Investment Science. Part I: Deterministic Cash Flow Streams. Dr. Xiaosong DING

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1 Investment Science Part I: Deterministic Cash Flow Streams Dr. Xiaosong DING Department of Management Science and Engineering International Business School Beijing Foreign Studies University , Beijing, People s Republic of China Dr. DING (xiaosong.ding@hotmail.com) Investment Science 1 / 174

2 Outline 1 Principal and Interest 2 Present Value 3 Present and Future Value of Streams 4 Internal Rate of Return 5 Evaluation Criteria 6 Applications and Extensions Dr. DING (xiaosong.ding@hotmail.com) Investment Science 2 / 174

3 Outline Principal and Interest 1 Principal and Interest 2 Present Value 3 Present and Future Value of Streams 4 Internal Rate of Return 5 Evaluation Criteria 6 Applications and Extensions Dr. DING (xiaosong.ding@hotmail.com) Investment Science 3 / 174

4 Principal and Interest Example If you invest $1.00 in a bank account that pays 8% interest per year, then at the end of 1 year you will have in your account the principal (your original amount) at $1.00 plus interest at $0.08 for a total of $1.08. What if you invest a larger amount, say A dollars, in the bank? What if the interest rate is r? Dr. DING (xiaosong.ding@hotmail.com) Investment Science 4 / 174

5 Principal and Interest Simple Interest Under a simple interest rule, money invested for a period different from 1 year accumulates interest proportional to the total time of the investment. If an amount A is left in an account at simple interest r, the total value after n years is V = (1 + rn)a. If the proportional rule holds for fractional years, then after any time t (measured in years), the account value is V = (1 + rt)a. The account grows linearly with time. The account value at any time is just the sum of the principal and the accumulated interest, which is proportional to time. Dr. DING (xiaosong.ding@hotmail.com) Investment Science 5 / 174

6 Principal and Interest Compound Interest Example Consider an account that pays interest at a rate of r per year. If interest is compounded yearly, then after 1 year, the first year s interest is added to the original principal to define a larger principal base for the second year. What is the account value after n years? The account earns interest on interest! Under yearly compounding, after n years, such an account will grow to V = (1 + r) n A. This is the analytic expression for the account growth under compound interest. This expression is said to exhibit geometric growth because of its nth-power form. Dr. DING (xiaosong.ding@hotmail.com) Investment Science 6 / 174

7 Principal and Interest Rule (The seven-ten rule) Money invested at 7% per year doubles in approximately 10 years. Also, money invested at 10% per year doubles in approximately 7 years. Dr. DING (xiaosong.ding@hotmail.com) Investment Science 7 / 174

8 Principal and Interest Exercise (The 72 rule) The number of years n required for an investment at interest rate r to double in value must satisfy By Taylor series, we have (1 + r) n = 2. ln (1 + x) = x x x x n ( 1)n 1 n +... ( 1 < x 1). Using ln 2 = 0.69 and the approximation ln(1 + r) r valid for small r, show that n 69/i, where i is the interest rate percentage, i.e., i = 100r. Using the better approximation ln(1 + r) r r 2 /2, show that for r 0.08, there holds n 72/i. Dr. DING (xiaosong.ding@hotmail.com) Investment Science 8 / 174

9 Principal and Interest Compounding at Various Intervals Example (Quarterly compounding) Quarterly compounding at an interest rate of r per year means that an interest rate of r/4 is applied every quarter. Hence, money left in the bank for 1 quarter will grow by a factor of 1 + (r/4) during that quarter. If the money is left in for another quarter, then that new amount will grow by another factor of 1 + (r/4). What is the account value after 1 year? ( 1 + r 4) 4 > 1 + r, r > 0. Right or not, why? What is the meaning? Dr. DING (xiaosong.ding@hotmail.com) Investment Science 9 / 174

10 Principal and Interest Definition Effective interest rate is the equivalent yearly interest rate that would produce the same result after 1 year without compounding. Example An annual rate of 8% compounded quarterly will produce an increase of (1.02) ; hence the effective interest rate is 8.24%. The basic yearly rate (8% in this example) is termed the nominal rate. Compounding can be carried out with any frequency. The general method is that a year is divided into a fixed number of equally spaced periods, say, m periods. The effective interest rate is the number r that satisfies ( 1 + r = 1 + r ) m. m Dr. DING (xiaosong.ding@hotmail.com) Investment Science 10 / 174

11 Principal and Interest Continuous Compounding Imagine dividing the year into infinitely small periods and use ( 1 + r ) m = e r, m lim m where e = is the base of the natural logarithm. The effective interest rate is the number r that satisfies 1 + r = e r. Dr. DING (xiaosong.ding@hotmail.com) Investment Science 11 / 174

12 Principal and Interest To calculate how much an account will have grown after any arbitrary length of time t k/m (measured in years), we use ( 1 + r ) k ( = 1 + r ) mt [( lim 1 + r ) m ] t = e rt. m m m m Continuous compounding leads to the familiar exponential growth curve. Dr. DING (xiaosong.ding@hotmail.com) Investment Science 12 / 174

13 Principal and Interest Debt and Money Markets Exactly the same thing happens to debt. If I borrow money from the bank at an interest rate r and make no payments to the bank, then my debt increases according to the same formulas. Specifically, if my debt is compounded monthly, then after k months my debt will have grown by a factor of [1 + (r/12)] k. Dr. DING (xiaosong.ding@hotmail.com) Investment Science 13 / 174

14 Outline Present Value 1 Principal and Interest 2 Present Value 3 Present and Future Value of Streams 4 Internal Rate of Return 5 Evaluation Criteria 6 Applications and Extensions Dr. DING (xiaosong.ding@hotmail.com) Investment Science 14 / 174

