ACCA paper f9 financial management study materials F9FM session04 d08

SESSION 04 – DISCOUNTED CASH FLOW TECHNIQUES OVERVIEW Objective To apply the time value of money to investment decisions INTEREST SIMPLE COMPOUND DISCOUNTING Single sum Annuities Effective Annual Interest Rates (EAIR) “Compounding in reverse” Points to note DISCOUNTED CASH FLOW (DCF) TECHNIQUES Procedure Meaning Cash budget pro forma Tabular layout Annuities Perpetuities Definition and decision rule Perpetuities Annuities Uneven cash flows Unconventional cash flows Time value of money DCF techniques NET PRESENT VALUE (NPV) INTERNAL RATE OF RETURN (IRR) NPV vs IRR Comparison 0401 SESSION 04 – DISCOUNTED CASH FLOW TECHNIQUES SIMPLE INTEREST Interest accrues only on the initial amount invested Illustration If $100 is invested at 10% per annum (pa) simple interest: Year Amount on deposit (year beginning) $100 $110 $120 Interest 0.1 × 100 = 10 0.1 × 100 = 10 0.1 × 100 = 10 Amount on deposit (year end) $110 $120 $130 A single principal sum, P invested for n years at an annual rate of interest, r (as a decimal) will amount to a future value FV Where FV = P (1 + nr) COMPOUND INTEREST Interest is reinvested alongside the principal 2.1 Single sum Illustration If Zarosa placed $100 in the bank today (t0) earning 10% interest per annum, what would this sum amount to in three years time? Solution In year’s time, $100 would have increased by 10% to $110 In years’ time, $110 would have grown by 10% to $121 In years’ time, $121 would have grown by 10% to $133.10 Or FV = P (1 + r) n where P = initial principal r = annual rate of interest (as a decimal) n = number of years for which the principal is invested 0402 SESSION 04 – DISCOUNTED CASH FLOW TECHNIQUES Example $500 is invested in a fund on 1.1.X1 Calculate the amount on deposit by 31.12.X4 if the interest rate is (a) 7% per annum simple (b) 7% per annum compound Solution The $500 is invested for a total of years (a) Simple interest FV = P (1 + nr) FV = (b) Compound interest FV = P (1 + r)n FV = Example $1,000 is invested in a fund earning 5% per annum on 1.1.X0 $500 is added to this fund on 1.1.X1 and a further $700 is added on 1.1.X2 How much will be on deposit by 31.12.X2? Solution Date Amount × invested $ 1.1.X0 1.1.X1 1.1.X2 1,000 500 700 Compound interest factor Amount on deposit = Compounded cashflow $ _ = _ 0403 SESSION 04 – DISCOUNTED CASH FLOW TECHNIQUES 2.2 Annuities Many saving schemes involve the same amount being invested annually There are two formulae for the future value of an annuity Which to use depends on whether the investment is made at the end of each year or at the start of each year (i) first sum paid/received at the end of each year (ii) first sum paid/received at the beginning of each year  (1 + r )n −   (i) FV = a   r   where a r n    (1 + r )n + −   − 1 (ii) FV = a      r    = annuity (i.e annual sum) = interest rate (interest payable annually in arrears) = number of years annuity is paid/invested Commentary These formula will not be provided in the examination Illustration Andrew invests $3,000 at the start of each year in a high interest account offering 7% pa How much will he have to spend after a fixed year term? Solution    (1.07 )6 −   −  = $3,000 × 6.153 = $18,460 FV = $3,000 ×      0.07    2.3 Effective Annual Interest Rates (EAIR) Where interest is charged on a non-annual basis it is useful to know the effective annual rate Foe example interest on bank overdrafts (and credit cards) is often charged on a monthly basis To compare the cost of finance to other sources it is necessary to know the EAIR Formula + R = (1 + r) n R = annual rate r = rate per period (month/quarter) n = number of periods in year 0404 SESSION 04 – DISCOUNTED CASH FLOW TECHNIQUES Illustration Borrow $100 at a cost of 2% per month How much (principal + interest) will be owed after a year? Using FV = P (1 + r)n ⇒ = £100 × (1.02)12 = £100 × 1.2682 * = £126.82 EAIR is 26.82% DISCOUNTING 3.1 “Compounding in reverse” Discounting calculates the sum which must be invested now (at a fixed interest rate) in order to receive a given sum in the future Illustration If Zarosa needed to receive $251.94 in three years time (t3), what sum would she have to invest today (t0) at an interest rate of 8% per annum? Solution The formula for compounding is: FV = P (1 + r) n Rearranging this: P = FV × (1 + r ) n or alternatively PV = CF × where PV r n (1 + r ) n = the present value of a future cash flow (CF) = annual rate of interest/discount rate = number of years before the cash flow arises In this case PV = $251.94 × = $200 (1.08) The present value of $251.94 receivable in three