Principles of Banking and Finance Revision Session Section B Jason Laws Jason Laws Principles of Banking and Finance Warning ❑ Please don’t treat these slides as an extra resource to work through – that is the role of the study guide. ❑ I share these with you so you can see examples and slides I used in the revision sessions. ❑ They include numerous past examination questions and past prelim examination questions. I will not go through the 2021 prelim paper as the solutions are available. Jason Laws 2 of 218 Sample Question 5. A firm is considering two investment projects, Gamma and Kappa. These projects are NOT mutually exclusive. Assume the firm is not capital constrained. The initial costs and cash flows for these projects are: ❑ The key point here that many students over looked is the words NOT mutually exclusive. Mutually exclusive in this context means that only one project could be chosen, for example, one use of a scarce resource or one production technique etc. Here more than one can be chosen if they have a positive NPV. Jason Laws 3 of 218 The NPV Rule ❑ Accept any project if its NPV > 0 or if NPV=0 ❑ Reject a project if its NPV < 0 ❑ Suppose a project has a positive NPV, but the NPV is small, say, only a few hundred dollars then the firm should still undertake that project if there are no alternative projects with higher NPV as a firms wealth is increased every time it undertakes a positive NPV project. ❑ A small NPV, as long as it is positive, is net of all input costs and financing costs so even if the NPV is low it still provides additional returns. ❑ A firm that rejects a positive NPV project is rejecting wealth. ❑ Observation – you are not forced to accept any of the projects. If you accept a project with a negative NPV then you are destroying shareholder wealth. Jason Laws 4 of 218 NPV decision rule ❑ Activity: Please draw NPV of a standard project, i.e. cash out followed by cash in, against the discount rate. NPV Discount rate Jason Laws 5 of 218 Present Values Example Assume that the cash flows from the construction and sale of an office building are as follows. Given a 7% required rate of return, create a present value worksheet and show the net present value. Year 0 Year 1 Year 2 − 150,000 − 100,000 + 300,000 Jason Laws 6 of 218 Present Values Example - continued Period 0 1 2 Discount Factor 1.0 1 1.07 = .935 1 = .873 (1.07 )2 Cash Present Flow Value − 150,000 − 150,000 − 100,000 − 93,500 + 300,000 + 261,900 NPV = Total = $18,400 Jason Laws 7 of 218 NPV Example ❑ Consider three alternative projects, A, B and C. ❑ They all cost $1,000,000 to set up but project’s A and C returns $800,000 per year for two years starting one year from set up. Projects B also returns $800,000 per year for two years, but the cash flows begin three years after set up , i.e. in year 3. ❑ Whilst project C costs $1,000,000 to set up it requires $500,000 initially and $500,000 at termination (a clean up cost for example). ❑ If the firm uses a discount rate of 20% which is the better project? ❑ If you can rank these projects without doing the calculations then you really understand NPV. Jason Laws 8 of 218 NPV Example ❑ Project A: interest rate Year Cash Flow discount factor PV NPV= 20% 0 1 2 -$1,000,000 $800,000 $800,000 1.000 0.833 0.694 -$1,000,000.00 $666,666.67 $555,555.56 3 $0 0.579 $0.00 4 $0 0.482 $0.00 $222,222.22 ❑ Project B: interest rate Year Cash Flow discount factor PV NPV= 20% 0 -$1,000,000 1.000 -$1,000,000.00 1 $0 0.833 $0.00 2 $0 0.694 $0.00 3 4 $800,000 $800,000 0.579 0.482 $462,962.96 $385,802.47 -$151,234.57 Jason Laws 9 of 218 NPV Example ❑ Project C: interest rate Year Cash Flow discount factor PV 20% 0 1 2 -$500,000 $800,000 $300,000 1.000 0.833 0.694 -$500,000.00 $666,666.67 $208,333.33 NPV= $375,000.00 Jason Laws 3 $0 0.579 $0.00 4 $0 0.482 $0.00 10 of 218 NPV Example ❑ Project C has the highest NPV and therefore if only one project can be undertaken it should be C. ❑ However if more than one project can be undertaken then both A and C should be selected since they both have positive NPV’s. ❑ Project B should be rejected since it has a negative NPV and would therefore destroy wealth. ❑ It makes sense that project C should have the highest NPV, since its cash outflows are deferred relative to the other projects, and its cash flows are early. ❑ In contrast project B has all the costs up front but the cash inflows are deferred. Jason Laws 11 of 218 Sample Question (a) Using a discount rate of 12% calculate the net present value for each project. What decision would you make based on your calculations? (4 marks) Jason Laws 12 of 218 Sample Question (b) How would your decision change if the discount rate used for calculating the net present value is 20%? (4 marks) Jason Laws 13 of 218 Internal Rate of Return ❑ The IRR of a project can be defined as the rate of discount which, when applied to the projects cash flows, produces a zero NPV. ❑ The IRR decision rule is then: ❑ “invest in any project which has an IRR greater than or equal to some predetermined cost of capital”. ❑ The comparison rate is usually the cost of capital, i.e. the discount rate we would have used in a NPV analysis. ❑ Observation: At 12% we find the NPV of both projects to be positive, and at 20% they are both negative. Hence the IRR must be between 10% and 20%. A 12% Gamma’s NPV is only 831 so we can expect that Gamma’s NPV tends to zero before Kappa. Jason Laws 14 of 218 Internal Rate of Return Example ❑ You can purchase a turbo powered machine tool gadget for $4,000. The investment will generate $2,000 and $4,000 in cash flows for two years, respectively. What is the IRR on this investment? 2,000 4,000 NPV = −4,000 + + =0 1 2 (1 + IRR) (1 + IRR) IRR = 28.08% ❑ Hint: You may recognize that if you multiply through by (1+IRR)2 you have a quadratic problem that can be solved. Jason Laws 15 of 218 Internal Rate of Return: Example ❑ If you don’t see it then here is the solution: 2,000 4,000 − 4,000 + + =0 1 2 (1 + IRR) (1 + IRR) − 4,000(1 + IRR) 2 + 2,000(1 + IRR) + 4,000 = 0 2000 : − 2(1 + IRR) 2 + (1 + IRR) + 2 = 0 x = 1 + IRR : − 2x2 + x + 2 = 0 Does this look familiar? Jason Laws 16 of 218 Internal Rate of Return: Example − 2x2 + x + 2 = 0 − b b 2 − 4ac x= , a = −2, b = 1, c = 2 : 2a − 1 12 − 4(−2)(2) − 1 1 + 16 x= = 2(−2) −4 1 + 17 1 − 17 x=− or − −4 −4 x = −0.78 or + 1.28 x = 1 + IRR, IRR = x − 1 x = −0.78 − 1 = −1.78 or x = 1.28 − 1 = 0.28 = 28% Or if you are lazy: http://www.mathsisfun.com/ quadratic-equationsolver.html Jason Laws 17 of 218 Internal Rate of Return 2500 2000 IRR=28% 1000 500 10 0 90 80 70 60 50 40 30 -500 20 0 10 NPV (,000s) 1500 -1000 -1500 -2000 Discount rate (%) Jason Laws 18 of 218 Internal Rate of Return ❑ The internal rate of return calculation is a trivial one to undertake assuming of course that you have a pc at your disposal. ❑ In 2023 I doubt there is anyone that undertakes an IRR calculation by hand though it is possible: ❑ IRR = i0+ (NPV0/(NPV0+│NPV1│))*(i1 – i0) ❑ So we must take two guesses of NPV at two different interest rates and linearly interpolate between the two. ❑ As the line is “non-linear” the closer to zero (either side the more accurate the answer). Jason Laws 19 of 218 Internal Rate of Return ❑ We take a guess at 25%: ir= 0.25 Time 0 1 2 CF $ -4,000.00 $ 2,000.00 $ 4,000.00 df 1 0.80 0.640 PV $ -4,000.00 $ 1,600.00 $ 2,560.00 NPV= $ 160.00 ❑ The we guess again at 35%: ir= Time CF df PV NPV= 0.35 0 1 2 $ -4,000.00 $ 2,000.00 $ 4,000.00 1 0.74 0.549 $ -4,000.00 $ 1,481.48 $ 2,194.79 $ -323.73 Jason Laws 20 of 218 Internal Rate of Return ❑ Plugging the numbers in the formula: NPV 0 IRR = i0 + (i1 − i0 ) NPV 0 + NPV1 160 IRR = 0.25 + (0.35 − 0.25) = 160 + − 323.73 IRR = 0.25 + 0.1 0.3281 IRR = 28.28% Jason Laws 21 of 218 Sample Question 5. A firm is considering two investment projects, Gamma and Kappa. These projects are NOT mutually exclusive. Assume the firm is not capital constrained. The initial costs and cash flows for these projects are: ❑ The key point here that many students overlooked is the words NOT mutually exclusive. Mutually exclusive in this context means that only one project could be chosen, for example, one use of a scarce resource or one production technique etc. Here more than one can be chosen if they have a positive NPV. Jason Laws 22 of 218 Sample Question (c) Calculate an approximate IRR for each project. Assume the hurdle rate is 12%. What decision would you make based on your calculations? (6 marks) ❑ Applying the IRR decision rule - “invest in any project which has an IRR greater than or equal to some predetermined cost of capital” – as the IRR is greater than the hurdle rate of 12% we accept BOTH projects. Jason Laws 23 of 218 Payback ❑ This investment appraisal method calculates the time period taken to payback the initial investment. ❑ When there are mutually exclusive investments or where ranking is required, the project that has the earliest payback will be selected. ❑ Example: You are looking at a new project and you have estimated the following cash flows: ❑ Capital Investment =£40,000 ❑ Cashflows = £16,000 per annum for three years, £12,000 per annum for the 4th year. ❑ Project Life = 4 years ❑ Salvage Value = Nil ❑ What is the payback? Jason Laws 24 of 218 Payback Year Cash Flow (£) Cummulative Cash Flow (£) 0 Cost -£40.00 -£40.00 1 Savings £16.00 -£24.00 2 Savings £16.00 -£8.00 3 Savings £16.00 £8.00 4 Savings £16.00 £24.00 Payback Period = 2 + 8/16 = 2.5 years ❑ Observation: Unless advised otherwise payback assumes that cash flow occurs throughout the year. Above we note that after two years there is still a £8,000 shortfall. In year three the cash flows are £16,000. if the cash flow is received throughout the year then £8,000 is received 0.5 through the year. Jason Laws 25 of 218 Payback ❑ Disadvantages: ❑ Ignores cash flows beyond the payback period ❑ Does not consider the profile of cash flows within the payback period ❑ Ignores discounting – but can use modified packback ❑ Advantages: ❑ Assumes cash flows after payback are so risky as to be of no value ❑ Easy to understand and calculate – useful screening and communication device ❑ Observation: Modified payback uses the PV of cash flows rather than the raw cash flows. As PV is less than the raw cash flow modified payback must be longer than “normal” payback. Jason Laws 26 of 218 Sample Question (d) Calculate the payback period for each project. The company looks to select investment projects paying back in 2 years. What decision would you make based on your calculations? (2 marks) £6,000/£17,000 = 0.35, so payback = 2 + 0.35 = 2.35. Decision: As the payback of both projects is greater than the companies payback policy then both projects should be rejected. Jason Laws 27 of 218 Sample Question (e) Explain what is meant by the ‘opportunity cost of capital’ in the context of the NPV method. (4 marks) Observation: Many students made the observation that this is the return on an alternative investment and/or that it is a market determined rate of return. However, very