15 Present Value Example Consider two situations. 1 You will receive $110 in 1 year. 2 You receive $100 now and deposit it in a bank account for 1 year at 10% interest. Clearly, these situations are identical after 1 year! We say that the $110 to be received in 1 year has a present value of $100. In general, $1 to be received a year in the future has a present value of $1/(1 + r), where r is the interest rate. The process of evaluating future obligations as an equivalent present value is alternatively referred to as discounting. The factor by which the future value must be discounted is called the discount factor. The 1-year discount factor is d 1 = r. Dr. DING (xiaosong.ding@hotmail.com) Investment Science 15 / 174

16 Present Value The formula for present value depends on the interest rate that is available from a bank or other source. Example Suppose that the annual interest rate r is compounded at the end of each of m equal periods each year; and suppose that a cash payment of amount A will be received at the end of the kth period. Then the appropriate discount factor is d k = [ ( 1 + r ) ] k 1, m and thus the present value of a payment of A to be received k periods in the future is PV = d k A. Dr. DING (xiaosong.ding@hotmail.com) Investment Science 16 / 174

17 Outline Present and Future Value of Streams 1 Principal and Interest 2 Present Value 3 Present and Future Value of Streams 4 Internal Rate of Return 5 Evaluation Criteria 6 Applications and Extensions Dr. DING (xiaosong.ding@hotmail.com) Investment Science 17 / 174

18 Present and Future Value of Streams The Ideal Bank An ideal bank applies the same rate of interest to both deposit, and loans, and it has no service charges or transactions fees. Example The interest rate applies equally to any size of principal. Separate transactions in an account are completely additive in their effect on future balances. Interest rates for all transactions may not be identical. 1 A 2-year CD might offer a higher rate than a 1-year CD. 2 A 2-year CD must offer the same rate as a loan that is payable in 2 years. If an ideal bank has an interest rate that is independent of the length of time for which it applies, and that interest is compounded according to normal rules, it is said to be a constant ideal bank. Dr. DING (xiaosong.ding@hotmail.com) Investment Science 18 / 174

19 Present and Future Value of Streams Future Value Theorem (Future value of a stream) Given a cash flow (x 0, x 1,..., x n ), and interest rate r each period, the future value of a stream is Example FV = n x i (1 + r) n i = x 0 (1 + r) n + x 1 (1 + r) n x n. i=0 Consider the cash flow stream ( 2, 1, 1, 1) when the periods are years and the interest rate is 10%. The future value is Dr. DING (xiaosong.ding@hotmail.com) Investment Science 19 / 174

20 Present and Future Value of Streams Present Value Theorem (Present value of a stream) Given a cash flow (x 0, x 1,..., x n ), and interest rate r each period, the present value of a stream is PV = n i=0 x i (1 + r) i = x 0 + x r + x 2 (1 + r) x n (1 + r) n. Example Consider the cash flow stream ( 2, 1, 1, 1). Using an interest rate of 10%, the present value is The relationship between PV and FV is PV = FV (1 + r) n. Dr. DING (xiaosong.ding@hotmail.com) Investment Science 20 / 174

21 Present and Future Value of Streams Frequent and Continuous Compounding Suppose that r is the nominal annual interest rate and interest is compounded at m equally spaced periods per year. Furthermore, suppose that cash flows occur initially and at the end of each period for a total of n periods, forming a stream (x 0, x 1,..., x n ). Then PV = n k=0 x k [1 + (r/m)] k. Suppose now that the nominal interest rate r is compounded continuously and cash flows occur at times t 0, t 1,..., t n. Denote by x(t k ) the cash flow at time t k. Then n PV = x(t k )e rt k. k=0 Dr. DING (xiaosong.ding@hotmail.com) Investment Science 21 / 174

22 Present and Future Value of Streams Present Value and an Ideal Bank In general, if an ideal bank can transform the stream (x 0, x 1,..., x n ) into the stream (y 0, y 1,..., y n ), it can also transform in the reverse direction. Definition Two streams that can be transformed into each other are said to be equivalent streams. Theorem (Main theorem on present value) The cash flow streams x = (x 0, x 1,..., x n ) and y = (y 0, y 1,..., y n ) are equivalent for a constant ideal bank with interest rate r if and only if the present values of two streams, evaluated at the bank s interest rate, are equal. Proof. Since x (v x, 0,..., 0) and y (v y, 0,..., 0), the result is obvious. Dr. DING (xiaosong.ding@hotmail.com) Investment Science 22 / 174

23 Outline Internal Rate of Return 1 Principal and Interest 2 Present Value 3 Present and Future Value of Streams 4 Internal Rate of Return 5 Evaluation Criteria 6 Applications and Extensions Dr. DING (xiaosong.ding@hotmail.com) Investment Science 23 / 174

24 Internal Rate of Return Definition Let (x 0, x 1,..., x n ) be a cash flow stream. Then the internal rate of return of this stream is a number r satisfying the equation 0 = n i=0 x i (1 + r) i = x 0 + x r + x 2 (1 + r) x n (1 + r) n. Equivalently, it is a number r satisfying 1/(1 + r) = c, where c satisfies the polynomial equation 0 = n x i c i = x 0 + x 1 c + x 2 c x n c n. i=0 Example Consider again the cash flow sequence ( 2, 1, 1, 1). Its internal rate of return is r = 0.23 with c = Dr. DING (xiaosong.ding@hotmail.com) Investment Science 24 / 174

25 Internal Rate of Return Exercise (Newton s method) Suppose that we define f (λ) = a 0 + a 1 λ + a 2 λ a n λ n, where all a i s are positive and n > 1. Here is an iterative technique that generates a sequence λ 0, λ 1, λ 2,..., λ k,... of estimates that converges to the root λ > 0, solving f (λ) = 0. Start with any λ 0 > 0 close to the solution. Assuming λ k has been calculated, we use f (λ k ) = a 1 + 2a 2 λ k + 3a 3 λ 2 k na nλ n 1 k, λ k+1 = λ k f (λ k) f (λ k ). Try the above procedure on the function f (λ) = 1 + λ + λ 2 with the starting points λ 0 = ±1, ±10,... Dr. DING (xiaosong.ding@hotmail.com) Investment Science 25 / 174

