attached to this saying. Naely, $1 today is worth ore than $1 toorrow or in a year's tie, for that atter. This is because having $1 now instead of later, gives you the opportunity to ake even ore oney by investing it wisely. For exaple, if you invest your $1 with a bank for one year, you expect to be copensated for not having this $1 during that tie by deanding interest to be paid to you by the bank, on a regular basis, in addition to getting back your $1 in a year's tie. Interest Rates In real life, there are any different interest rates for different users based on credit ratings and they are function of the tie period for which they apply. So, in general an interest rate can be represented by a variable r T where T represents the tie period for which this rate applies, for exaple T = 3M, 6M, 1Y, 0Y etc. This dependence of interest rates on tie is called ter structure of interest rates. Although rates vary with tie, for now we are going to assue that rates are constant and independent of tie. Let's denote an interest rate by r and keep in ind that interest rates are typically given as percentage rates per annu. There is siple and copound interest. Siple interest is based only on the initial principal aount while in the case of copound interest, you also get interest on the interest. Exaple of Siple Interest Under the siple interest rule, interest is coputed based only on the original principal aount and is proportional to the tie period the investent is held. The proportionality applies not only to whole but also fractional years. In general, if an aount A is deposited in an account at a siple interest rate of r (expressed as a decial), the total value after t years (t need not be a whole nuber) is V t = A C A$r$t = A$ 1 C r$t So, the deposit grows linearly with tie. For exaple, if A =$1000, r = 5 % and t =.5 years, then the original principal grows to 1000$ 1 C 0.05 #.5 = If a bank tells you they will give you 10% annual interest rate for your one year deposit, then this inforation is incoplete because we also need to know the copounding frequency. Exaple of Copound Interest Suppose that an initial aount of deposit A is invested for n years (n is an integer) with annual interest rate r such that interest is copounded ties per year. Then the siple interest rate for each of the copounding periods in one year is r so that the interest accrued over the first period is A$r and is

2 added to the principal A to get a total value at the end of the first period A C A$r = A$ 1 C r. Next, since interest also gets interest, this total aount is used as the new principal to copute the interest at the end of the second period and the total aount in the deposit grows to A$ 1 C r C A$ 1 C r $ r = A$ 1 C r $ 1 C r = A$ 1 C r Siilarly, at the end of the first year, that is, at the end of the th period, the total aount in the deposit grows to A$ 1 C r and since n (wh ole) years have n$ copounding periods, then after n years the initial deposit grows to a terinal value of V n : V n = A$ 1 C r $n When the nuber of years is not a whole nuber but a decial nuber, say t = 1.6 years, we can ake the following approxiation for the terinal value V t : V t z A$ 1 C r $t, t in years is decial We can illustrate this approxiation by considering the case t = 1.6 years and = (quarterly copounding). Since we copound interest every 3 onths there is a whole nuber of periods up to 1.5 years, which is 6 periods and we can use the exact forula for a whole nuber of periods to get 6 V 1.5 = A$ 1 C r the total aount after 1.5 years. However, there is 0.1 year left and for this tie we accrue interest under the siple rule, proportionally to tie, naely the total aount after 1.6 years is V 1.5 C V 1.5 # r # 0.1 = V C 0. r z V C r 0. = A$ 1 C r 6. Notice that $t = # 1.6 = 6. and the approxiation that we used coes fro 1 C x t z 1 C t x for sall x, where in our case t = 0. and x = r is sall indeed. Noinal vs. Effective Interest Rate The effect of copounding on yearly growth is highlighted by stating an effective annual interest rate, which is the equivalent annual interest rate that would produce the sae result after 1 year without copounding. For exaple, an annual rate of 8% copounded quarterly will produce a growth factor for one year 1 C 0.08 = ; by definition, the siple effective rate r eff should produce the sae result: 1 C r eff = 1.08 thus r eff = 8. %. The copounded rate of 8% is called the noinal rate.