few students note that “this is the return on an alternative investment with the same level of risk” Jason Laws 28 of 218 Sample Question (f) Explain the additivity property of the NPV method. (5 marks) ❑ Mutually exclusive projects are a set of projects of which only one can be chosen at a given time. Firms have to choose one project among several that do the same job: for example, a manually controlled machine versus a computer controlled machine. ❑ Given that present values obey the additivity principle, it follows that the NPV also possesses the additivity property. Assume that a firm has only two projects (X and Y); the NPV of projects X and Y is equal to the NPV of project X plus the NPV of project Y. (Note that the additivity property holds because present values are all measured in today’s dollars.) ❑ This can be written as: ❑ NPV (X +Y ) = NPV (X ) + NPV (Y ) Jason Laws 29 of 218 Sample Question ❑ The additivity property implies that the value of the firm is simply the sum of the values of the separate projects. Because of the additivity property, when there are mutually exclusive projects, the NPV method indicates that the project with the largest positive NPV should be adopted. ❑ The reason for this is that the project with the largest NPV generates the largest NPV of the firm’s aggregated cash flows. ❑ One example clarifies the point that the choice of project relies on the additivity property. Assume that project X is a positive NPV project, while project Y is a negative NPV. The joint project (X+Y) will have a lower NPV than project X on its own. The NPV enables managers to avoid choosing bad projects just because they are packaged with good ones. Jason Laws 30 of 218 The time value of money ❑ When people undertake to set aside money for investment something has to be given up now, e.g. If someone buys shares in a firm now or lends to a business there is a sacrifice of current consumption. ❑ Hence compensation is required to induce people to sacrifice consumption. ❑ Compensation will be required for at least three things: ❑ Impatience to consume – individuals generally prefer to have £1000 today than £1000 in say five years time. The utility of £1000 now is greater than £1000 in five years time. The rate of exchange between certain future consumption and certain current consumption is the pure rate of interest. This would occur even in the absence of inflation and risk. ❑ Inflation – in addition to the above compensation for time investors will also have to be compensated for the loss in purchasing power. ❑ Risk – the promise of a receipt of a sum of money on the future generally carries with it and element of risk – the payout may not take place or the amount may be less than anticipated. Jason Laws 31 of 218 The time value of money ❑ Example ❑ Consider an investor considering a £1,000 one-year investment and requires compensation for these three elements: ❑ 2% is required to compensate for the pure time value of money, ❑ Inflation is anticipated to be 3% over the year ❑ To compensate the investor for impatience to consume and inflation the investment needs to generate a return of: ❑ (1 + 0.02) x (1+.0.03) – 1 = .0506 = 5.06% ❑ The 5.06% may be regarded as the risk-free return (RFR), the interest rate which is sufficient to induce investment assuming no uncertainty about cash flows. ❑ Lending to governments, through the purchase of bonds and bills is typically considered risk free. Jason Laws 32 of 218 The time value of money ❑ However different investments carry different degrees of uncertainty about the outcome of the investment. ❑ For instance, an investment on the Russian stock market, with its high volatility may be regarded as more risky than the purchase of a share in say Exxon Mobil. ❑ Investors require different risk premiums on top of the risk free return to reflect the perceived level of extra risk: ❑ Required Return (or time value of money) = RFR + Risk Premium ❑ Activity: List any models that allow you do determine the risk adjusted rate of return? Jason Laws 33 of 218 The time value of money ❑ In order to compare like with like it is important to value cash flows at the same point in time, that could be the current time period or some time period in the future. ❑ The conversion process is achieved by discounting all future cash flows by the time value of money, thereby expressing the cash flows as an equivalent amount received at time zero: ❑ F = P(1 + i)n ❑ Where F = future value, P = present value, i = interest rate, n = number of years ❑ Example: if a saver deposited £100 in a bank account paying interest at 8 per cent per annum, after three years the account will contain: ❑ 100 x (1 + 0.08)3 = £125.97 Jason Laws 34 of 218 The time value of money ❑ The formula can be inverted so we can ask the question “how much money must I deposit now to receive £125.97 in three years?” ❑ P = F/(1 + i)n ❑ P = £125.97/(1 + .08)3 = £100 ❑ Here we have discounted the £125.97 back to a present value of £100. ❑ We can also say that we have discounted the future cash flow back to “year zero”. ❑ More formally, in order to find the present value of some future cash flow we multiply by the discount factor: DF = 1 (1+ r ) t Jason Laws 35 of 218 The time value of money ❑ Discount Factors can be used to compute the present value of any cash flow: Ct PV = DF Ct = t (1 + r ) ❑ Where Ct is the cash flow received in time period t, and r is the required return/time value of money. Jason Laws 36 of 218 The time value of money ❑ Present Values can be added together to evaluate multiple cash flows: PV = C1 + (1+ r ) 2 + .... C2 (1+ r ) 1 ❑ Each individual cash flow is measure in “year zero” money and so they can be added together, this is the additivity property of discounted cash flow. Jason Laws 37 of 218 The time value of money ❑ Consider how the discount factor varies with r and t. T/r 1 2 3 4 5 6 7 8 9 10 1% 0.9901 0.9803 0.9706 0.961 0.9515 0.942 0.9327 0.9235 0.9143 0.9053 2% 0.9804 0.9612 0.9423 0.9238 0.9057 0.888 0.8706 0.8535 0.8368 0.8203 5% 0.9524 0.907 0.8638 0.8227 0.7835 0.7462 0.7107 0.6768 0.6446 0.6139 Jason Laws 10% 0.9091 0.8264 0.7513 0.683 0.6209 0.5645 0.5132 0.4665 0.4241 0.3855 15% 0.8696 0.7561 0.6575 0.5718 0.4972 0.4323 0.3759 0.3269 0.2843 0.2472 20% 0.8333 0.6944 0.5787 0.4823 0.4019 0.3349 0.2791 0.2326 0.1938 0.1615 38 of 218 The time value of money ❑ Sometimes there are shortcuts that make it very easy to calculate the present value of an asset that pays out in different periods. These tools allow us to cut through the calculations quickly. ❑ Perpetuity - Financial concept in which a cash flow is theoretically received forever. cash flow C1 PV of Cash Flow = PV = discount rate r Think of a preference share ❑ Annuity - An asset that pays a fixed sum each year for a specified number of years. 1 1 PV of annuity = C − t r ( ) r 1 + r Jason Laws 39 of 218 Other time value of money topics ❑ The time value of money, i.e. a dollar today is worth more than a dollar tomorrow, is fundamental to many topics in finance: a) NPV b) Dividend Discount Model c) Bond pricing ❑ Only a) and c) were covered on the prelim paper. Here we cover topic b) which has been covered on many PBF exam papers in the past. Jason Laws 40 of 218 Valuation of Equities ❑ The dividend discount model is based on the premise that the market value of ordinary shares represents the sum of expected future dividends, to infinity, discounted to time zero. ❑ Consider a shareholder who intends holding a share for one year. A single dividend will be paid at the end of the holding period, d1 and the share will be sold at a price p1 in one year. ❑ To derive the value of a share at time 0 to this investor the future cash flows d1 and p1 need to be discounted at a rate which includes an allowance for the risk of the share, k: d1 p1 p0 = + 1+ k 1+ k Jason Laws 41 of 218 Valuation of Equities ❑ Where does p1 come from? ❑ Consider a second investor who expects to hold the share for a further year and sell at time 2 for P2, the price, p1, will be: d2 p2 p1 = + 1+ k 1+ k ❑ Substituting into the equation for p0 we get: d1 d2 p2 p0 = + + 2 2 1+ k (1+ k ) (1+ k ) Jason Laws 42 of 218 Valuation of Equities ❑ If a series of one year investors bought this share, and we in turn solved for p2, p3 etc. we would find: d1 d2 d¥ p0 = + +.... 2 ¥ 1+ k (1+ k ) (1+ k ) ❑ The terminal stock price can effectively be ignored as its present value is zero. However, it value feeds in to p∞-1 etc. And so by iteration it enters into p0. Jason Laws 43 of 218 Example: Stock 1st ❑ Consider stock 1st that has just paid a dividend of $10 and is expected to pay this dividend forever. If the cost of capital, k, is 5% what is the value of stock 1st? ❑ We could use Excel and find the PV the future dividends. k= Time 1 2 3 4 5 6 7 8 9 10 5% $ $ $ $ $ $ $ $ $ $ Div 10.00 10.00 10.00 10.00 10.00 10.00 10.00 10.00 10.00 10.00 PV(Div) $ 9.52 $ 9.07 $ 8.64 $ 8.23 $ 7.84 $ 7.46 $ 7.11 $ 6.77 $ 6.45 $ 6.14 Jason Laws Sum (years 1 - 500)= $ 200.00 44 of 218 Example: Stock 1st ❑ But of course this is analogous to a perpetuity so we could have also found the price using: ❑ P0 = Div/k = $10/0.05 = $200 ❑ In this example the PV tends towards zero, in fact it is less than 10 cents after 90 (ish) years, and 1 cent after 150 (ish) years. ❑ Of course the bigger is k, the faster the PV tends towards zero. Jason Laws 45 of 218 Example: Stock 2nd ❑ Consider stock 2nd who will not pay a dividend for the next five years but in year six it will pay a dividend of $10 and it is expected to pay this dividend forever. ❑ If the cost of capital, k, is 5%, what is the value of stock 2nd? ❑ Again, we can map out the future dividends in Excel, find the PV’s and sum them together. k= Time 1 2 3 4 5 6 7 8 9 10 5% $ $ $ $ $ $ $ $ $ $ Div 10.00 10.00 10.00 10.00 10.00 PV(Div) $ $ $ $ $ $ 7.46 $ 7.11 $ 6.77 $ 6.45 $ 6.14 Jason Laws Sum (years 1 - 500)= $ 156.71 46 of 218 Example: Stock 2nd ❑ Clearly the answer is lower than that of Stock 1st. We could have found the answer by taking the value of Stock 1st and subtracting the sum of the PV’s from years 1 to 5 (in fact this is an annuity, for which there is a formula). ❑ Or we can think laterally... ❑ Figuratively speaking, standing at the end of year 5 we know that the Stock 2nd will pay a dividend of $10 and pay this dividend forever. ❑ In “year five money” the value of Stock 2nd is: ❑ P5 = $10/0.05 = $200 ❑ In order to get the value in “year zero money” we need to discount at 5% over 5 years: ❑ P0 = $200/1.055= $156.71 Jason Laws 47 of 218 Valuation of Equities ❑ According to the discounted cash flow model, the equilibrium price of a share is equal to the present discounted cash flow of expected future dividend payments. Dt +i Pt = Et i i =1 (1 + k ) ❑ However, in order to compute this value, one has to make assumptions about future dividends. ❑ We have considered a constant dividend but what if dividends grow at some constant rate, g? Jason Laws 48 of 218 Valuation of Equities ❑ So that:❑ Dt+1 = Dt(1+g) ❑ Et(Dt+2) = Dt+1(1+g) ❑ Et(Dt+3) = Et(Dt+2)(1+g) = Dt+1(1+g)2 ❑ Et(Dt+i) = Et(Dt+i-1)(1+g) = Dt+1(1+g)i-1 ❑ Substituting this result in the equation on the previous slide we get:- Dt +i Dt +1 (1 + g ) Pt = Et = i i i =1 (1 + k ) i =1 (1 + k ) Jason Laws i −1 49 of 218 Valuation of Equities ❑ If we can get the summation sign from zero to infinity rather than from 1 to infinity we can treat the result as a geometric series. ❑ Note, if we start summing from 1 we must increase the index within the summation by 1 to give:- Dt +1 (1 + g ) Pt = i (1 + k ) i =1 i −1 ( 1+ g ) = Dt +1 i +1 i = 0 (1 + k ) i ( Dt +1 1+ g ) 1+ g = Dt +1 = i 1 (1 + k ) i =0 1 + k i = 0 (1 + k ) (1 + k ) i Jason Laws i 50 of 218 Valuation of Equities ❑ The last expression on the right hand side is a geometric expression. ❑ If g < k so that the entire bracketed term < 1 we can use the following rule: 1 a = if a 1 1− a i =0 i ❑ Then:- Dt +1 Dt +1 Dt +1 1 1+ g Pt = = = (1 + k ) i =0 1 + k (1 + k ) 1 − 1 + g k − g 1+ k i Jason Laws 51 of 218 Example: Stock 3rd ❑ Consider stock 3rd that has just paid a dividend of $10 and is expected to grow dividends at 2% forever. ❑ If the cost of capital, k, is 5% what is the value of stock 3rd ❑ Again, we can map out the future dividends in Excel, find the PV’s and sum them together k= g= Time 1 2 3 4 5 6 7 8 9 10 $ $ $ $ $ $ $ $ $ $ 5% 2% Div 10.20 10.40 10.61 10.82 11.04 11.26 11.49 11.72 11.95 12.19 D0= $ 10.00 PV(Div) $ 9.71 $ 9.44 $ 9.17 $ 8.91 $ 8.65 $ 8.40 $ 8.16 $ 7.93 $ 7.70 $ 7.48 Jason Laws Sum (years 1 - 500)= $ 340.00 52 of 218 Example: Stock 3rd ❑ Or we can use the formula: ❑ Pt = $10 x 1.02/(0.05 – 0.02) = $10.2/0.03 = $340. ❑ Activity: ❑ Stock A is expected to pay a dividend of £10 forever; ❑ Stock B is expected to pay a dividend of £8 next year with dividend growth expected to be 3% per annum thereafter. ❑ Stock C just paid a dividend of £6 with dividend growth expected to be 4% p.a. thereafter. ❑ If the required return on similar equities is 10%, calculate the value of each stock. Jason Laws 53 of 218 Valuation of Equities ❑ Given the expected next period dividend, the anticipated dividend growth rate and the discount rate we can calculate the theoretical equilibrium price of the security. ❑ Alternatively, given the current price, expected end of period dividends and the anticipated growth rate we can obtain the implicit discount rate. ❑ This is the cost of equity capital facing the company. ❑ 𝑃𝑡 = 𝐷𝑡+1 𝑘−𝑔 →𝑘= 𝐷𝑡+1 𝑃𝑡 +𝑔 Jason Laws 54 of 218 Dividend Growth Model II ❑ What if we do not anticipate a constant rate of growth of dividends ? ❑ In this case we would need to forecast all future dividend payments - impossible ! ❑ One possible solution is to assume that dividends grow at a constant rate over a certain period and then grow at a different rate over a second period etc. ❑ Clearly the simplest model to solve is the two period growth model in which dividends grow at g1 in periods 1 to N and that at g2 in periods N+1 to infinity. Jason Laws 55 of 218 Example: Stock 4th ❑ Consider stock 4th that has just paid a dividend of $10 and is expected to maintain this stable dividend for the next five years and then grow dividends at 2% forever. ❑ If the cost of capital, k, is 5% what is the value of stock 4th? ❑ Again, we can map out the future dividends in Excel, find the PV’s and sum them together. k= 5% D0= $ 10.00 g= Time 1 2 3 4 5 6 7 8 9 10 $ $ $ $ $ $ $ $ $ $ 2% Div 10.00 10.00 10.00 10.00 10.00 10.20 10.40 10.61 10.82 11.04 PV(Div) $ 9.52 $ 9.07 $ 8.64 $ 8.23 $ 7.84 $ 7.61 $ 7.39 $ 7.18 $ 6.98 $ 6.78 Jason Laws Sum (years 1 - 500)= $ 309.69 56 of 218 Example: Stock 4th ❑ Or again we can think laterally. ❑ In “year five money” the value of Stock 4th is: ❑ P5 = ($10x 1.02)/(0.05-0.02) = $340 ❑ In order to get the value in “year zero money” we need to discount at 5% over 5 years: ❑ P0 = $340/1.055= $266.40 ❑ But then of course we have the PV of the dividends received in years 1 through 5: Time 1 2 3 4 5 $ $ $ $ $ Div 10.00 10.00 10.00 10.00 10.00 PV(Div) $ 9.52 $ 9.07 $ 8.64 $ 8.23 $ 7.84 $ 43.29 ❑ Giving a total of $309.69 ($266.40 + $43.29) Jason Laws 57 of 218 Example: Stock 5th ❑ Consider stock 5th that has just paid a dividend of $10 and is expected to grow this dividend at a rate of 10% p.a. for the next five years and then grow dividends at 2% forever. ❑ If the cost of capital, k, is 5% what is the value of stock 5th? ❑ Think intuitively... Jason Laws 58 of 218 A more complex example ❑ Vornado is a company that has patent rights for a new mobile phone technology that is expected to enable it to generate growth in earnings of 20% for the next 3 years. After that (from the start of year 4) the company expects to see earnings growth drop to a constant rate of 5%. ❑ Assuming that the company pays out 60% of earnings as dividends and that the last dividend payment made by the company was $2.20, calculate an estimate of the current price of Vornado. Assume the required return on equity is 8%. Jason Laws 59 of 218 A more complex example ❑ This is rather a difficult exercise as it has a number of complexities relative to previous examples. ❑ Was the reference to the company paying out 60% of earnings as dividends and growing earnings by 20% a “red-herring”. ❑ Let’s assume some Earnings and Share values. ❑ ❑ ❑ ❑ Year 0 Earnings = £3,666,666.67 Number of shares = 1,000,000 What is the total dividend and the dividend per share? Jason Laws 60 of 218 A more complex example ❑ Dividend = 60% x £3,666,666.67 = £2,200,000 ❑ Dividend per share = £2.2 ❑ Note we arrive at the same answer if the number of shares are 500,000 and the Earnings are £1,833,333.33. ❑ If the firm grows earnings at 20% per annum find the new earnings level, total dividend and dividend per share. ❑ ❑ ❑ ❑ ❑ Year 1 Earnings= Dividends= Number of shares= Dividend per share= £4,400,000.00 £2,640,000.00 1000000 £2.64 Jason Laws 61 of 218 A more complex example ❑ Dividends have increased from £2.20 to £2.64, an increase of: ❑ [£2.64 - £2.20]/£2.20 = 20% ❑ So if a firm is to pay a constant proportion of earnings then earnings growth and dividend growth are the same. ❑ D0= cE/n ❑ D1 = cE(1+g)/n ❑ (D1-D0)/D0 = g Jason Laws 62 of 218 A more complex example ❑ The example can now be expressed as: ❑ Vornado is a company that has patent rights for a new mobile phone technology that is expected to enable it to generate growth in DIVIDENDS of 20% for the next 3 years. After that (from the start of year 4) the company expects to see DIVIDENDS growth drop to a constant rate of 5%. ❑ The last dividend payment made by the company was £2.20, calculate an estimate of the current price of Vornado. ❑ Assume the required return on equity is 8%. Jason Laws 63 of 218 c Earnings DPS0 = n c Earnings (1 + g ) DPS1 = n c Earnings (1 + g ) c Earnings − DPS1 − DPS 0 n n = = c Earnings DPS 0 n c Earnings (1 + g − 1) n =g c Earnings n Jason Laws 64 of 218 A more complex example ❑ Now we need to map out the future stream of dividends: Year Dividend 1 £2.64 2 £2.64 + 20% = £3.17 3 £3.17 + 20% = £3.80 4 £3.80 + 5% = £3.99 ❑ In year 4 dividend growth drops to 5% and continues to grow at 5% into infinity. Jason Laws 65 of 218 A more complex example ❑ We are applying: Dt +i Pt = Et i i =1 (1 + k ) • Finding the sum over the first three periods is trivial: • £2.64/1.08 + £3.17/1.082 + £3.80/1.083=£8.18. • But how do we find the value, in today’s terms, of the dividends received after year 3? • Well if I asked you at the end of year 3, when the dividend of £3.80 had just been paid, and dividends were forecast to grow at 5%, what the price of the share was then I hope you would say: Jason Laws 66 of 218 A more complex example Dt +1 £3.80 1.05 Pt = = = £133.00 k−g .08 − .05 ❑ However that is in “year 3 money” and so to find the value now we must find its present value: ❑ £133.00/1.083 = £105.58 ❑ Adding this to the PV of the dividends from years 1 to 3 we get a current share price of: ❑ £8.18 + £105.58 = £113.76 Jason Laws 67 of 218 Some dividend data ❑ The dividend per share of British American Tobacco since 2002 is: 18 𝑔= 208.55 − 1 = 10.79% 33 Jason Laws Year Div 2002 33 2003 36.3 2004 39.7 2005 43.2 2006 48.7 2007 58.8 2008 58.8 2009 2010 69.7 89.5 2011 104.8 2012 119.1 2013 130.6 2014 144.9 2015 150 2016 155.9 2017 174.6 2018 190 2019 201.05 2020 208.55 68 of 218 Yield to Maturity ❑ A bond’s yield to maturity (ytm) is the interest rate implied by the payment structure, ❑ i.e. the interest rate at which the PV of the income stream equals the current bond price. ❑ Yields are always quoted on an annual basis. ❑ Let T be the maturity of the bond and C(1), C(2) … C(T) be the future cash flows, the ytm is the rate of return which satisfies:- C (1) C (2) C (T ) P= + + ... + 2 1 + y (1 + y ) (1 + y )T ❑ It is clear that there is an inverse relationship between the price of a security and its ytm. If the ytm increases the market price of the bond will decreases Jason Laws 69 of 218 Bond Pricing ❑ When the yield is below the coupon rate, the bond will be priced at a premium to its par value. ❑ When the yield is above the coupon rate, the bond will be priced at a discount to its par value. ❑ Hint: Remember this relationship and so you have an idea of what the price will be before calculating a bond price. ❑ The price-yield relationship is not a straight line; rather it is convex. As yields decline, the price increases at an increasing rate; as the yield increases, the price declines at a declining rate. Jason Laws 70 of 218 Price Yield Curve 400 This is a price yield chart drawn for a 30 year bond with a face value of $100 paying a $10 coupon each year 350 300 Price 250 200 150 100 50 0 0 5 10 15 20 25 Yield to maturity Jason Laws 71 of 218 Zero coupon Bonds ❑ A bond that pays a single cash flow at maturity is referred to as a single coupon bond. ❑ The calculation of the yield to maturity for such bonds is very easy. ❑ For example, consider a zero-coupon bond maturing in ten years with a maturity value of $1,000. ❑ If the current price of the bond is $311.80 then the yield to maturity is: $311.80 = $1,000/(1+ ytm)10 ytm = 12% Jason Laws 72 of 218 Duration ❑ Definition – how sensitive the price is to the change in yield; % change in price for a given small change in the ytm. P (1 + r ) D=− r P ( T Ct 1+ r) T −t =− T =− Ct (1 + r ) t Y t =1 (1 + r ) Y t =1 Ct t ( ) 1 + r t =1 (1 + r ) Ct t ( ) 1 + r t =1 T T tCt t T T ( ) ( ) tC 1 + r 1 + r −t −1 t =1 (1 + r ) t = + tCt (1 + r ) T = = t +1 P T Ct ( ) 1 + r Ct t =1 t =1 t t ( ) ( ) 1 + r 1 + r t =1 t =1 Jason Laws 73 of 218 Duration ❑ Expanding out the summation sign: C1 C2 CT 1 + 2 + ... + T 2 T ( 1+ i) ( ( 1+ i) 1+ i) D= P ❑ The duration can therefore be calculated by computing the Present Value (PV) of the cash flows, and then multiplying them by the time indices. ❑ The Duration of a zero coupon bond is simply its Maturity. CT T T ( 1+ i) D= =T P Jason Laws 74 of 218 Duration ❑ The effective duration of a coupon bond is lower than its stated maturity. ❑ For example a 10 year coupon bond with an 8% ytm. Jason Laws 75 75 of 218 Sample Question 6. (a) Explain the concept of Macaulay duration and explain why a coupon paying bond has a duration less than its maturity. (4 marks) ❑ Observation – Many students simply regurgitated the guide and even included material on the relationship between duration and yield to maturity which was not relevant. Jason Laws 76 of 218 Sample Question (b) Calculate the Macaulay duration of a four year 5% coupon bond where the market interest rate is 4%. Assume the par value of the bond is $1000. And coupons are paid annually (4 marks) Jason Laws 77 of 218 Modified Duration ❑ This measure of Duration can also be used to obtain the percentage change in a bonds price as a function of a basis point change in the yield to maturity. ❑ It can be shown that:- P0 (1 + i ) = −D (1 + i ) P0 ❑ Security firms tend to divide D by (1+i) and call the result modified duration. So that:- D Dm = 1+ i P0 (1 + i ) (1 + i ) = −D = − Dm (1 + i ) = − Dm (1 + i ) (1 + i ) (1 + i ) P0 Jason Laws 78 of 218 Modified Duration P0 = − Dm (1 + i ) P0 ❑ D (Macaulays) = 3.73 years. ❑ Current Price = $1036.30 ❑ Current ytm = 4% ❑ Activity: ❑ Draw bond price against interest rate. ❑ Estimate the new price of the bond if interest rates change to 4.25%, 4.5%, 5%. Jason Laws 79 of 218 Characteristics of Macaulays Duration 1. 