26 Internal Rate of Return Theorem (Main theorem of internal rate of return) Suppose the cash flow stream (x 0, x 1,..., x n ) has x 0 < 0 and x k 0 for all k, k = 1, 2,..., n, with at least one term being strictly positive. Then there is a unique root to the equation 0 = n x i c i = x 0 + x 1 c + x 2 c x n c n. i=0 Furthermore, if n k=0 x k > 0 (meaning that the total amount returned exceeds the initial investment), then the corresponding IRR is positive. Dr. DING (xiaosong.ding@hotmail.com) Investment Science 26 / 174

27 Outline Evaluation Criteria 1 Principal and Interest 2 Present Value 3 Present and Future Value of Streams 4 Internal Rate of Return 5 Evaluation Criteria 6 Applications and Extensions Dr. DING (xiaosong.ding@hotmail.com) Investment Science 27 / 174

28 Evaluation Criteria Net Present Value Definition Net present value is the present value of the benefits minus the present value of the costs. Example Suppose that you have the opportunity to plant trees that later can be sold for lumber. This project requires an initial outlay of money to purchase and plant the seedlings. No other cash flow occurs until the trees are harvested. However, you have a choice as to when to harvest: after 1 year or after 2 years. If you harvest after 1 year, you get your return quickly; but if you wait an additional year, the trees will have additional growth and the revenue generated from the sale of the trees will be greater. (r = 10%) 1 ( 1, 2), cut early, NPV = ( 1, 0, 3), cut later, NPV = Dr. DING (xiaosong.ding@hotmail.com) Investment Science 28 / 174

29 Evaluation Criteria Internal Rate of Return Provided that it is greater than the prevailing interest rate, the higher the internal rate of return, the more desirable the investment. Example (contd.) Let us use the internal rate of return method to evaluate the two tree harvesting proposals. The equations for the internal rate of return in the two cases are c = 0 c = 1 2 = r r = c 2 = 0 c = 3 3 = r r = 0.7. In other words, for cut early, the internal rate of return is 100%, whereas for cut late, it is about 70%. Hence under the internal rate of return criterion, the best alternative is to cut early. Dr. DING (xiaosong.ding@hotmail.com) Investment Science 29 / 174

30 Evaluation Criteria Discussion of the Criteria There is considerable debate as to which of the two criteria, NPV or IRR, is the most appropriate for investment evaluation. Both have attractive features, and both have limitations. Net present value is simplest to compute; does not have the ambiguity associated with the several possible roots of the internal rate of return equation; and can be broken into component pieces, unlike internal rate of return. However, internal rate of return has the advantage that it depends only on the properties of the cash flow stream, and not on the prevailing interest rate (which in practice may not be easily defined). Dr. DING Investment Science 30 / 174

31 Which criterion to choose? Evaluation Criteria 1 In the situation where the proceeds of the investment can be repeatedly invested in the same type of project but scaled in size, it makes sense to select the project with the highest internal rate of return in order to get the greatest growth of capital. 2 In one-time opportunity, the net present value method is the appropriate criterion, since it compares the investment with what could be obtained through normal channels (which offer the prevailing rate of interest). Many other factors, e.g., risk-free interest rate, rates for borrowing and lending, cost of capital, and rate of return, can influence NPV analysis. Definition In business decisions, it is common to use a figure called the cost of capital as the baseline rate. This figure is the rate of return that the company must offer to potential investors in the company; that is, it is the cost the company must pay to get additional funds. Or sometimes it is taken to be the rate of return expected on alternative desirable projects. Dr. DING (xiaosong.ding@hotmail.com) Investment Science 31 / 174

32 Evaluation Criteria However, some of these cost of capital figures are derived from uncertain cash flow streams and are not really appropriate measures of a risk-free interest rate. For NPV analysis, it is best to use rates that represent true interest rates, since we assume that the cash flows are certain. Another factor to consider is that NPV by itself does not reveal much about the rate of return. Example (NPV and rate of return) Two alternative investments might each have a net present value of $100, but one might require an investment of $100, whereas the other requires $1,000,000. Clearly these two alternatives should be viewed differently. Dr. DING (xiaosong.ding@hotmail.com) Investment Science 32 / 174

33 Outline Applications and Extensions 1 Principal and Interest 2 Present Value 3 Present and Future Value of Streams 4 Internal Rate of Return 5 Evaluation Criteria 6 Applications and Extensions Dr. DING (xiaosong.ding@hotmail.com) Investment Science 33 / 174

34 Net Flows Applications and Extensions Example (SimpIico gold mine) The Simplico gold mine has a great deal of remaining gold deposits, and you are part of a team that is considering leasing the mine from its owners for a period of 10 years. Gold can be extracted from this mine at a rate of up to 10,000 ounces per year at a cost of $200 per ounce. This cost is the total operating cost of mining and refining, exclusive of the cost of the lease. Currently the market price of gold is $400 per ounce. Assume that the price of gold, the operating cost, and the interest rate r = 10% remain constant over the 10-year period. What is the present value of the lease? PV = 10 k= ($400 $200) 1.1 k = $12.29M. Dr. DING (xiaosong.ding@hotmail.com) Investment Science 34 / 174

35 Applications and Extensions Cycle Problems Example (Automobile purchase) You are contemplating the purchase of an automobile and have narrowed the field down to two choices. Car A costs $20,000, is expected to have a low maintenance cost of $1,000 per year (payable at the beginning of each year after the first year), but has a useful mileage life that for you translates into 4 years. Car B costs $30,000 and has an expected maintenance cost of $2,000 per year (after the first year) and a useful life of 6 years. Neither car has a salvage value and r = 10%. Which car should you buy? Assume that similar alternatives will be available in the future, and assume a planning period of 12 years, corresponding to three cycles of car A and two of car B. PV A =? PV A3 = PV A (? ), PV B =? PV B2 = PV B (? ). Dr. DING (xiaosong.ding@hotmail.com) Investment Science 35 / 174