3 Continuous Copounding As the copounding frequency tends to infinity, the liit is known as continuous copounding. When we have an integer nuber of years, that is n ;, we can use the faous liit fro Calculus: li /N 1 C r $n = e r $n, where e = is the irrational (infinitely any non-repeating decial digits) Euler nuber. For an arbitrary, fractional nuber of years t, the approxiate forula that we derived above becoes an exact forula as the nuber of periods tends to infinity. The reason is that we can approxiate better and better the given fractional tie by a whole nuber of periods. Therefore, even in the ore general case of a fractional nuber of years t, we get the exact forula for the growth factor with continuous copounding: li /N 1 C r $t = e r $t Therefore, an aount A invested for t years t = at the continuously copounded rate r grows to V t = A$e r $t. For ost practical purposes, continuous copounding can be though of as being equivalent to daily copounding, as one can see fro the table below. Copounding oney at a continuously copounded rate r for T years aounts to ultiplying by a growth factor of e r $T. Discounting oney at a continuously copounded rate r for T years involves ultiplying by a discount factor of e Kr $T. We can always convert a discretely copounded rate R into the equivalent continuously copounded rate r and vice versa; based on 1 year tie and $1 deposit: e r = 1 C R / r = $ln 1 C R and R = e K 1 r Fro the first equation above, we get the first forula for r by taking the natural logarith on both sides and the second forula we get by taking both sides to the power of 1 or siply taking the th root. Reeber that y = e x is equivalent to x = ln y. Exaple An interest rate is quoted as 10% per annu with sei-annual copounding. Find the equivalent rate with continuous copounding. Let r be the wanted rate with continuous copounding. On the one hand, $1 will grow to 1 C 0.1 after t = 1 year, under the discrete (sei-annual) copounding and on the other hand this $1 under the equivalent continuous copounding at rate r should grow to the sae final aount: 1 C 0.1 = e r $1, r = $ln 1 C 0.1 = r = or r = 9.76 %

4 Here is a table showing the effect of copounding frequency for 10% annual rate on the value of $10,000 deposit at the end of 1 year n = 1 Copounding frequency Value of $10,000 at the end of 1 year Annually = 1 1 = Seiannually = = Quarterly = = Monthly = = Weekly = = Daily = = Continuous Copounding 10000$e 0.1 = Reading Assignents: Read Chapter, sections.1 and. fro Options, Futures and Other Derivatives 6th ed. by Hull. nd Part of Hoework 1, Due Thursday February 11, 010 Tie Value of Money 1. An investor receives $1,100 in one year in return for an investent of $1,000 now. Calculate the rate of return per annu with a) Annual copounding b) Sei-annual copounding c) Monthly copounding and d) Continuous copounding.. An interest rate is quoted as 5% per annu with sei-annual copounding. What is the equivalent rate with a) annual copounding b) onthly copounding and c) continuous copounding. 3. (Loan Calculation) Suppose you have borrowed $10,000 fro a bank. The ters of the loan are that the annual interest is 1% copounded onthly. You are to ake equal onthly payents as to repay (aortize) this loan over 5 years. How uch are your onthly payents? How uch is the total interest you you will pay for this loan? Hint: Either work period by period to find the fixed onthly payent that would give a net value of zero after 60 onths or think about discounting the periodic fixed onthly payents so that the total present value is equal to the loan of $10,000.

5 The Matheatics of Bond Pricing In finance, a bond is a debt security, such that the issuer of the bond owes the holder of the bond a debt and is obliged to pay interest (called the coupon) and to repay the principal (called the face value) at aturity. A bond is thus a foral contract to repay borrowed oney with interest at fixed tie intervals, specified in the contract. Thus a bond is like a loan: the issuer of the bond is the borrower, the holder of the bond is the lender, and the coupon is the interest. Bonds provide the issuer with external funds to finance long-ter investents or in the case of governent bonds, to finance current governent expenditure. Bonds and stocks are both securities, but the ajor difference between the two is that stockholders have an equity stake in the copany (i.e., they own a piece of the copany), whereas bondholders have a creditor stake in the copany (i.e., they are lenders to the copany). Another difference is that bonds usually have a defined ter, or aturity, after which the bond is redeeed, whereas stocks ay be outstanding indefinitely. Debt securities with a ter of less than one year are generally designated oney arket instruents rather than bonds. Certificates of deposit (CDs) offered by coercial banks are considered to be oney arket instruents and not bonds. Most bonds have a ter of up to thirty years. In the arket for U.S. Treasury securities, there are three groups of bond aturities: Short ter (Treasury Bills): aturities up to one year; Mediu ter (Treasury Notes): aturities between one and ten years; Long ter (Treasury Bonds): aturities greater than ten years. The bonds issued by the US governent are considered risk-free since the probability that the US governent will default is very low, partly because the federal governent can always print ore oney if required. Corporate bonds are considered risky and their probabilities of default (that is the inability of the corporations to eet their financial obligations) are captured by the credit ratings provided by credit rating agencies such as Standard & Poor's, Moody's, etc. Bonds have a long history: Zero-Coupon Bonds A zero-coupon bond (also called a discount bond) is a bond bought at a price lower than its face value, with the face value repaid at the tie of aturity. It does not ake periodic interest payents, the so-called "coupons," hence the ter zero-coupon bond. Exaples of zero-coupon bonds include U.S. Treasury bills and U.S. savings bonds. Zero Rates The n-year zero-coupon rate is the rate of interest earned on an investent that starts today and lasts for n years. There is only one payent at the end of n years. There are no interediate payents. The n-year zero coupon rate is also referred to as the n-year spot rate, the n-year zero rate or siply the n-year zero. Treasury zero rates are deterined by coupon-bearing Treasury bonds by the bootstrap ethod (hoework #3 is a siple exaple of this approach). Exaple