2. 3. 4. 5. The Duration of a coupon bond will always be less than its term to maturity. An inverse relationship exists between coupon and duration – a bond with a larger coupon will have a shorter duration as more of the total cash flows come earlier. A bond with no coupon payments will have a duration equal to its term to maturity. A positive relationship generally holds between term to maturity and duration, but duration increases at a decreasing rate with maturity. There is an inverse relationship between ytm and duration. Jason Laws 80 of 218 Jason Laws 81 of 218 Duration versus Beta ❑ If you expected a large drop in interest rates would you prefer to hold high or low duration bonds? Why? P0 = − Dm (1 + i ) P0 ❑ The larger is Duration the more a given change in interest rates is amplified. ❑ Analogously the larger the Beta of a portfolio then the more a movement in the mart is amplified. ❑ Note that just like Beta’s, Durations are linearly additive. N D p = wi Di i =1 Jason Laws 82 of 218 Sample Question (c) An investor decides to construct a bond portfolio made up of $10,000 in the four year 5% coupon bond (Par = $1000, ytm = 4%) and $30,000 in a three year zero coupon bond (par value = $1,000). What is the Macaulay duration of this bond portfolio? (4 marks) Jason Laws 83 of 218 Sample Question (c) An investor decides to construct a bond portfolio made up of $10,000 in the four year 5% coupon bond (in (b) above) and $30,000 in a three year zero coupon bond (par value = $1,000). What is the Macaulay duration of this bond portfolio? (4 marks) Jason Laws 84 of 218 Sample Question (d) For the four year 5% coupon bond described in (b) above, would the Macaulay duration of this bond increase or decrease if the market interest rate increases from 4% to 5%? Explain your answer. (3 marks) Tip: Duration is approximately the slope of the price-yield curve. As interest rates rise the price yield curve gets flatter, hence Duration falls. C1 C2 C2 C4 1´ +2´ + 3´ +4´ 2 3 4 1+ i 1+ i) 1+ i) 1+ i) ( ) ( ( ( D= C1 C2 C2 C4 + + + 2 3 4 1+ i ( ) (1+ i) (1+ i) (1+ i) Jason Laws 85 of 218 Sample Question ❑ As i increases the denominator falls – note negative relationship between price and interest rate. ❑ However in the numerator the weight on the 4, 3, 2 and 1 gets smaller so the numerator gets smaller overall. Hence overall Duration decreases. Jason Laws 86 of 218 Sample Question (e) Estimate, using modified duration, the change in the price of the four year 5% coupon bond if the market interest rate decreases from 4% to 3%. (4 marks) ❑ Here we use the following relationship: P0 (1 + i ) = −D (1 + i ) P0 ❑ P/P =( -3.73 x -0.01)/1.04 = + 0.0373/1.04 = 0.03587 = 3.59% ❑ New P = (1 + 0.03587) x 1036.30 = $1,073.47. ❑ Observation: Many students simply re-estimate rather than using Duration. Jason Laws 87 of 218 Sample Question (f) Estimate, using modified duration, the change in the price of the three year zerocoupon bond if the market interest rate decreases from 4% to 3%. (3 marks) ❑ Here we use the following relationship: P0 (1 + i ) = −D (1 + i ) P0 ❑ ❑ ❑ ❑ P/P =( -3 x -0.01)/1.04 = + 0.03/1.04 = 0.0288 = 2.88% We cannot find the new price without the original price. Original price = $1000/1.043 = $889.00 New P = (1 + 0.0288) x 889.00 = $914.60 Jason Laws 88 of 218 Sample Question (g) Explain why the modified duration measure only gives good estimates when the change in the market interest rate being considered is small. (3 marks) ❑ Note back to the equation to find the price of a bond. It includes squared terms, cubed terms etc. This creates a curved relationship between price and yield. ❑ The greater the maturity of the bond the more curved this line will be. ❑ Not further that Duration is essentially the slope of the line at the current yield to maturity. If we draw a tangent at the current yield we will note that the tangent gets further away from the price-yield curve the further along the ling we go. ❑ Hence, due to the convex relationship between price and yield, if we use Duration to estimate new bond prices, for a large change in interest rates, our estimate will differ substantially compared to the “real price” – see diagram.. Jason Laws 89 of 218 Convexity Price Yield Curve Error in Estimating Price based only on Duration 400 This is a price yield chart drawn for a 30 year bond with a face value of $100 paying a $10 coupon each year 350 300 Price 250 200 Actual Price 150 100 50 0 0 5 10 15 20 25 Yield to maturity Jason Laws 90 of 218 Sample Question ❑ Calculate the price and Macaulay duration of the following two bonds. Note that both bonds pay annual coupons, have par values of £100 and the current market interest rate is 8%. Bond Maturity Coupon A 7 years 3% B 4 years 10% Jason Laws 91 of 218 Sample Question ❑ Bond A Time 1 2 3 4 5 6 7 CF £ 3.00 £ 3.00 £ 3.00 £ 3.00 £ 3.00 £ 3.00 £ 103.00 Price= PV £ 2.78 £ 2.57 £ 2.38 £ 2.21 £ 2.04 £ 1.89 £ 60.10 £ 73.97 D= Mod D= Jason Laws t x PV 2.78 5.14 7.14 8.84 10.20 11.34 420.70 £ 466.14 6.30 5.83 92 of 218 Sample Question ❑ Bond B Time 1 2 3 4 CF PV t x PV £ 10.00 £ 9.26 9.26 £ 10.00 £ 8.57 17.14 £ 10.00 £ 7.94 23.82 £ 110.00 £ 80.85 323.40 Price= £ 106.62 £ 373.62 D= 3.500 Mod D= 3.24 Jason Laws 93 of 218 Sample Question ❑ (b) Use the modified duration formula to estimate the % change in price of bond A and bond B if interest rates were to rise by 1%. P0 = − Dm (1 + i ) P0 • Modified Duration is 5.83 and 3.24 years respectively. • The % change in price would therefore be: • Bond A: -5.83 x 0.01 = -5.83% • Bond B -3.24 x 0.01 - -3.24% Jason Laws 94 of 218 Sample Question ❑ (c) For an investor looking for the lowest risk of capital loss on their investment which bond would you recommend? ❑ Since Bond B has the lowest modified duration then an investor holding this bond will suffer the least in the event of rise in interest rates. A risk averse investor should therefore choose Bond B. Jason Laws 95 of 218 Sample Question ❑ Explain, using appropriate diagrams, why the modified duration formula only provides an estimate of the interest rate sensitivity of a bond ❑ See price-yield relationships for Bond A – note the value of the bond when the coupon = yield = 4%. Jason Laws 96 of 218 Bond Convexity 140 Predicted Change using duration 120 Actual Change - less than predicted due to convex nature of price yield relationship. 100 80 Price A 60 40 20 Change in interest rates 0.245 0.23 0.215 0.2 0.17 0.185 Jason Laws 0.155 0.14 0.125 0.11 0.095 0.08 0.065 0.05 0.035 0.02 0.005 0 97 of 218 Alternative Exercise ❑ It is the the end of June 2021 a UK corporate bond has a coupon rate of 3.5%, par (face) value of £1,000 and will mature in June 2024. ❑ Using the data given above and assuming semi-annual coupons and a discount rate equal to 2.5% p.a., calculate the value of the corporate bond. Jason Laws 98 of 218 Alternative Exercise ❑ It is the end of June 2021 a UK corporate bond has a coupon rate of 3.5%, par (face) value of £1,000 and will mature in June 2024. ❑ Using the data given above and assuming semi-annual coupons and a discount rate equal to 2.5% p.a., calculate the value of the corporate bond. Pcb, sem = PV = 17.5 17.5 17.5 17.5 17.5 1,017.5 + + + + + 1.0125 (1.0125 )2 (1.0125) 3 (1.0125) 4 (1.0125) 5 (1.0125) 6 = £1,028.73 Jason Laws 99 of 218 Alternative Exercise ❑ Calculate the duration of the UK corporate bond assuming annual coupons and annual discount rate. T CF DF CFxDF CFxDFxT 1 35 0.961538 33.65385 33.65385 2 35 0.924556 32.35947 64.71893 3 35 0.888996 31.11487 93.34462 4 35 0.854804 29.91815 119.6726 5 35 0.821927 28.76745 143.8372 6 1035 0.790315 817.9755 4907.853 973.7893 5363.08 ❑ D= 5.51 years Jason Laws 100 of 218 Alternative Exercise ❑ Assume annual interest rates rise by 1% from 4% to 5%. What will be the approximate percentage change in the value of the UK bond assuming annual coupon and annual discount rate? %P − D x i 0.01 = − 5.51x = −0.05298 1+ i 1.04 Jason Laws 101 of 218 Alternative Exercise ❑ Activity: ❑ Compare the problems of estimating future cash flows for government bonds, corporate bonds and common stock. ❑ Write an essay plan include aspects of this essay that you would include. Jason Laws 102 of 218 The term structure of interest rates ❑ Compare and contrast expectations theory, liquidity premium theory and market segmentation in explaining the term structure of interest rates. ❑ A yield curve plots the yields of bonds with different maturity but the same risk. ❑ Usually the yield curve is constructed from government securities – same low default risk, ❑ Yield = yield to maturity = IRR (discount rate) of the bond. Jason Laws 103 of 218 The term structure of interest rates ❑ Example yield curves: Jason Laws 104 of 218 The term structure of interest rates ❑ The current US yield curve is shown below: Jason Laws 105 of 218 The term structure of interest rates ❑ The expectations theory of the term structure of interest rates states that in equilibrium, the long-term rate is a geometric average of today’s short term rate and expected short-term rates in the future. ❑ (1+R)2 = (1+r1) (1+r2) ❑ Where: ❑ ❑ R = annual yield on a 2 year bond r1 = annual return on a 1 year bond r2 = one year forward rate beginning in 1 years time ❑ Arbitrage ensures the LHS = RHS Jason Laws 106 of 218 The term structure of interest rates ❑ Suppose that the yield on a two-year government bond, R is 9% p.a. and the yield on an equivalent one year bond, r1 is 8% p.a. The yield implied on a one year bond held during year two of the two year bond’s life, r2, is given as (assuming £1,000 invested): ❑ £1,000 x (1.09) x (1.09) = £1,188.10 = £1,000 x (1.08) x (1 + r2) ❑ r2 = 10.01% ❑ So when interest rates expected to rise (i.e. r2 > r1) then the