36 Applications and Extensions Example (Machine replacement) A specialized machine essential for a company s operations costs $10,000 and has operating costs of $2,000 the first year. The operating cost increases by $1,000 each year thereafter. Assume that these operating costs occur at the end of each year, the machine has no salvage value, and r = 10%. How long should the machine be kept until it is replaced by a new identical machine? Suppose that the machine is replaced every year. Then we have PV = PV 1.1, since after the first machine is replaced, the stream from that point looks identical to the original one, except that this continuing stream starts 1 year later and hence must be discounted by the effect of 1 year s interest. Dr. DING (xiaosong.ding@hotmail.com) Investment Science 36 / 174

37 Applications and Extensions Example (contd.) We may do the same thing assuming 2-year replacement, then 3 years, and so forth. The general approach is based on the following equation ( ) 1 k PV total = PV 1cycle + PV total, 1.1 where k is the length of the basic cycle. Dr. DING (xiaosong.ding@hotmail.com) Investment Science 37 / 174

38 Taxes Applications and Extensions Example (Depreciation) Suppose a firm purchases a machine for $10,000. This machine has a useful life of 4 years and its use generates a cash flow of $3,000 each year. The machine has a salvage value of $2,000 at the end of 4 years. The government does not allow the full cost of the machine to be reported as an expense the first year, but instead requires that the cost of the machine be depreciated over its useful life. There are several depreciation methods, each applicable under various circumstances, but for simplicity we shall assume the straight-line method. In this method a fixed portion of the cost is reported as depreciation each year. Hence corresponding to a 4-year life, one-fourth of the cost (minus the estimated salvage value) is reported as an expense deductible from revenue each year. Dr. DING (xiaosong.ding@hotmail.com) Investment Science 38 / 174

39 Applications and Extensions Example (contd.) If we assume a combined federal and state tax rate of 43%, we obtain the cash flows, before and after tax, shown in the following table. The salvage value is not taxed (since it was not depreciated). The present values for the two cash flows with r = 10% are also shown. Tax rules convert an otherwise profitable operation into an unprofitable one! Dr. DING (xiaosong.ding@hotmail.com) Investment Science 39 / 174

40 Inflation Applications and Extensions Definition Inflation rate f : Prices 1 year from now will on average be equal to today s prices multiplied by 1 + f. Inflation compounds much like interest does. Constant (real) dollars is defined relative to a given reference year. These are the (hypothetical) dollars that continue to have the same purchasing power as dollars did in the reference year. Actual (nominal) dollars are what we really use in transactions. Real interest rate r 0 is the rate at which real dollars increase if left in a bank that pays the nominal rate. 1 + r 0 = 1 + r 1 + f r 0 = r f 1 + f. Dr. DING (xiaosong.ding@hotmail.com) Investment Science 40 / 174

41 Applications and Extensions Example (Inflation) Suppose that inflation is 4%, the nominal interest rate is 10%, and we have a cash flow of real (or constant) dollars as shown in the second column of the following table (it is common to estimate cash flows in constant dollars, relative to the present, because ordinary price increases can then be neglected in a simple estimation of cash flows). To determine the present value in real terms, we use the real rate of interest, r 0 = 5.77%. Dr. DING (xiaosong.ding@hotmail.com) Investment Science 41 / 174

42 Outline 7 The Market for Future Cash 8 Value Formulas 9 Bond details 10 Yield 11 Duration 12 Immunization 13 Convexity Dr. DING (xiaosong.ding@hotmail.com) Investment Science 42 / 174

43 Outline An interest rate is a price, or rent, for the most popular of all traded commodities money. The market interest rate provides a ready comparison for investment alternatives that produce cash flows arising from transactions between individuals, associated with business projects, or generated by investments in securities. Definition Financial instruments: Vast assortments of bills, notes, bonds, annuities, futures contracts, and mortgages are part of the well-developed markets for money. They are traded only as pieces of paper, or as entries in a computer database rather than real goods in the sense of having intrinsic value. If there is a well-developed market for an instrument, so that it can be traded freely and easily, then that instrument is termed a security. Fixed-income securities are financial instruments that are traded in well-developed markets and promise a fixed (i.e., definite) income to the holder over a span of time. Dr. DING (xiaosong.ding@hotmail.com) Investment Science 43 / 174

44 Outline Fixed-income securities are important to an investor because they define the market for money, and most investors participate in this market; important as additional comparison points when conducting analyses of investment opportunities that are not traded in markets, such as a firm s research projects, oil leases, and royalty fights. The only uncertainties about the promised stream from a fixed-income security were associated with whether the issuer of the security might default in which case the income would be discontinued or delayed. However, some fixed-income securities promise cash flows whose magnitudes are tied to various contingencies or fluctuating indices. payment levels on an adjustable-rate mortgage may be tied to an interest rate index; or corporate bond payments may in part be governed by a stock price. Dr. DING (xiaosong.ding@hotmail.com) Investment Science 44 / 174

45 Outline The Market for Future Cash 7 The Market for Future Cash 8 Value Formulas 9 Bond details 10 Yield 11 Duration 12 Immunization 13 Convexity Dr. DING (xiaosong.ding@hotmail.com) Investment Science 45 / 174

46 The Market for Future Cash Savings Deposits: Demand deposit pays a rate of interest that varies with market conditions. A time deposit must be maintained for a given length of time, or else a penalty for early withdrawal is assessed. A similar instrument is a certificate of deposit (CD). Money Market Instruments Definition Money market refers to the market for short-term (1 year or less) loans by corporations and financial intermediaries, including, for example, banks. Definition Commercial paper is used to describe unsecured loans (that is, loans without collateral) to corporations. Eurodollar deposits; Eurodollar CDs. A banker s acceptance. If company A sells goods to company B, company B might send a written promise to company A that it will pay for the goods within a fixed time. Some bank accepts the promise by promising to pay the bill on behalf of company B. Company A can then sell the banker s acceptance to someone else at a discount before the time has expired. Dr. DING (xiaosong.ding@hotmail.com) Investment Science 46 / 174