years time is $200 0405 SESSION 04 – DISCOUNTED CASH FLOW TECHNIQUES 3.2 Points to note is known as the “simple discount factor” and gives the present value of $1 (1 + r) n receivable in n years at a discount rate, r A present value table is provided in the exam The formula for simple discount factors is provided at the top of the present value table For a cash flow arising now (at t0) the discount factor will always be t1 is defined as a point in time exactly one year after t0 Always assume that cash flows arise at the end of the year to which they relate (unless told otherwise) Example Find the present value of (a) 250 received or paid in years time, r = 6% pa (b) 30,000 received or paid in 15 years time, r = 9% pa Solution (a) From the tables: r = 6%, n = 5, discount factor = Present value = (b) From the tables: r = 9%, n = 15, discount factor = Present value = 0406 SESSION 04 – DISCOUNTED CASH FLOW TECHNIQUES DISCOUNTED CASH FLOW (DCF) TECHNIQUES 4.1 Time value of money Investors prefer to receive $1 today rather than $1 in one year This concept is referred to as the “time value of money” There are several possible causes: Liquidity preference – if money is received today it can either be spent or reinvested to earn more in future Hence investors have a preference for having cash/liquidity today Risk – cash received today is safe, future cash receipts may be uncertain Inflation – cash today can be spent at today’s prices but the value of future cash flows may be eroded by inflation DCF techniques take account of the time value of money by restating each future cash flow in terms of its equivalent value today 4.2 DCF techniques DCF techniques can be used to evaluate business projects i.e for investment appraisal Two methods are available: NET PRESENT VALUE INTERNAL RATE OF RETURN 0407 SESSION 04 – DISCOUNTED CASH FLOW TECHNIQUES NET PRESENT VALUE (NPV) 5.1 Procedure Forecast the relevant cash flows from the project Estimate the required return of investors i.e the discount rate The required return of investors represents the company’s cost of finance, also referred to as its cost of capital Discount each cash flow (receipt or payment) to its present value (PV) Sum present values to give the NPV of the project If NPV is positive then accept the project as it provides a higher return than required by investors 5.2 Meaning NPV shows the theoretical change in the $ value of the company due to the project It therefore shows the change in shareholders’ wealth due to the project The assumed key objective of financial management is to maximise shareholder wealth Therefore NPV must be considered the key technique in business decision making 5.3 Cash budget pro forma Time $000 $000 $000 $000 Capital expenditure Cash from sales Materials Labour Overheads Advertising Grant (X) – (X) – – (X) – _ – X (X) (X) (X) – X _ – X (X) (X) (X) (X) – _ X X – (X) (X) – – _ Net cash flow (X) _ X _ X _ X _ 1 1+ r (1 + r ) (1 + r )3 (X) X X X r% discount factor Present value NPV = X 0408 SESSION 04 – DISCOUNTED CASH FLOW TECHNIQUES 5.4 Tabular layout Time 1–10 0–9 1–10 10 Cash flow $000 Discount factor @ r% Present value $000 (X) X (X) (X) (X) (X) X X x x x x x x x (X) X (X) (X) (X) (X) X X _ CAPEX Cash from sales Materials Labour and overheads Advertising Advertising Grant Scrap value Net present value X _ Example Elgar has $10,000 to invest for a five-year period He could deposit it in a bank earning 8% pa compound interest He has been offered an alternative: investment in a low-risk project that is expected to produce net cash inflows of $3,000 for each of the first three years, $5,000 in the fourth year and $1,000 in the fifth Required: Calculate the net present value of the project Solution Time Description Cash flow $ Investment (10,000) Net inflow 3,000 Net inflow 3,000 Net inflow 3,000 Net inflow 5,000 Net inflow 1,000 8% DF PV $ _ NPV = _ 0409 SESSION 04 – DISCOUNTED CASH FLOW TECHNIQUES 5.5 Annuities An annuity is a stream of identical cash flows arising each year for a finite period of time The present value of an annuity is given as CF × 1   1 − r  (1 + r) n  where CF is the cash flow received each year commencing at t1 1   is known as the “annuity factor” or “cumulative discount factor” It is 1 − r  (1 + r) n  simply the sum of a geometric progression The formula is given in the exam as - (1 + r) −n r Annuity factor tables are also provided in the exam Remember that the formula and tables are based on the assumption that the cash flow starts after one year Illustration Calculate the present value of $1,000 receivable each year for years if interest rates are 10% Time Description t1–3 