long rate, R is greater than the short rate r1 - Hence yield curve is upward sloping. ❑ If r2 < r1 (yields expected to fall) then R < r1 ❑ - Hence downward sloping yield curve ❑ So expectations of future changes in interest rates determine the yield curve ❑ However, the expectations theory does not help explain why the yield curve we observe is normally upward sloping. ❑ (i.e. interest rates not always expected to rise) Jason Laws 107 of 218 The term structure of interest rates ❑ Liquidity premium theory asserts that, in a world of uncertainty, investors and lenders will want to hold assets which can be converted into cash quickly. Therefore they will demand a liquidity premium for holding long term debt. ❑ (1+R)2 = (1+r1) (1+r2 + L) ❑ where L = liquidity premium ❑ - Hence long term rates will normally be greater than short term rates Jason Laws 108 of 218 The term structure of interest rates ❑ The market segmentation theory, suggests that the bond market is actually made up of a number of separate markets distinguished by time to maturity, each with their own supply and demand conditions. ❑ Hence no relationship between yields for different maturities ❑ i.e. long rates determined in market for long maturity bonds, short rates determined in market for short maturity bonds ❑ The theory that best describes the normal upward sloping yield curve = liquidity preference Jason Laws 109 of 218 Default Risk ❑ We have just discussed a number of reasons why the rates of return on bonds may differ across maturity. ❑ It is also the case that bonds of the same maturity offer a different rate of return. ❑ Why ? ❑ Due to the risk of default. ❑ The difference between the return on “safe” bonds (i.e. government) and “risky” bonds is called a default premium. ❑ How do you assess risk ? ❑ Moodys and S&P - evidence suggest that low ranked bonds promise higher returns. Jason Laws 110 of 218 Bond Ratings ❑ Since the primary function of bonds as an investment vehicle is to make fixed payments, it's essential that the company or government issuing the debt has the ability to make all payments on time and in full. Bond ratings evaluate the debt issuer to determine the risk of default. ❑ The leading rating agencies, Standard & Poor's and Moody's Investors Services, assign ratings when a bond is first issued, and that rating helps determine how high the bond's interest rate will be. If the agencies assign a high rating, that means there's little risk of default, so the issuer can obtain a lower interest rate. ❑ While the rating systems of Moody's and S&P differ somewhat, they're more alike than different. Both agencies have investment-grade ratings, which connote a high level of creditworthiness, and speculative ratings, which mean higher risk levels and merit higher interest rates. ❑ Here are Moody's ratings, from highest to lowest. Investment grade: Aaa, Aa1, Aa2, Aa3, A1, A2, A3, Baa1, Baa2, Baa3. Speculative grade: Ba1, Ba2, Ba3, B1, B2, B3, Caa1, Caa2, Caa3, Ca, C1. ❑ Here are S&P's ratings. Investment grade: AAA, AA+, AA, AA-, A+, A, A-, BBB+, BBB, BBB-. Speculative grade: BB+, BB, BB-, B+, B, B-, CCC+, CCC, CCC-, CC, D. Jason Laws 111 of 218 Interest rate risk management ❑ Income Gap Analysis ❑ - the difference between interest sensitive assets and interest sensitive liabilities ❑ The Fed classifies assets and liabilities into the following maturity buckets: ❑ Overnight ❑ 1 day to 3 months ❑ 3 months to 6 months ❑ 12 months to 5 years ❑ Over 5 years Jason Laws 112 of 218 Interest rate risk management ❑ Under the income gap analysis banks report the gap in each maturity bucket: ❑ GAP = RSA – RSL ❑ A positive gap implies sensitive assets > sensitive liabilities: ❑ So if interest rates rise the banks interest revenue will be rising faster than interest cost and net interest margin and income will rise. ❑ Vice versa for a fall in interest rates. ❑ I = GAP x i Jason Laws 113 of 218 Interest rate risk management ❑ Income Gap analysis is essentially a book value accounting cash flow analysis of the gap between interest revenues and interest costs over a period of time. ❑ It therefore ignores the time value of money. ❑ When interest rates change there is a market value effect - change in PV of the cash flows of assets and liabilities and a income effect (interest received or paid) ❑ Even rate insensitive assets and liabilities have a component that is rate sensitive. ❑ A bank receives a runoff cash flow from these rate insensitive items that can be reinvested at the current market rates. ❑ e.g. A fixed rate mortgage repaid within one year. Jason Laws 114 of 218 Interest rate risk management ❑ Can try and estimate the size of the run off cash flows. ❑ Income Gap Analysis also ignores the effects of the changes in interest rates on off balance sheet items. Jason Laws 115 of 218 Interest rate risk management ❑ Consider the following balance sheet (in millions). ❑ Assume further that 20% of fixed rate mortgages are repaid within the year and 20% of savings deposits are rate sensitive. Assets Liabilities and Equity Variable Rate Mortgages 20 Money Market Deposits 5 Fixed Rate Mortgages 25 Savings deposits 20 Commercial Loans 50 Variable Rate CD (< 1 year) 30 Physical Capital 5 Equity 45 Total Assets 100 Total Liabilities 100 Jason Laws 116 of 218 Interest rate risk management ❑ Assume further that 20% of fixed rate mortgages are repaid within the year and 20% of savings deposits are rate sensitive. ❑ RSA = 20 + .2 x 25 + 50 = 75m ❑ RSL = 5 + .2 x 20 + 30 = 39m ❑ GAP = RSA – RSL = 75 – 39 = 36m ❑ If interest rates increase from 8 per cent to 9.5 per cent: ❑ I = GAP x i = 36 x (.015) = 0.54m ❑ Or: ❑ Increase in income on assets = 0.015 x 75 = 1.125m ❑ Increase on payments on liabilities = 0.015 x 39 = 0.585m ❑ Increase in net income = 0.54m Jason Laws 117 of 218 Interest rate risk management ❑ A market value -based model of measuring and managing interest rate risk is the so called duration gap analysis. ❑ Under duration gap analysis banks are able to take into account the effects of changes in interest rates on both income and market value. ❑ Banks are also able to immunise their balance sheets against interest rate risk. ❑ In order to implement duration analysis bank managers need to determine the duration of all assets and liabilities. ❑ Fortunately, like beta (to follow) and NPV duration is linearly additive. N D p = wi Di i =1 Jason Laws 118 of 218 Interest rate risk management ❑ The overall duration gap can then be calculated as: DURgap L = DURassets − DURliabilitie s A ❑ Where L = market value of liabilities and A = market value of assets. ❑ Note that when the gap is zero then that part of the banks balance sheet is said to be “immunised against unexpected changes in interest rates” ❑ Immunisation therefore allows banks to lock into a fixed yield. Jason Laws 119 of 218 Interest rate risk management ❑ Duration gap analysis can be used to calculated the change in the market value of net worth as a percentage of total assets induced by a change in interest rates: NW i = − DURgap A 1+ i ❑ Note assumes a parallel “shift” in the term structure of interest rates. ❑ Term structure could be upward sloping/downward sloping or flat as long as shift is parallel. Jason Laws 120 of 218 ❑ The following balance sheet information is available for Bank Plus (amount in $millions and duration is years). DURgap L = DURassets − DURliabilitie s A NW i = − DURgap A 1+ i Amount Duration Assets Capital 2400 Residential Mortgages Variable Rate 1600 9.1 Fixed Rate 1400 5.1 Commercial Loans 5600 3.5 Money market deposits 3500 1.3 Savings deposits 2800 2.3 Variable Rate CD’s (> 1year) 1200 3.1 Equity 3500 Liabilities If interest rates changed from 4% to 3.5% what is the change in net wealth as a proportion of equity? Jason Laws 121 of 218 Interest rate risk management ❑ Total Assets (inc. Capital) = 11,000 ❑ Total Liabilities (exc. Equity) = 7,500 ❑ Duration of Assets = (1600/11000) x 9.1 + (1400/11000) x 5.1 + (5600/11000) x 3.5 = 3.75 ❑ Duration of Liabilities = (3500/7500) x 1.3 + ( 2800/7500) x 2.3 + (1200/7500) x 3.1 = 1.96 ❑ Duration Gap = 3.75 - (7500/11000) x 1.96 = 2.42 years ❑ If interest rates changed from 4% to 3.5% then the change in net wealth as a proportion of equity is: ❑ -2.42 x (-0.005)/(1.04) = 1.16% Jason Laws 122 of 218 Sample Question ❑ The following balance sheet is available (amounts in $m and duration in years) for ABC bank: Amount Duration Loans 3,400 4.6 T-bonds 600 2.1 Deposits 3,300 2.3 Equity 700 ❑ What is the average duration of all the assets? What is the average duration of the liabilities? Jason Laws 123 of 218 Sample Question ❑ Total Assets ❑ Total Liabilities (exc. Equity) = = 4000 3300 ❑ Duration of Assets = [(3400)/4000 x 4.6]+[(600/4000)x 3.1] = = 0.85 x 4.6 + 0.15 x 3.1 = 4.38 ❑ Duration of Liabilities= 2.30 Jason Laws 124 of 218 Sample Question ❑ What is the duration gap for ABC bank? Explain what this duration gap implies for ABC bank. DURgap L = DURassets − DURliabilitie s A ❑ Gap = 4.38 – (3300/4000) x 2.3 = 2.478. ❑ What does 2.478 mean? ❑ The average duration of assets is greater than the average duration of liabilities, thus asset values change by more than liability values. Jason Laws 125 of 218 Sample Question ❑ Positive DGAP ❑ Indicates that assets are more price sensitive than liabilities, on average. ❑ Thus, when interest rates rise (fall), assets will fall proportionately more (less) in value than liabilities and equity will fall (rise) accordingly. ❑ Negative DGAP ❑ Indicates that weighted liabilities are more price sensitive than weighted assets. ❑ Thus, when interest rates rise (fall), assets will fall proportionately less (more) in value that liabilities and the equity will rise (fall). Jason Laws 126 of 218 Sample Question A+L=E So : é ù é Di Di DEquity = ê-DA ´ ´ Aú - ê-DL ´ ´ (1 + i) (1 + i) ë û ë Dividing through by A: é DEquity Di ù é Di Lù = ê-DA ´ ´ ú ú - ê-DL ´ A (1 + i) û ë (1 + i) Aû ë é DEquity Di Lù Di =´ êDA - DL ´ ú = -DGAP ´ A (1 + i) ë Aû (1 + i) Jason Laws ù Lú û 127 of 218 Sample Question ❑ (c) What is the forecast impact on the market value of ABC bank resulting from a 0.25% decrease in interest rates from 3.5% to 3.25%? ❑ ❑ The potential loss/gain to equity holders’s net worth/Equity (as a percentage of assets is): NW i = − DURgap A 1+ i ❑ ∆NW/A = -Durgap * ∆i / (1+i) = -2.478*(-0.0025) /(1+0.035) = +0.00598 ❑ ∆NW = 0.00598*A = -0.00598*4,000 = 23.92 ❑ i.e. NW rises to 700+23.92 = 723.92 Jason Laws 128 of 218 Sample Question ❑ (e) How can income gap analysis help banks manage interest rate risk? Discuss the weaknesses of income gap analysis. What is runoff cashflow and how does this affect the income gap analysis? ❑ Banks identify the difference between rate sensitive assets and rate sensitive liabilities for a particular time horizon. This difference (the income gap) is then multiplied by the forecast change in interest rate for the time horizon considered. This will identify the net change in income (net interest) over the forecast time horizon. Jason Laws 