47 The Market for Future Cash Definition U.S. Government Securities 1 U.S. Treasury bills are issued in denominations of $10,000 or more with fixed terms to maturity of 13, 26, and 52 weeks, and are sold on a discount basis. 2 U.S. Treasury notes have maturities of 1 to 10 years and are sold in denominations as small as $1,000. The owner of such a note receives a coupon payment every 6 months until maturity. 3 U.S. Treasury bonds are issued with maturities of more than 10 years and make coupon payments. Some Treasury bonds are callable, meaning that at some scheduled coupon payment date, the Treasury can force the bond holder to redeem the bond at that time for its face (par) value. 4 U.S. Treasury strips are bonds the U.S. Treasury issue in stripped form, offering minimal risk and some tax benefits in certain states. Each interest payment and the principal payment becomes a separate zero-coupon security. Each component has its own identifying number and can be held or traded separately. Dr. DING (xiaosong.ding@hotmail.com) Investment Science 47 / 174

48 The Market for Future Cash Definition A security generates a single cash flow with no intermediate coupon payments is called a zero-coupon bond. Other Bonds 1 Municipal bonds are issued by agencies of state and local governments. Two main types are general obligation bonds and revenue bonds. 2 Corporate bonds are issued by corporations for the purpose of raising capital for operations and new ventures. A bond carries with it an indenture. Some of the features might be: 1 Callable bonds: A bond is callable if the issuer has the right to repurchase the bond at a specified price. 2 Sinking funds: Rather than incur the obligation to pay the entire face value of a bond issue at maturity, the issuer may establish a sinking fund to spread this obligation out over time. 3 Debt Subordination: To protect bond holders, limits may be set on the amount of additional borrowing by the issuer. Dr. DING (xiaosong.ding@hotmail.com) Investment Science 48 / 174

49 The Market for Future Cash Mortgages 1 The standard mortgage is structured so that equal monthly payments are made throughout its term, which contrasts to most bonds that have a final payment equal to the face value at maturity. 2 There may be modest sized periodic payments for several years followed by a final balloon payment that completes the contract. 3 Adjustable-rate mortgages adjust the effective interest rate periodically according to an interest rate index. Mortgages are not usually thought of as securities since they are written as contracts between two parties! However, mortgages are typically bundled into large packages and traded among financial institutions. These mortgage-backed securities are quite liquid. Dr. DING (xiaosong.ding@hotmail.com) Investment Science 49 / 174

50 The Market for Future Cash Annuities Definition An annuity is a contract that pays the holder (the annuitant) money periodically, according to a predetermined schedule or formula, over a period of time, e.g., pension. There are numerous variations, for example, 1 Sometimes the level of the annuity payments is tied to the earnings of a large pool of funds from which the annuity is paid. 2 Sometimes the payments vary with time. Annuities are not really securities since they are not traded! Annuities are, however, considered to be investment opportunities that are available at standardized rates. Hence from an investor s viewpoint, they serve the same role as other fixed-income instruments. Dr. DING (xiaosong.ding@hotmail.com) Investment Science 50 / 174

51 Outline Value Formulas 7 The Market for Future Cash 8 Value Formulas 9 Bond details 10 Yield 11 Duration 12 Immunization 13 Convexity Dr. DING (xiaosong.ding@hotmail.com) Investment Science 51 / 174

52 Value Formulas Perpetual Annuities Definition A perpetual annuity, or perpetuity, pays a fixed sum periodically forever. Suppose an amount A is paid at the end of each period, starting at the end of the first period, and suppose the per-period interest rate is r. Then A P = (1 + r) k = A 1 + r + A (1 + r) k = A 1 + r + P 1 + r. k=1 Formula (Perpetual annuity formula) The present value P of a perpetual annuity that pays an amount A every period, beginning one period from the present, is k=2 P = A r, where r is the one-period interest rate. Dr. DING (xiaosong.ding@hotmail.com) Investment Science 52 / 174

53 Value Formulas Finite-Life Streams Suppose that the stream consists of n periodic payments of amount A, starting at the end of the current period and ending at period n. Figure: Time indexing. The present value of the finite stream relative to the interest rate r per period is n A P = (1 + r) k = A r A r(1 + r) n. k=1 Dr. DING (xiaosong.ding@hotmail.com) Investment Science 53 / 174

54 Value Formulas Formula (Annuity formulas) Consider an annuity that begins payment one period from the present, paying an amount A each period for a total of n periods. The present value P, the one-period annuity amount A, the one-period interest rate r, and the number of periods n of the annuity are related by P = A r [ 1 1 (1 + r) n ], or equivalently, A = r(1 + r)n P (1 + r) n 1. Figure: Finite stream from two perpetual annuities. Dr. DING (xiaosong.ding@hotmail.com) Investment Science 54 / 174

55 Value Formulas Definition The annuity formula is frequently used in the reverse direction; that is, A as a function of P. This determines the periodic payment that is equivalent (under the assumed interest rate) to an initial payment of P. This process of substituting periodic payments for a current obligation is referred to as amortization. Example (Loan calculation) Suppose you have borrowed $1,000 from a credit union. The terms of the loan are that the yearly interest is 12% compounded monthly. You are to make equal monthly payments of such magnitude as to repay (amortize) this loan over 5 years. How much are the monthly payments? Five years is 60 months, and 12% a year compounded monthly is 1% per month. Hence we use the formula for n = 60, r = 1%, and P = $1, 000. We find that the payments A are $22.20 per month. Dr. DING (xiaosong.ding@hotmail.com) Investment Science 55 / 174