Annuity Cash flow $ 1,000 10% Annuity factor   1− = 2.486 0.1  1.1  Note: An annuity received for the next three years is written as t1–3 Example Calculate the present value of $2,000 receivable for each of 10 years commencing three years from now Assume interest at 7% 0410 PV $ 2,486 SESSION 04 – DISCOUNTED CASH FLOW TECHNIQUES Solution 5.6 Perpetuities A perpetuity is a stream of identical cash flows arising each year to infinity As n → ∞ (1 + r)n → ∞ →0 (1 + r) n 1 1 −  r (1 + r ) n  →  r  is known as the “perpetuity factor” r The present value of a perpetuity is given as CF × r where CF is the cash flow received each year The formula is based on the assumption that the cash flow starts after one year Illustration Calculate the present value of $1,000 receivable each year in perpetuity if interest rates are 10% Solution Time Description t1–∞ Perpetuity Cash flow $ 1,000 10% Annuity factor = 10 01 PV $ 10,000 0411 SESSION 04 – DISCOUNTED CASH FLOW TECHNIQUES Example Calculate the present value of $2,000 receivable in perpetuity commencing in 10 years time Assume interest at 7% Solution INTERNAL RATE OF RETURN (IRR) 6.1 Definition and decision rule IRR is the discount rate where NPV = IRR represents the average annual % return from a project It therefore shows the highest finance cost that can be accepted for the project If IRR > cost of capital, accept project If IRR < cost of capital, reject project 6.2 Perpetuities If a project has equal annual cash flows receivable in perpetuity then IRR = Annual cash inflows × 100% Initial investment Illustration An investment of $1,000 gives income of $140 per annum indefinitely, the return on the investment is given by IRR = 140/1000 × 100% = 14% 0412 SESSION 04 – DISCOUNTED CASH FLOW TECHNIQUES Example An investment of $15,000 now will provide $2,400 each year to perpetuity Required: Calculate the return inherent in the investment Solution 6.3 Annuities To give an NPV of zero, the present value of the cash inflows must equal the initial cash outflow i.e annual ash inflow × Annuity factor = Cash outflow Annuity factor = Cash outflow Cash inflow Once the annuity factor is known the discount rate can be established from the appropriate table Illustration An investment of $6,340 will yield an income of $2,000 for four years Calculate the internal rate of return of the investment Solution Year 1-4 Description Initial investment Annuity NPV AF1-4 years = CF (6,340) 2,000 DF AF1-4 years PV (6,340) 6,340 _ Nil _ ,340 = 3.17 ,000 From the annuity table, the rate with a four year annuity factor closest to 3.17 is 10% and this is therefore the approximate IRR for this investment 0413 SESSION 04 – DISCOUNTED CASH FLOW TECHNIQUES Example An immediate investment of $10,000 will give an annuity of $1,000 for the next 15 years Required: Calculate the internal rate of return of the investment Solution Time 1-15 Description Investment Annuity Cash flow $ Discount factor (10,000) 1,000 PV $ 6.4 Uneven cash flows Method Calculate the NPV of the project at a chosen discount rate If NPV is positive, recalculate NPV at a higher discount rate (i.e to get closer to IRR) If NPV is negative, recalculate at a lower discount rate The IRR can be estimated using the formula: IRR ~ A + Where A B NA NB NA (B − A) NA − NB = = = = Lower discount rate Higher discount rate NPV at rate A NPV at rate B This method is known as “linear interpolation” 0414 SESSION 04 – DISCOUNTED CASH FLOW TECHNIQUES Illustration 10 The NPVs of a project with uneven cash flows are as follows Discount rate NPV £ 10% 20% 64,237 (5,213) Estimate the IRR of the investment Solution IRR ~ A + NA (B – A) NA − NB IRR ~ 10% + 64 ,237 (20 – 10)% 64 ,237 − ( −5,213) IRR ~ 19% Graphically NPV IRR using formula (interpolated) NA A NB Actual IRR B Discount rate Actual NPV as discount rate varies 0415 SESSION 04 – DISCOUNTED CASH FLOW TECHNIQUES Example An investment opportunity with uneven cash flows has the following net present values $ At 10% At 15% Required: Estimate the IRR of the investment Solution Formula IRR ~ A + NA (B – A) NA − NB IRR ~ Graphically 0416 71,530 4,370 SESSION 04 – DISCOUNTED CASH FLOW TECHNIQUES 6.5 Unconventional cash flows If there are cash outflows, followed by inflows are then more outflows (e.g suppose at the end of the project a site had to be decontaminated), the situation of “multiple yields” may arise – i.e more than one IRR NPV Actual NPV as discount rate varies IRR2 IRR1 Discount rate Actual IRR The project appears to have two different IRR’s – in this case IRR is not a reliable method of decision making However NPV is reliable, even for unconventional projects NPV vs