129 of 218 Sample Question ❑ Problems: ❑ (1) Ignores the effect of interest rate changes on market values of assets (through discounting) ❑ (2) Even rate insensitive assets and liabilities will have a component that is sensitive e.g. a proportion of fixed rate mortgages will be repaid within the forecast time horizon – these repaid mortgages can be re-invested at a (rate sensitive) interest rate. Jason Laws 130 of 218 Sample Question ❑ (3) Is applied to the balance sheet therefore ignores the income effects of changes in interest rates on off-balance sheet items e.g. overdrafts – a proportion of which may become balance sheet items over the forecast period). ❑ Runoff cashflow is the estimate of the cash flow from rate insensitive assets and liabilities that will become rate sensitive over the forecast period. These runoff cash flows can be reinvested at the current interest rate. Examples should be given to illustrate. Jason Laws 131 of 218 Sample Question 7. (a) Briefly explain each of the following terms: (i) efficient frontier (ii) feasible region (iii) security market line (iv) tangency portfolio (8 marks) ❑ Observation: Many students did not draw diagrams. For these definitions they were essential. Evidence of confusion between the security market line and capital market line was also evident. Jason Laws 132 of 218 Historical record ❑ We now examine the performance of three important US financial investments. ❑ Standard and Poor 500 – an index of the 500 largest US companies. ❑ Long term US treasury bonds. ❑ US treasury bills – a portfolio of 3 month US treasury bills. ❑ See https://www.credit-suisse.com/media/assets/corporate/docs/aboutus/research/publications/credit-suisse-global-investment-returns-yearbook-2020summary-edition.pdf Jason Laws 133 of 218 Historical Record Jason Laws 134 of 218 Historical Record Jason Laws 135 of 218 Historical Record Jason Laws 136 of 218 Historical record ❑ Given the historical record, why would any investor buy anything other than common stocks? ❑ If you look closely at the figure you will see the answer ❑ Risk. ❑ The long-term government bond portfolio grew more slowly than did the stock portfolio, but it also grew much more steadily. ❑ The common stocks ended up on top, but as you can see, they grew more erratically much of the time. Jason Laws 137 of 218 Return variability Frequency chart of SPX monthly returns (1928-2021) 350 300 250 200 150 100 50 0 Stock Market Crash of 1987= -21.8% Jason Laws 138 of 218 Risk and Return ❑ As you can see a variety of return outcomes are possible. ❑ Accordingly the return measure of a risk asset is considered to be a random variable. ❑ For analytical purposes the uncertainty of the return is measured by the expected (average) return. ❑ To provide a quantitative measure of the expected we normally use the weighted average of all possible returns where the weights are the probabilities of the occurrence of that return: ❑ E(R) = p1R1 + p2R2 + … + pnRn ❑ Ri = return of state of nature i. n = number of possible states of nature; pi = probability of occurrence of return Ri. ❑ However in reality probabilities are not known so instead we use the historical returns and the number of past observations. ❑ The variation around the expected return is a measure of the risk of a security. Jason Laws 139 of 218 Risk and Return ❑ To provide a quantitative measure of the degree of possible deviations we use the variance, and its square root – standard deviation. ❑ Following the previous notation: = p1 (R1 − E ( R) )2 + p2 (R2 − E ( R) )2 + ... pn (Rn − E ( R))2 ❑ e.g. Consider a stock second class with a return of 18% half the time and 10% the other half of the time. The Expected Return is 14% and the standard deviation is: = 0.5(18 − 14)2 + 0.5(10 − 14)2 = 0.5(4)2 + 0.5(− 4)2 = 0.5 16 + 0.5 16 = 16 = 4 Jason Laws 140 of 218 The Risk and Return of a Portfolio ❑ A portfolio is a combination of different assets held by an investor. ❑ The share of each individual asset over the total value of the portfolio is referred to as the portfolio weight. ❑ Example ❑ You have 10,000 shares in Singapore Telecom (price – 2.53) and 5,000 shares in Singapore Airlines (price – 11). What is the weight of each asset in the portfolio? Jason Laws 141 of 218 The Risk and Return of a Portfolio ❑ A portfolio is a combination of different assets held by an investor. ❑ The share of each individual asset over the total value of the portfolio is referred to as the portfolio weight. ❑ Example ❑ You have 10,000 shares in Singapore Telecom (price – 2.53) and 5,000 shares in Singapore Airlines (price – 11). What is the weight of each asset in the portfolio? ❑ Answer ❑ Portfolio Value = 10000 x 2.53 + 5000 x 11 = 25,300 + 55,000 = 80,300 ❑ SAIR weight = 55,000/80,300 = 68.5%, ❑ TELC weight = 31.5% Jason Laws 142 of 218 The Risk and Return of a Portfolio ❑ The expected return of a portfolio is simply the weighted average of the expected returns of the individual stocks of which the portfolio is composed, where the weights are the portfolio weights: ❑ E(R) = w1E(R1) + w2E(R2) + … + wnE(Rn) ❑ Where wi is the weight of each stock. Jason Laws 143 of 218 Myth Fact In order to reduce portfolio risk I need the correlation between assets to be negative. If I form a portfolio between two assets with a correlation of -1 I can obtain a risk free portfolio. Is the return zero? The minimum variance portfolio is optimal for everybody? Jason Laws 144 of 218 Myth In order to reduce portfolio risk I need the correlation between assets to be negative. If I form a portfolio between two assets with a correlation of -1 I can obtain a risk free portfolio. Is the return zero? The minimum variance portfolio is optimal for everybody? Fact x x x Jason Laws 145 of 218 The Risk and Return of a Portfolio ❑ When computing the variance we also have to concern ourselves with how the asset returns vary together - the covariance. ❑ If returns tend to move in opposite directions then this reduces the overall variability of the portfolio. ❑ But if returns tend to move in the same direction then the variability of the portfolio is increased. ❑ In the analysis that follows 12 refers to the covariance between assets 1 and 2. Jason Laws 146 of 218 The Risk and Return of a Portfolio ❑ If 12 is positive, when the return on the first asset is greater than the mean value then the return on the second asset, is also, on average, greater than its mean value. ❑ And vice versa when the return on the first asset is less than the mean value. ❑ If 12 is negative, the returns on assets 1 and 2 tend to move in opposite directions and offset each other. ❑ If the return on the asset is above the mean, the return on the second will, on average, be below the mean. ❑ The fact that the returns on the assets move in opposite directions reduces the variability of the portfolio. Jason Laws 147 of 218 The Risk and Return of a Portfolio ❑ In general, the variance of a portfolio of assets is given by:N N N 2 2 2 p p i i i k ik i =1 i =1 k =1 k i ❑ and its standard deviation is P = P2. ❑ A measure of the association between two assets which is always in the range +1 to -1 is the correlation coefficient. This is defined as:- = Var( R ) = X + X X 12 12 = 1 2 Jason Laws The covariance and correlation coefficient always have the same sign 148 of 218 Mean, Variance and Covariance of a Portfolio ❑ When there are two assets in a portfolio the variance of a portfolio is give by: ❑ Or: p2 = x12 12 + x22 22 + 2 x1 x2 12 p2 = x12 12 + x22 22 + 2 x1 x2 12 1 2 ❑ Since: 12 12 = 1 2 Jason Laws 149 of 218 Negative correlation ❑ When correlation between two assets is “-1” you have the ultimate in diversification benefits and a risk-free portfolio. ❑ Perfect negative correlation gives a mean combined return for two securities over time equal to the mean for each of them, so the returns for the portfolio show no variability. ❑ Any returns above and below the mean for each of the assets are completely offset by the return for the other asset, so there is no variability in total returns, that is, no risk, for the portfolio. Jason Laws 150 of 218 Role of correlation ❑ Consider the following market data:- Ave= sd= GM_% 19.99% 39.3% correl= 0.122 WMT_% 15.20% 20.3% ❑ graphically we consider the diversification effects of combining these two assets under different assumptions about the correlation coefficient. Jason Laws 151 of 218 Jason Laws 152 of 218 Correlation = - 1 Is there a combination of assets with zero risk ? Jason Laws 153 of 218 Jason Laws 154 of 218 A B C Jason Laws 155 of 218 Efficient Portfolios ❑ Despite correlation between the two assets being positive diversification can still reduce the risk of the portfolio. ❑ On the diagram on the previous slide we can identify a number of “inefficient portfolios”. ❑ The portfolios which lie on the line from B to C are inefficient, since for each one of them we can find an alternative portfolio, on the line from B to A, which has the same standard deviation but a higher expected return. ❑ The set of efficient portfolios is now formed on the line from B to A. Jason Laws 156 of 218 Portfolio Risk Exercise ❑ Consider the following information about two stocks: Stock Expected Return Standard Deviation 1 8% 15% 2 5% 11% ❑ Calculate the risk and returns of the following portfolios (correlation = +0.2): Portfolio Proportion in 1 Proportion in 2 1 0 100 2 30 70 3 75 25 4 100 0 Jason Laws 157 of 218 Portfolio Risk Exercise ❑ E(RP) = w1*E(R1) + w2*E(R2) ❑ ❑ ❑ ❑ E(R1) = 5% E(R2) = 0.3*8 + 0.7*5 = 5.9% E(R3) = 0.75*8 + 0.25*5 = 7.25% E(R4) = 8% Jason Laws 158 of 218 Portfolio Risk Exercise 22 = wX2 * x2 + wY2 * 2 + Y2 + 2 * WX * WY * X * Y*CORRXY 1 = 11% 22 = .32 * .152 + .72 * .112 + 2 * .3 * .7 * .15* .11* .2 = 0.00934 2 = 0.0966 = 9.66% 32 = .752 * .152 + .252 * .112 + 2 * .75 * .25 * .15* .11* .2 = 0.01465 3 = 0.1203 = 12.10% 4 = 15% Jason Laws 159 of 218 Minimum Variance Portfolio ❑ With the exception of the perfect correlation diagram we can see that there is one point where standard deviation is minimised. ❑ Calculus is appropriate here. ❑ Recall:- = X + (1 − X 1 ) + 2 X 1 (1 − X 1 ) 12 1 2 2 P 2 1 2 2 1 2 2 ❑ multiplying out the brackets: = X + − 2 X 1 + X + 2 P 2 1 2 1 2 2 2 2 2 1 2 2 2 X 1 12 1 2 − 2 X 12 ! 2 2 1 Jason Laws 160 of 218 Minimum Variance Portfolio ❑ We should now differentiate 2p with respect to X1. 