56 Value Formulas Definition The annual percentage rate (APR) is the rate of interest that, if applied to the loan amount without fees and expenses, would result in a monthly payment of A. Example (APR) Consider a mortgage corresponding to the first listing and calculate the total fees and expenses. Using the APR of 7.883%, a loan amount of $203,150, and a 30-year term, A = $1, 474. Using an interest rate of 7.625% and the monthly payment calculated, the total initial balance is $208,267. Total of fees and expenses is $208, 267 $203, 150 = $5, 117. The loan fee is 1 point, or $2,032. Hence other expenses are $3, 085. Dr. DING (xiaosong.ding@hotmail.com) Investment Science 56 / 174

57 Value Formulas Running Amortization Example Consider the loan of $1,000, which you will repay over 5 years at 12% interest (compounded monthly). Suppose you took out the loan on January 1, and the first payment is due February 1. The running balance account procedure is consistent with reamortizing the loan each month. What will happen if one needs to amortize the balance of $ at 12% on June 1 over a period of 55 months? Dr. DING (xiaosong.ding@hotmail.com) Investment Science 57 / 174

58 Annual Worth Value Formulas The value A below is the annual worth (over n years) of the project. (x 0, x 1, x 2,..., x n ) (v, 0, 0,..., 0) (0, A, A,..., A). Example (A capital cost) The purchase of a new machine for $100,000 (at time zero) is expected to generate additional revenues of $25,000 for the next 10 years starting at year 1. Suppose the interest rate r = 16%. Is this a profitable investment? We simply need to determine how to amortize the initial cost uniformly over 10 years, that is, we need to find the annual payments at 16% that are equivalent to the original cost. Using the annuity formula, this corresponds to $20,690 per year. Hence the annual worth of the project is $25, 000 $20, 690 = $4, 310, and thus the investment is profitable. Dr. DING (xiaosong.ding@hotmail.com) Investment Science 58 / 174

59 Outline Bond details 7 The Market for Future Cash 8 Value Formulas 9 Bond details 10 Yield 11 Duration 12 Immunization 13 Convexity Dr. DING (xiaosong.ding@hotmail.com) Investment Science 59 / 174

60 Bond details Definition A bond is an obligation by the bond issuer to pay money to the bond holder according to rules specified at the time the bond is issued. A bond pays a specific amount, its face value or, equivalently, its par value such as $1,000 or $10,000 at the date of maturity. Most bonds pay periodic coupon payments. The last coupon date corresponds to the maturity date, so the last payment is equal to the face value plus the coupon value. The coupon amount is described as a percentage of the face value. The bid price is the price dealers are willing to pay for the bond. The ask price is the price at which dealers are willing to sell the bond. The issuer of a bond initially sells the bonds to raise capital immediately, and then is obligated to make the prescribed payments. Usually bonds are issued with coupon rates close to the prevailing general rate of interest so that they will sell at close to their face value. Dr. DING (xiaosong.ding@hotmail.com) Investment Science 60 / 174

61 Bond details Dr. DING Investment Science 61 / 174

62 Bond details Definition The accrued interest (AI) is AI = number of days since last coupon coupon amount. number of days in current coupon petiod Example (Accrued interest calculation) Suppose we purchase on May 8 a U.S. Treasury bond that matures on August 15 in some distant year. The coupon rate is 9%. Coupon payments are made every February 15 and August 15. The accrued interest is computed by noting that there have been 83 days since the last coupon and 99 days until the next coupon payment. Hence, AI = This 2.05 would be added to the quoted price, expressed as a percentage of the face value. For example, $20.50 would be added to the bond if its face value were $1,000. Dr. DING (xiaosong.ding@hotmail.com) Investment Science 62 / 174

63 Quality Ratings Bond details Although bonds offer a supposedly fixed-income stream, they are subject to default if the issuer has financial difficulties or falls into bankruptcy. To characterize the nature of this risk, bonds are rated by rating organizations. Bonds that are either high or medium grade are considered to be investment grade. Bonds that are in or below the speculative category are often termed junk bonds. Dr. DING (xiaosong.ding@hotmail.com) Investment Science 63 / 174

64 Outline Yield 7 The Market for Future Cash 8 Value Formulas 9 Bond details 10 Yield 11 Duration 12 Immunization 13 Convexity Dr. DING (xiaosong.ding@hotmail.com) Investment Science 64 / 174

65 Yield Yield to Maturity (YTM): The interest rate at which the present value of the stream of payments (consisting of the coupon payments and the final face-value redemption payment) is exactly equal to the current price. Yield to maturity is just the internal rate of return of the bond at the current price! Definition Suppose that a bond with face value F makes m coupon payments of C/m each year and there are n periods remaining. The coupon payments sum to C within a year. Suppose also that the current price of the bond is P. Then the yield to maturity is the value of λ such that P = F n [1 + (λ/m)] n + C/m [1 + (λ/m)] k. k=1 Dr. DING (xiaosong.ding@hotmail.com) Investment Science 65 / 174

66 Yield Formula (Bond price formula) The price of a bond, having exactly n coupon periods remaining to maturity and a yield to maturity of λ, satisfies { } P = 1, F [1 + (λ/m)] n + C λ 1 [1 + (λ/m)] n where F is the face value of the bond, C is the yearly coupon payment, and m is the number of coupon payments per year. As in any calculation of internal rate of return, to derive λ generally requires an iterative procedure, easily carried out by a computer! Dr. DING (xiaosong.ding@hotmail.com) Investment Science 66 / 174

67 Yield Qualitative Nature of Price-Yield Curves Figure: Price-yield curves and coupon rate (30 years). 1 What are the prices at YTM = 0, and for a par bond? 2 Price and yield have an inverse relation. The price of the bond must tend toward zero as the yield increases. The shape is convex. 3 With a fixed maturity date, the price-yield curve rises as the coupon rate increases. Dr. DING (xiaosong.ding@hotmail.com) Investment Science 67 / 174