IRR 7.1 Comparison NPV IRR An absolute measure ($) A relative measure (%) If NPV ≥ ,accept If IRR ≥ target %, accept If NPV ≤ 0, reject If IRR ≤ target %, reject Shows $ change in value of company/wealth of shareholders Does not show absolute change in wealth A unique solution i.e a project has only one NPV May be a multiple solution Always reliable for decision making Not always reliable 0417 SESSION 04 – DISCOUNTED CASH FLOW TECHNIQUES Key points Discounted cash flow techniques are arguably the most important methods used in financial management DCF techniques have two major advantages (i) they focus on cash flow, which is more relevant than the accounting concept of profit (ii) they take into account the time value of money NPV must be considered a superior decision-making technique to IRR as it is an absolute measure which tells management the change in shareholders’ wealth expected from a project FOCUS You should now be able to: explain the difference between simple and compound interest rate and calculate future values; calculate future values including the application of annuity formulae; calculate effective interest rates; explain what is meant by discounting and calculate present values; apply discounting principles to calculate the net present value of an investment project and interpret the results; calculate present values including the application of annuity and perpetuity formulae; explain what is meant by, and estimate the internal rate of return, using a graphical and interpolation approach, and interpret the results; identify and discuss the situation where there is conflict between these two methods of investment appraisal; compare NPV and IRR as decision making tools 0418 SESSION 04 – DISCOUNTED CASH FLOW TECHNIQUES EXAMPLE SOLUTION Solution — 7% simple and compound interest The $500 is invested for a total of years (a) Simple interest FV = P (1 + nr) FV = 500 (1 + × 0.07) = 500 × 1.28 = $640 (b) Compound interest FV = P (1 + r)n FV = 500 (1 + 0.07)4 = 500 × 1.3108 = $655.40 Solution — 5% compound interest Date 1.1.X0 1.1.X1 1.1.X2 Amount invested $ × 1,000 500 700 Compound interest factor = (1 + 0.05)3 (1 + 0.05)2 (1 + 0.05)1 Amount on deposit Compounded cashflow $ 1,157.63 551.25 735.00 _ = 2,443.88 _ Solution — Present value (a) From the tables: r = 6%, n = 5, discount factor = 0.747 Present value = 250 × 0.747 = $186.75 (b) From the tables: r = 9%, n = 15, discount factor = 0.275 Present value = 30,000 × 0.275 = $8,250 0419 SESSION 04 – DISCOUNTED CASH FLOW TECHNIQUES Solution — Net present value Time Description Cash flow $ 8% DF PV $ Investment (10,000) (10,000) Net inflow 3,000 (1.08) 2,778 Net inflow 3,000 (1.08) 2,572 Net inflow 3,000 (1.08) 2,381 Net inflow 5,000 (1.08) 3,675 Net inflow 1,000 (1.08) 681 NPV = _ 2,087 _ Solution — Annuity Time Description t3-12 Annuity Cash flow $ 7% Annuity factor PV $ 2,000 6.135 (W) 12,270 WORKING Cdf3-12 @ 7% = CDF1-12 @ 7% – CDF1-2 @ 7% = 7.943 – 1.808 (per tables) = 6.135 Solution — Perpetuity 0420 Time Description Cash flow $ 7% Annuity factor PV $ t10-∞ Perpetuity 2,000 7.771 (W) 15,542 SESSION 04 – DISCOUNTED CASH FLOW TECHNIQUES WORKING = Cdf10-∞ @ 7% = CDF1-∞ @ 7% - CDF1-9 @ 7% – 6.515 (per tables) 0.07 14.286 – 6.515 = 7.771 = Solution — IRR (perpetuity) IRR = 2,400 × 100 = 16% 15,000 Solution — IRR (annuity) Time Description 1-15 Investment Annuity Cash flow $ (10,000) 1,000 Discount factor PV $ Cdf1-15 = 10 (βal) (10,000) 10,000 Nil From the annuity table the rate with a 15 year annuity factor of 10 lies between 5% and 6% Thus if $10,000 could be otherwise invested for a return of 6% or more, this annuity is not worthwhile Solution — IRR (uneven cash flows) Formula Commentary The formula always works but take care with + and – signs IRR ~ A + NA (B – A) NA − NB 71,530   IRR ~ 10 +   (15 – 10)  71,530 − ,370  IRR ~ 10 + 5.325 say 15.4% (rounded up) 0421 SESSION 04 – DISCOUNTED CASH FLOW TECHNIQUES Graphically NPV £ Actual NPV 71,530 Actual IRR 4,370 10 15 IRR using formula (extrapolated) 0422 Discount rate (%) ... therefore shows the change in shareholders’ wealth due to the project The assumed key objective of financial management is to maximise shareholder wealth Therefore NPV must be considered the key technique... Key points Discounted cash flow techniques are arguably the most important methods used in financial management DCF techniques have two major advantages (i) they focus on cash flow, which is more... $000 (X) X (X) (X) (X) (X) X X x x x x x x x (X) X (X) (X) (X) (X) X X _ CAPEX Cash from sales Materials Labour and overheads Advertising Advertising Grant Scrap value Net present value X _