2 2 2 2 P 1 1 2 1 2 1 = 2 X − 2 + 2 X + X 2 12 1 2 − 4 X 1 12 1 2 ❑ Setting this equal to zero and dividing through by 2. ¶s P2 2 2 2 = X1 s1 + X1 s 2 - 2X1 r12s 1s 2 = s 2 - r12s 1s 2 ¶X! P2 = X1 (12 + 22 − 2 12 1 2 ) = 22 − 12 1 2 X 1 Jason Laws 161 of 218 Minimum Variance Portfolio ❑ Hence:- 22 − 12 1 2 X1 = 2 2 1 + 2 − 2 12 1 2 ❑ and X2 = 1 – X1. ❑ A special case of this result is when we have perfect negative correlation between assets (12 = -1). Then:2 2 1 2 2 2 1 2 1 2 2 2 1 2 1 2 1 2 1 2 + ( + ) X = = = + + 2 ( + ) ( + ) ❑ Activity: Find the minimum variance portfolio for the previous exercise. Jason Laws 162 of 218 Indifference Curves ❑ Every individual will exhibit unique preferences for risk and return and so everyone has a unique set of indifference curves. ❑ Consider the following hypothetical set of indifference curves. ❑ Consider an individual holding portfolio W. ❑ How much extra return would they require in order to increase the risk to 20% (i.e. Portfolio Z)? ❑ All the risk-return combinations along the indifference curve offer the same level of desirability. Jason Laws 163 of 218 Indifference Curves ❑ Now let us consider a map of indifference curves. ❑ An individual would be indifferent between points W and Z but not between: ❑ W and S – as S gives a higher return for the same level of risk. ❑ Z and T – as T gives the same return but for a lower level of risk. ❑ The further “north west” the indifference curve, the higher the desirability. Jason Laws 164 of 218 Indifference Curves ❑ Other investors may be less risk averse than the individual we have just considered and therefore the increase in return required to compensate for risk may be less. The indifference curves will therefore have a lower slope. ❑ Alternatively they may less tolerant of risk and required large increases in return for small increases in risk. The indifference curves for these individuals will have a steep slope. Jason Laws 165 of 218 Indifference Curves ❑ In the absence of a risk free asset investors will identify the efficient frontier which is that part of the meanstandard deviation frontier that contains portfolios that give the highest expected return for a given risk. ❑ That is all portfolios to the right of the minimum variance portfolio on the mean standard deviation frontier. ❑ Investors will locate somewhere on this efficient frontier. ❑ An investor will locate on the efficiency frontier according where his/her indifference curve is just tangential to the efficient frontier. Jason Laws 166 of 218 Expected Return, kp IB I 2 B 1 Efficient Frontier Optimal Portfolio Investor B IA 2 IA 1 Optimal Portfolio Investor A Risk p Jason Laws 167 of 218 Expected Return, kp IB I 2 B 1 Efficient Frontier Optimal Portfolio Investor B IA 2 IA 1 Optimal Portfolio Investor A Risk p Jason Laws 168 of 218 Feasible Region/Set ❑ Investors first need to identify the mean standard deviation frontier for N risky assets using every possible combination of assets. This defines the feasible region. The outer points of this feasible region (left and up) define the mean standard deviation frontier. Expected Portfolio Return, p Efficient Set Feasible Set Jason Laws Risk, p 169 of 218 Diversification and Risk ❑ What happens to the variance of a portfolio as the number of assets increases ? ❑ Recall again the formula for calculating the variance of a portfolio:N N N 2p = Var ( R p ) = X i2 i2 + X i X k ik i =1 i =1 k =1 k i ❑ assume further that all assets are held in equal proportions such that Xi = 1/N. N 1 2 N N 1 1 1 1 = 2 i + ik = N N N i =1 N i =1 k =1 N 2 p k i Jason Laws 1 1 ( N − 1) N N + ik N N ( N − 1) i =1 k =1 i =1 N 2 i k i 170 of 218 Diversification and Risk ❑ We could write expressions for the average covariance and variances as:- −2 1 i = N N N N 1 2 ik i ; ik = N ( N − 1) i =1 k =1 i =1 _ k i ❑ note there are N(N-1) elements inside the double summation signs since each of the N assets can be combined with any of the remaining N-1 assets. ❑ Thus:- _ 1 N − 1 2 p = i + ik N N _ 2 Jason Laws 171 of 218 Diversification and Risk (N − 1) N 1 1 = 2 N ( N − 1) N ❑ Note that:❑ this is the “trick” used to write the last expression. ❑ Look at the last expression on the previous slide:❑ What happens to contribution to portfolio variance of the individual variances as N gets large ? ❑ It goes to zero ! ❑ Therefore, in the limit portfolio variance equals average covariance. ❑ Individual risks of securities can be diversified away but contribution to total risk of the covariance cannot. Jason Laws 172 of 218 Jason Laws 173 of 218 Even a little diversification can substantially reduce variability Unique risk Market risk Jason Laws 174 of 218 How Diversification reduces risk ❑ Market risk stems from the fact that there are many other economy-wide perils which threaten businesses. That is why stocks have a tendency to move together. ❑ That is why investors are exposed to market uncertainties, no matter how many stocks they hold. ❑ Unique risk may also be called unsystematic risk, residual risk, specific risk or diversifiable risk. ❑ Market risk may be called systematic risk or undiversifiable risk. Jason Laws 175 of 218 Market risk ❑ If you want to know the contribution of an individual security to the risk of a well diversified portfolio, it is no good thinking about how risky that security is in isolation – you need to measure its market risk. ❑ i.e. how sensitive it is to market movements. ❑ This sensitivity is called beta (). ❑ Stocks with betas greater than 1 tend to amplify the overall movements of the market. ❑ Stocks with betas between 0 and 1 tend to move in the same direction as the market, but not as far. ❑ The market is the portfolio of all stocks and so has a beta of 1. Jason Laws 176 of 218 Market Risk - Beta ❑ Some sample Beta’s for S&P 500 companies (source finance.yahoo.com – March 22nd 2021): ❑ KELLOGG 0.64 ❑ WAL-MART STORES 0.47 ❑ WALT DISNEY 1.21 ❑ Goodyear Tire Company 2.26 ❑ Verizon 0.47 ❑ What would be predicted to happen to the returns of each of these stocks if the market went up 10% or down 10%? Jason Laws 177 of 218 Methods for Estimating Betas ❑ Collect data on historical returns of all the securities which could enter the portfolio; ❑ Compute the average return and variance of each security; ❑ Collect data on the market index; ❑ Compute the average return and variance of the market index; ❑ Compute the covariance of each security with the market index; ❑ estimate the alpha and beta using:❑ Beta = im/2m ; ❑ alpha = ave(Ri) - i x ave(Rm) ❑ This is identical to fitting a trend line in Excel with the return on the stock on the vertical axis and the return on the market on the horizontal axis and hypothesising the relationship: ❑ Ri=a + Rm Jason Laws 178 of 218 Jason Laws 179 of 218 CAPM Assumptions ❑ The CAPM assumes that:❑ Investors rely on two factors in making their decisions: expected return and variance. ❑ Investors are rational and risk averse and subscribe to Markovitz methods of portfolio diversification. ❑ Investors all invest for the same period of time. ❑ They share all expectations about assets. ❑ There is a risk free investment, and investors can borrow and lend any amount at the risk-free rate. ❑ Capital markets are completely competitive and frictionless. Jason Laws 180 of 218 Capital Market Theory ❑ The major factor that allowed Portfolio theory to develop into capital market theory is the concept of the risk free asset. ❑ Risk free asset – asset with zero variance ❑ Risky Asset – one from which future returns are uncertain. ❑ Recall that covariance between two sets of returns is: 1 n ij = (Ri − E ( Ri ) )(R j − E ( R j ) ) n i =1 ❑ But because the returns for the risk free asset are certain, Ri – E(Ri) = 0. Hence the covariance (and correlation) will also be zero. Jason Laws 181 of 218 Capital Market Theory ❑ What happens to expected return and standard deviation of returns when you combine risky assets and risk free assets? ❑ Expected Return = wRFRF + (1-wRF) E(Ri) ❑ Variance = (1-wRF)2i2 ❑ standard deviation = (1-wRF)i ❑ Standard deviation is therefore a linear proportion of the standard deviation of the risky asset portfolio. ❑ Because both expected return and standard deviation are linear combinations a graph of possible risks and returns looks like a straight line between the two assets. Jason Laws 182 of 218 Capital Market Theory ❑ The introduction of a risk free security to the universe of risky securities allows investors to move to higher levels of utility as they can allocate their wealth to the risk free security and a portfolio of risky assets on the efficient frontier. This creates a new (linear) efficient frontier – the CML. ❑ The line from the risk free rate to the efficient frontier that is the steepest (i.e. where the line is just tangential to the (old) efficient frontier is the optimal new efficient frontier – as this will yield the set of combinations of risk free asset and efficient risky portfolio that delivers the highest expected return for a given risk. ❑ Every combination of the risk free asset and the Markowitz efficient portfolio (M) is shown on the Capital Market Line (CML). Jason Laws 183 of 218 CML P2 P1 Jason Laws 184 of 218 Capital Market Theory ❑ All the portfolios on the capital market line are feasible for the investor to construct. ❑ Portfolios to the left of M represent combinations of risky assets and the risk free asset. ❑ Portfolios to the right of M included purchases of risky assets made with funds borrowed at the risk free rate. ❑ Such a portfolio is called a leveraged portfolio. ❑ Compare portfolio P1, on the Markowitz efficient frontier, with portfolio P2 which is on the CML. ❑ Note that for the same risk the expected return is greater for P2 than P1. Jason Laws 185 of 218 Capital Market Theory ❑ A risk averse investor will prefer P2 to P1. This is true for all but one portfolio on the line: portfolio M. ❑ With the introduction of the risk free asset we must now modify the conclusion from portfolio theory such that we now say that an investor will select a portfolio on the line representing a combination of borrowing or lending at the risk free rate and purchases of the Markowitz efficient portfolio, M. ❑ The portfolio that includes all risky assets is referred as the market portfolio. It includes all risky assets – stocks, bonds, real estate, options etc. ❑ This result is known as the two-fund separation theorem. ❑ The tangent portfolio will therefore be the portfolio of risky assets that all investors will choose to invest in. ❑ Investors will locate somewhere on the new efficient frontier (CML) according to their preference for risk. Jason Laws 186 of 218 Decomposing Total Risk Using the Market Model ❑ Recall:❑ Ri = ai + iRm + ei ❑ Taking the variance of this:❑ Var(Ri) = i2 Var(Rm) + Var(ei) ❑ Such that the total risk is measured by:❑ Systematic/market risk = i2 Var(Rm) ❑ Unsystematic/unique risk = Var(ei) ❑ Another product of the statistical technique used to estimate beta is the percentage of systematic risk to total risk. ❑ In statistical terms this is measure by the coefficient of determination or the R-squared value. Jason Laws 187 of 218 The Security Market Line ❑ Noting that:- iM i = 2 M ❑ If this is substituted into our CML them we have the beta version of the SML or CAPM:- E ( Ri ) = RF + i E ( RM ) − RF ❑ This equation states that given the assumptions of the CAPM, the expected (or required) return on an individual asset is a linear function of its index of systematic risk as measured by beta. ❑ The higher the beta, the higher the expected return. Jason Laws 188 of 218 The Security Market Line ❑ Notice also that only an assets Beta determines its expected return. ❑ When Beta = 0:- E ( Ri ) = RF + 0 E ( RM ) − RF = RF ❑ When Beta = 1:- E ( Ri ) = RF + 1 E ( RM ) − RF = E ( RM ) Jason Laws 189 of 218 The Security Market Line Expected Return E(Rm) Rf 1 Jason Laws Beta 190 of 218 The Security Market Line ❑ Even firms within one industry have different levels of debt, and increasing debt increases leverage. Increasing leverage increases beta. ❑ We can "synthesize" a security with a beta of 1.3 by borrowing 30% of our wealth, and investing the total in an asset with a beta of one. ❑ Suppose, for instance, that investor A hold a portfolio of $100 invested in an S&P 500 index trust. ❑ In order to increase his expected return, investor B, who also has $100, borrows an additional $30 for one year at 0% interest, and invests $130 in the S&P 500 index trust. ❑ What will happen if the S&P goes up by next year? ❑ A will have $110, for a gain of 10%, ❑ while B will have $143 - $30, leaving a gain of 13%! Jason Laws 191 of 218 The Security Market Line ❑ What will happen if the market drops by 10% next year? ❑ A will have $90, a loss of -10%, ❑ while B will have a net loss of $87, a 13% loss. ❑ B's leverage increased his exposure to market risk. ❑ Leverage can be used by corporations as well as individuals to increase their expected returns, and in fact, this is exactly what some firms do. ❑ Even if they are in a low-beta business, such as a utility, they can increase expected return through leverage Jason Laws 192 of 218 Portfolio Risk Exercise Introducing a riskless asset ❑ e.g. a government bill ❑ The risk free rate is the certain return on the riskless asset: ❑Note σF = zero ❑ Assume investors can lend/borrow unlimited funds at risk free rate. ❑ An investor will therefore invest some funds in a portfolio of risky assets and lend/borrow at RFR. ❑ What portfolio of risky assets? Jason Laws 193 of 218 E(R port ) D M C RFR B A port B superior to A (in risk-return trade-off) Optimal point = M Jason Laws 194 of 218 Portfolio Risk Exercise ❑ To attain a higher expected return than is available at point M (in exchange for accepting higher risk) ❑ Either invest along the efficient frontier beyond point M, such as point D ❑ Or, add leverage to the portfolio by borrowing money at the risk-free rate and investing in the risky portfolio at point M Jason Laws 195 of 218 E(R port ) ing d n Le ing w o rr Bo L CM M RFR port Everybody will want to invest in Portfolio M and borrow or lend to be somewhere on the CML n Risk averse investors will locate to the left of M n Risk tolerant investors will locate to the right of M Therefore this portfolio M must include ALL RISKY ASSETS Jason Laws 196 of 218 Portfolio Risk Exercise Distinguish between the capital market line and the security market line ❑ The CML will contain all efficient portfolios and is defined in E(R), standard deviation space. ❑ The SML will contain all securities and portfolios and is defined in E(R), Beta space. ❑ Both are anchored at the riskless portfolio. ❑ Both contain the market portfolio. Jason Laws 197 of 218 Jason Laws 198 of 218 Beta of a portfolio ❑ For a portfolio of N assets the historical beta is simply:❑ a weighted average of the observed historical beta is simply a weighted average of the observed historical betas for the individual assets in the portfolio. N b P = åwi bi i=1 ❑ So for example, the historical beta for a portfolio consisting of 30% of Microsoft (beta = 0.81) and 70% of Sun Microsystems (beta =0.47) is:❑ .3 x 0.81 + .7 x .47 = 0.57. Jason Laws 199 of 218 Beta of a portfolio ❑ Example ❑ You own 200,000 shares of Sembcorp Industries (Beta = 0.887, Price = 1.78 ), 30,000 of DBS Group (Beta = 1.172, Price = 28.38), 100,000 of Wilmar Int’l (Beta = 0.866, Price = 5.39) and 200,000 of Comfort Del Gro (Beta = 1.114, Price = 1.7). ❑ What is the Beta of your portfolio? Jason Laws 200 of 218 Beta of a portfolio ❑ Example ❑ You own 200,000 shares of Sembcorp Industries (Beta = 0.887, Price = 1.78 ), 30,000 of DBS Group (Beta = 1.172, Price = 28.38), 100,000 of Wilmar Int’l (Beta = 0.866, Price = 5.39) and 200,000 of Comfort Del Gro (Beta = 1.114, Price = 1.7). ❑ What is the Beta of your portfolio? ❑ Solution Stock Sembcorp DBS Group Wilmar International Comfort Del Gro SP 1.78 28.38 5.39 1.7 N 200000 30000 100000 200000 Beta 0.887 1.172 0.866 1.114 Jason Laws N x SP SGD 356,000.00 SGD 851,400.00 SGD 539,000.00 SGD 340,000.00 SGD 2,086,400.00 wi 17.06% 40.81% 25.83% 16.30% wi x Beta 0.151348 0.47826 0.223722 0.181538 1.03 201 of 218 Sample Question ❑ Consider the following portfolio composed of three stocks (X, Y, Z): Stock Quantity Price Beta X 100 1.5 0.7 Y 120 1.7 0.95 X 210 1.1 1.05 ❑ What is the beta of this portfolio? [5 marks] Jason Laws 202 of 218 Sample Question Stock Quantity Price Beta X 100 £ 1.50 0.7 Y 120 £ 1.70 0.95 X 210 £ 1.10 1.05 Value of Holding £ 150.00 £ 204.00 £ 231.00 £ 585.00 Weight 25.64% 34.87% 39.49% 100.00% Weight x Beta 0.179487179 0.331282051 0.414615385 0.925 ❑ Hence the Beta of the portfolio is 0.925. Jason Laws 203 of 218 Sample Question if Stock market rises by 10% X rises by: 7.00% Y rises by: 9.5% Z rises by: 10.50% New Price= New Price= New Price= £ 1.61 New Value= £ 1.86 New Value= £ 1.22 New Value= New Total= £ 160.50 £ 223.38 £ 255.26 £ 639.14 or directly; Portfilio beta x 10%= 9.25% New Total= £ 639.14 Jason Laws 204 of 218 Sample Question ❑ You are given the following information: (a) A stock with a beta of 0 has an expected return of 6% (b) A portfolio made up of 50% invested at the risk free rate and 50% invested in the market portfolio has an expected return of 9%. ❑ What is the expected return of the market portfolio? ❑ Beta of zero implies that the risk free rate is 6%. From (b): ❑ 9% = 0.5 x 6% + 0.5 x E(Rm) ❑ 6% = 0.5 x E(Rm) ❑ E(Rm) = 12% Jason Laws 205 of 218 Alternative Exercise Securities I and J lie on the security market line: I J Expected return 14% 18% Beta 1 1.5 Assume the CAPM holds. I. What is the risk free rate of return and the risk premium on the market portfolio? II. Security K has an expected return of 24% and a Beta of 1.8. What is likely to happen to the return and price of security K? Explain your answer. Jason Laws 206 of 218 Alternative Exercise (i) CAPM: E(R*) = Rf + β[E(RM) – Rf] For I: 14 = Rf + 1 [E(RM) – Rf] (1) For J: 18 = Rf + 1.5 [E(RM) – Rf] (2) Subtracting equation (1) from (2) 4= 0.5 [E(RM) – Rf] [E(RM) – Rf] = 4/0.5 = 8% Jason Laws 207 of 218 Alternative Exercise Substituting this result into equation (1) gives: 14 = Rf + 1 * 8 Rf = 14 – 8 = 6% The risk free rate = 6% Jason Laws 208 of 218 Alternative Exercise (ii) For a Beta of 1.8 the CAPM predicts the expected return on security K will be: E(R*) = 6 + 1.8 [8] = 20.4 i.e. security K has an expected return that is higher than that predicted by the CAPM Investors will buy security K which will push up the price and drive down the expected return until it is line with the CAPM Jason Laws 209 of 218 Security K Jason Laws 210 of 218 Sample Question (d) Discuss the problems with empirically testing the Capital Asset Pricing Model. ❑ ❑ ❑ ❑ Some of the assumptions of the CAPM are unrealistic: e.g. Unlimited amounts can be borrowed/lent at the risk free rate However, the best way of judging a model is how well it predicts. However, Roll (1977) argues that the CAPM is not testable because the market portfolio is not observable and using proxies is measures it with error. ❑ Also we proxy expected returns with actual (historical) returns but actual returns may be poor proxies for expected returns. Jason Laws 211 of 218 Arbitrage ❑ In its simplest form arbitrage is the simultaneous buying and selling of securities at two different prices in two different markets. ❑ The arbitrageur profits without risk by buying cheap in one market and simultaneously selling at the higher price in the other market. ❑ Investors don’t hold their breath waiting for such situations to occur because they are rare. ❑ Less obvious situations occur where a package of securities can produce a payoff identical to another security that is priced differently. Jason Laws 212 of 218 Arbitrage ❑ This arbitrage relies on a fundamental principle of finance called the law of one price. ❑ i.e. a security must have the same price regardless of the means of creating that security. ❑ The law of one price implies:❑ that if the payoff of a security can be synthetically created by a package of other securities, the price of the package and the price of the security whose payoff it replicates must be equal. ❑ When this situation is discovered not to hold rational investors will trade these securities in such a way to restore equilibrium. Jason Laws 213 of 218 Assumptions of the APT ❑ The arbitrage pricing theory (APT) postulates that a security’s expected return is influenced by a variety of factors, as opposed to just the single market index of the CAPM. ❑ The APT in contrast states that the return on a security is linearly related to H “factors”. ❑ The APT does not specify what these factors are, but it is assume that the relationship between security returns and the factors is linear. Jason Laws 214 of 218 Derivation of the APT ❑ For now and to illustrate the APT model let us assume a simple world consisting of three securities and with two factors. ❑ I will use the following notation:❑ Ri – random rate of return on security i (i = 1,2,3) ❑ E(Ri) – the expected return on security i (i = 1,2,3) ❑ Fh – the h’th factor that is common to the returns of all three assets (h =1,2). ❑ i,h - the sensitivity of the i’th security to the h’th factor. ❑ ei – the unsystematic return for security i. Jason Laws 215 of 218 The APT equation ❑ Ross (1976) has shown that the following risk return relationship will result for security i. ❑ E(Ri) = RF + i,F1[E(RF1) – RF] + i,F2[E(RF2) – RF] ❑ Or, E(Ri) = RF + i,F1lF1+ i,F2lF2 ❑ Where:❑ i,Fj – the sensitivity of security i to the j’th factor. ❑ [E(RFj) – RF] – the excess return of the j’th systematic factor over the risk free rate – can be thought of as the price for the j’th systematic risk. ❑ This can of course be generalised to the case where there are H factors as follows:- Jason Laws 216 of 218 The APT equation ❑ E(Ri) = RF + i,F1[E(RF1) – RF] + i,F2[E(RF2) – RF] + ….. + i,FH[E(RFH) – RF] ❑ This is the APT model. ❑ It states that investors want to be compensated for all the factors that systematically affect the return of a security. ❑ The compensation is the sum of the products of each factors systematic risk (i,FH) and the risk premium assigned to it by the financial market [E(RFH) – RF]. ❑ As usual investors are not compensated for accepting unsystematic risk. Jason Laws 217 of 218 Alternative Exercise ❑ How does the APT differ from the CAPM? ❑ The advantage of the APT is that it does not require us to identify and measure the market portfolio ❑ This solves most of the theoretical limitations of the CAPM. ❑ The disadvantage is that it does not tell us what the underlying factors are (unlike the CAPM, which collapses all the macroeconomic factors into the market portfolio). Jason Laws 218 of 218