68 Yield Figure: Price-yield curves and maturity (10% coupon). 1 All of these bonds are at par when the yield is 10%; hence the three curves all pass through the common par point. 2 As the maturity is increased, the price-yield curve becomes steeper, which indicates that longer maturities imply greater sensitivity of price to yield. Dr. DING (xiaosong.ding@hotmail.com) Investment Science 68 / 174

69 Yield Exercise What will happen to the price if the yield of a 30-year par bond increases from 10% to 11%? What if it is a 3-year 10% par bond? Bond holders are subject to yield risk in the sense that if yields change, bond prices also change. The bond with 30-year maturity is much more sensitive to yield changes than the bond with 1-year maturity! Dr. DING (xiaosong.ding@hotmail.com) Investment Science 69 / 174

70 Other Yield Measures Yield Definition Current Yield (CY) is defined as CY = annual coupon payment bond price 100. which gives a measure of the annual return of the bond. Example Consider a 10%, 30-year bond. If it is selling at par (that is, at 100), then the current yield is 10, which is identical to the coupon rate and to YTM. If the same bond were selling for 90, then CY = 11.11, YTM = Definition Yield to Call (YTC) is defined as the internal rate of return calculated assuming that the bond is in fact called at the earliest possible date. Dr. DING (xiaosong.ding@hotmail.com) Investment Science 70 / 174

71 Outline Duration 7 The Market for Future Cash 8 Value Formulas 9 Bond details 10 Yield 11 Duration 12 Immunization 13 Convexity Dr. DING (xiaosong.ding@hotmail.com) Investment Science 71 / 174

72 Duration Rule of thumb: The prices of long bonds are more sensitive to interest rate changes than those of short bonds. Definition The duration of a fixed-income instrument is a weighted average of the times that payments (cash flows) are made. The weighting coefficients are the present values of the individual cash flows. D = n k=0 PV (t k )t k PV = PV (t 0)t 0 + PV (t 1 )t 1 + PV (t 2 )t PV (t n )t n, PV where PV (t k ) denotes the present value at the cash flow that occurs at time t k ; PV in the denominator is the total present value, which is the sum of the individual PV (t k ) values. Duration is a time intermediate between the first and last cash flows, i.e., t 0 D t n. Dr. DING (xiaosong.ding@hotmail.com) Investment Science 72 / 174

73 Macaulay Duration Duration Definition Suppose a financial instrument makes payments m times per year, with the payment in period k being c k, and there are n periods remaining. The Macaulay duration D is defined as D = 1 PV n k=1 ( ) k PV k = 1 m PV where λ is the yield to maturity and PV = n PV k = k=1 n k=1 n k=1 ( ) k c k m [1 + (λ/m)] k, c k [1 + (λ/m)] k. The factor k/m in the formula for D is time, measured in years! Dr. DING (xiaosong.ding@hotmail.com) Investment Science 73 / 174

74 Duration Example (A short bond) Consider a 7% bond with 3 years to maturity. Assume that the bond is selling at 8% yield. We can find the value and the Macaulay duration by the simple spreadsheet layout as shown below. The duration is years. Figure: Layout for calculating duration. Dr. DING (xiaosong.ding@hotmail.com) Investment Science 74 / 174

75 Explicit Formula Duration Formula (Macaulay duration formula) The Macaulay duration for a bond with a coupon rate c per period, yield y per period, m periods per year, and exactly n periods remaining, is Proof. PV = k=1 D = 1 PV D = 1 + y my 1 + y + n(c y) mc[(1 + y) n 1] + my. n c (1 + y) k + 1 (1 + y) n = c y [ n k c m (1 + y) k + n m k=1 [ 1 1 (1 + y) n 1 (1 + y) n ] = 1 PV ] + 1 (1 + y) n, [ c m S + n m ] 1 (1 + y) n. Dr. DING (xiaosong.ding@hotmail.com) Investment Science 75 / 174

76 Duration contd. S = ys = n k=1 n k=1 k n (1 + y) k, (1 + y)s = k (1 + y) k 1, 1 (1 + y) k 1 n (1 + y) n, y n 1 + y S = 1 (1 + y) k n (1 + y) n+1 k=1 = 1 [ ] 1 PV c (1 + y) n S = 1 [ PV (1 + y) c y k=1 n (1 + y) n+1, ] 1 y(1 + y) n 1 n y(1 + y) n. Dr. DING (xiaosong.ding@hotmail.com) Investment Science 76 / 174

77 Duration contd. c PV (1 + y) 1 S = m my my(1 + y) n 1 nc my(1 + y) n, D = 1 [ ] PV (1 + y) 1 + y + n(c y) PV my my(1 + y) n = 1 + y my 1 + y + n(c y) mc[(1 + y) n 1] + my. Example (Duration of a 30-year par bond) Consider the 10%, 30-year bond. Assume that it is at par; that is, the yield is 10%, At par, c = y, and D = 1 + y [ ] 1 1 my (1 + y) n D = Dr. DING (xiaosong.ding@hotmail.com) Investment Science 77 / 174

78 Duration Qualitative Properties of Duration 1 As the time to maturity increases to infinity, the durations do not also increase to infinity, but instead tend to a finite limit that is independent of the coupon rate. 2 The durations do not vary rapidly with respect to the coupon rate. The fact that the yield is held constant tends to cancel out the influence of the coupons. Dr. DING (xiaosong.ding@hotmail.com) Investment Science 78 / 174

79 Duration A general conclusion: Very long durations (of, say, 20 years or more) are achieved only by bonds that have both very long maturities and very low coupon rates. Exercise (Duration limit) For Macaulay duration formula, show that the limiting value of duration as maturity is increased to infinity is D 1 + (λ/m). λ For the bonds in the previous table (where λ = 0.05 and m = 2), we obtain D Note that for large λ, this limiting value approaches 1/m, and hence the duration for large yields tends to be relatively short. Dr. DING (xiaosong.ding@hotmail.com) Investment Science 79 / 174