6 Let the 10-year zero rate with continuous copounding be quoted as 7% per annu. This eans that $1000, if invested for 10 years grows to 1000$e 0.07$10 = Treasury zero rates are iplied fro arket prices of Treasury coupon bonds. The arket prices of Treasury bonds are deterined at auctions. Exaple Consider a zero-coupon bond with a face value of $100 and aturity T years fro today (tie 0). If the T- year zero rate with continuous copounding is known to be r T then the price P 0 of this bond at tie 0 is Kr $T P 0 = 100$e T We say that we siply discount back to tie 0, the tie T cash flow of $100 using the T-zero rate and Kr T $T e we call the discount factor associated with the T-spot rate. We'll always use continuous copounding fro now on. Keep in ind that the tie 0 value of this bond, that is P 0 should grow by a r $T r $T Kr $T r $T factor of e T after tie T to P 0 $e T = 100$e T $ e T = 100 as expected. Discounting P 0 = 100$e Kr T $T $100 Bond Features 0 T 0 Bond's principal also known as face value or par value is the aount on which interest is paid periodically in the for of coupons and the principal is received by the bond holder at the end of the bond's life. Most bonds provide coupons on a sei-annual basis and the coupon is given as a fixed percentage rate of the principal, called the coupon rate. Exaple of Bond Pricing Consider a -year Treasury bond with a principal of $100 paying coupons sei-annually with a coupon rate of 6% per annu. To calculate the present value of all the cash flows given in the diagra below, we need the spot rates for all payent ties. The theoretical price of a bond can then be calculated as the present value of all the cash flows that will be received by the owner of the bond. To do this correctly, we have to use the zero rates fro the table above to discount the cash flows for the different payent ties. For exaple, the first coupon will be paid in 6 onths, so we need to use the 0.5Y-zero rate of 5% for discounting. But what is the exact coupon? The coupons are calculated based on the principal and using the siple interest rule. Since the coupon rate is 6% per annu, for 6 onths the coupon rate is then 6%(0.5) = 3%. Reeber that the siple rule gives us linear dependence on tie and if the annual rate is R, then the siple rate for tie t years is R$t, in our case t = 0.5 years. Therefore, the coupon is $100$3 % = $3 and is paid twice a year. Also, reeber that the principal is received by the bond holder at the end of the

7 bond's life and so it has to be included in calculating the present value of the bond cash flows. So, at tie t = years, we have a coupon of $3 plus $100 principal and the total aount $103 has to be discounted at the Y-zero rate of 6.8% (see the diagra below). The table below provides these spot rates: Maturity (years) Treasury Zero rate (%) cont. copounded Therefore, the theoretical price of the bond, as the present value of all the cash flows, is $ Note that this price is very close but less than the face value of $100. We say it's priced at discount: V = 3$e K0.05$0.5 C 3$e K0.058$1 C 3$e K0.06$1.5 C 103$e K0.068$ = we discount all cash flows back to tie 0 to find the price Cash in-flow Price= $98.39 $3 $3 $3 $103 Cash out-flow 0 t 1 t t 3 t 1.0y Reading Assignents: Read Chapter, sections.3 and. fro Options, Futures and Other Derivatives 6th ed. by Hull. 3nd Part of Hoework 1, Due Thursday February 11, Suppose that 6-onth, 1-onth, 18-onth, -onth and 30-onth zero rates are %,.%,.%,.6%,.8% per annu with continuous copounding respectively. Estiate the price of a bond with a face value of 100 that will ature in 30 onths and pays a coupon of % per annu seiannually.. A 10-year, 8% coupon bond currently sells for $90. A 10-year, % coupon bond currently sells for $80. What is the 10-year zero rate? Hint: Consider taking a long position in two of the % coupon bonds and a short position in one of the 8% coupon bonds. 3. The prices of 6-onth and 1-year Treasury bills are $9.0 and $89.0. A 1.5-year bond that will pay coupons of $ every 6 onths currently sells for $9.8. A -year bond that will pay coupons of $5 every 6 onths currently sells for $97.1. Calculate the 6-onth, 1-year, 1.5-year and -year zero rates.