80 Duration Duration and Sensitivity In the case where payments are made m times per year and yield is based on those same periods, we have PV k = The derivative with respect to λ is c k [1 + (λ/m)] k. dpv k dλ = (k/m)c k [1 + (λ/m)] k+1 = (λ/m) Now apply this to the expression for price, P = PV = n k=1 PV k dp dλ = n k=1 where D M is called the modified duration. ( ) k PV k. m dpv k dλ = (λ/m) D P = D MP, Dr. DING (xiaosong.ding@hotmail.com) Investment Science 80 / 174

81 Duration Formula (Price sensitivity formula) The derivative of price P with respect to λ of a fixed-income security is dp dλ = D MP, or equivalently, where D M = D/[1 + (λ/m)] is the modified duration. 1 dp P dλ = D M, By dp/dλ P/ λ, the price sensitivity formula can be used to estimate the change in price due to a small change in yield (or vice versa), Example (A zero-coupon bond) P D M P λ. Consider a 30-year zero-coupon bond. Suppose its current yield is 10%. Then we have D = 30 and D M 27. Suppose that yields increase to 11%. According to the price sensitivity formula, the relative price change is approximately equal to 27%. This is a very large loss in value. Because of their long durations, zero-coupon bonds have very high interest rate risk. Dr. DING (xiaosong.ding@hotmail.com) Investment Science 81 / 174

82 Duration Example (A 10% bond) Consider the price-yield curve for a 30-year, 10% coupon bond, whose duration at the par point (where price is 100) is D = Therefore, D M = 9.94/( /2) = The slope of the price-yield curve at that point is equal to dp/dλ = 947. If the yield changes to 11%, then P by estimating the change in price as P D M P λ = Dr. DING (xiaosong.ding@hotmail.com) Investment Science 82 / 174

83 Duration Duration of a Portfolio Consider a portfolio that is the sum of two bonds A and B. D A = n k=0 t kpv A k P A D B = n k=0 t kpv B k P B Formula (Duration of a portfolio) D = PA D A + P B D B P A + P B. Suppose there are m fixed-income securities with prices and durations of P i and D i, respectively, i = 1, 2,..., m, all computed at a common yield. The portfolio consisting of the aggregate of these securities has price P and duration D given by P = m P i, D = i=1 m w i D i = i=1 m i=1 P i D i, i = 1, 2,..., m. P Dr. DING (xiaosong.ding@hotmail.com) Investment Science 83 / 174

84 Outline Immunization 7 The Market for Future Cash 8 Value Formulas 9 Bond details 10 Yield 11 Duration 12 Immunization 13 Convexity Dr. DING (xiaosong.ding@hotmail.com) Investment Science 84 / 174

85 Immunization Example Suppose that you wish to invest money now that will be used next year for a major household expense. If you invest in 1-year Treasury bills, you know exactly how much money these bills will be worth in a year (little risk). If, on the other hand, you invested in a 30-year zero-coupon bond, the value of this bond a year from now would be quite variable, depending on what happens to interest rates during the year (high risk). Suppose that you are saving money to pay off an obligation that is due in 10 years (reverse situation). The 10-year zero-coupon bond will provide completely predictable results (little risk). The 1-year Treasury bill will impose reinvestment risk since the proceeds will have to be reinvested after 1 year at the then prevailing rate (high risk). Dr. DING (xiaosong.ding@hotmail.com) Investment Science 85 / 174

86 Immunization Example Suppose now that you face a series of cash obligations and you wish to acquire a portfolio that will be used to pay these obligations as they arise. 1 One way is to purchase a set of zero-coupon bonds that have maturities and face values exactly matching the separate obligations. However, this simple technique may be infeasible if corporate bonds are used since there are few corporate zero-coupon bonds. 2 You may instead acquire a portfolio having a value equal to the present value of the stream of obligations. You can sell some of your portfolio whenever cash is needed to meet a particular obligation. If your portfolio delivers more cash than needed at a given time (from coupon or face value payments), you can buy more bonds. Provided the yield does not change, the value of your portfolio will, throughout this process, continue to match the present value of the remaining obligations. Dr. DING (xiaosong.ding@hotmail.com) Investment Science 86 / 174

87 Immunization A problem with this present-value-matching technique arises if the yields change, the value of your portfolio and the present value of the obligation stream will both change in response, but probably by amounts that differ from one another. Your portfolio will no longer be matched! Immunization immunizes the portfolio value against interest rate changes, at least approximately, by matching durations and present values. If yields increase, the present value of the asset portfolio will decrease, but the present value of the obligation will decrease by approximately the same amount. If yields decrease, the present value of the asset portfolio will increase, but the present value of the obligation will increase by approximately the same amount. Dr. DING (xiaosong.ding@hotmail.com) Investment Science 87 / 174

88 Immunization Example (The X Corporation) The X Corporation has an obligation to pay $1 million in 10 years. It wishes to invest money now that will be sufficient to meet this obligation. The X Corporation is planning to select from the three corporate bonds, whose durations are D 1 = 11.44, D 2 = 6.54, and D 3 = 9.61, respectively. The present value of the obligation PV = $414, 643 is computed at 9% interest compounded every six months. The immunized portfolio is found by solving the two equations PV 1 + PV 2 = PV, D 1 PV 1 + D 2 PV 2 = 10PV. Dr. DING (xiaosong.ding@hotmail.com) Investment Science 88 / 174

89 Immunization Example (contd.) The solution is PV 1 = $292, and PV 2 = $121, The number of bonds to be purchased is then found by dividing each value by the respective bond price (par value is $100). These numbers are then rounded to integers to define the portfolio. Dr. DING (xiaosong.ding@hotmail.com) Investment Science 89 / 174