Introduction to Real Estate Investment Appraisal Maths of Finance Present and Future Values Pat McAllister
INVESTMENT APPRAISAL: INTEREST Interest is a reward or rent paid to a lender or investor who has provided an asset (capital, principal) to another party the borrower bank lending e.g. a mortgage a share in a company government or corporate borrowing, bonds property investment Interest may appear as income profit dividend or rent depending on the asset / deal What determines interest rates? The Fisher model: Time Preference or Impatience Expected or Anticipated Inflation The Risk Associated with the Investment Other assets: opportunity cost, cost of capital
INVESTMENT APPRAISAL: SIMPLE INTEREST Place 00 into an account paying 8% interest. After one year it will contain the original 00 plus 8 interest. 00 * ( + 0.08) = 08. In the second year, it earns a further 8% on the original sum invested. 00 * ( + 2 * 0.08) = 6 In general C ( + ni) = A where C = initial capital invested I = interest rate n = term A = final accumulated amount e.g invest 00 for 5 years at 8% A = 00 ( + 5 *.08) = 00 *.4 = 40 What would a rational investor do, however?
INVESTMENT APPRAISAL: COMPOUND INTEREST The rational investor withdraws the 08 at the end of year one and reinvests it at 8% At year two, she has 08 ( +.08) = 6.64 The extra 0.64 is interest earned on earlier interest, or COMPOUND interest Deposit 00 earning 0% compound interest. At end Year 00 (.) = 0.00 end Year 2 00 (.) (.) = 2.00 or 00 (.) 2 end Year 3 00 (.) (.) (.) = 33.0 or 00 (.) 3 And so on. In general: A n = C( + i) n A n is the Future Value of C at i% interest.
INVESTMENT APPRAISAL: COMPOUND INTEREST In Property Valuation, the An with the capital set at is known as the amount of one pound, written A There are tables (e.g. Parry s Tables) which give values of A it acts as a multiplier for other sums invested. If you invest 00 in an account bearing 5.5% compound interest, how much will the account hold in exactly five years? 00 ( + 0.055) 5 = 00(.3070) = 30.70
INVESTMENT APPRAISAL: DISCOUNTING Reverse the process: you need,000 in three years and can invest at 2% (I wish). How much should you deposit? n n An C( i)...so...c An A( i) n ( i) C = 000 (.2) -3 = 000 (0.78) = 7.80 7.80 is the Present Value of,000 received in three years time, discounted at 2% 00 received today is more valuable than 00 received in four years. Why? Inflation erodes spending power (inflation) Could invest the 00 and let it grow (opp. Cost) Will we actually receive the 00 later? (risk) WE WANT IT NOW NOW NOW (impatience) Could pay off other debt (cost of capital)
INVESTMENT APPRAISAL: DISCOUNTING Discounting pulls back all future cashflows to a common base period today. In Property Valuation ( + i) -n is the Present Value of - written PV In actuarial notation, this is written This is a useful, universal notation. It is NOT a formula In summary: v n i Future Value (+i) n C 0 0 n Discounting (+i) -n A n
INVESTMENT APPRAISAL: INTEREST RATE CHANGES Suppose an investment earns 8% for three years then 0% for two years (assuming compound interest). Future value is: A n = C (.08) (.08) (.08) (.) (.) = C (.08) 3 (.) 2 = C (.2597) (.2) = C (.5243) What would the average rate be over the whole period? A n = C( + i*) n So ( + i*) 5 =.5243 So ( + i*) = (.5243) (/5) So i* = (.5243) (/5) - = 0.088 or 8.8% Check: (.088) 5 =.5246 (rounding error only).
INVESTMENT APPRAISAL: INTEREST PAID LESS THAN ANNUALLY What is Annual Rate of Interest Charged at 2% per month? (+.02) 2 = 26.8% per annum NOT 24% NB However, what is 0% per annum credited quarterly? In fact, this is 2.5% per quarter. (.025) 4 - = 0.38% (and not 0%) Distinguish between NOMINAL & EFFECTIVE rates the published APR is an Effective Rate In general, where i (p) is the nominal rate payable ( convertible ) p thly (so p = 2 is monthly, p = 4 quarterly) then the annual effective rate is given by: i ( p) i p [ ] p
INVESTMENT APPRAISAL: INTEREST PAID LESS THAN ANNUALLY So a loan repayable monthly at a nominal 4.5% has an effective rate of: [.45 ] 2 2.55 5.5% Remember that i (p) is not a formula, it is a number the p is the number of times that interest is added in the year. For commercial property, in the UK rent is payable quarterly. In many other countries, it is paid monthly. This means it is important to be able to carry out calculations that are not annual in nature.
INVESTMENT APPRAISAL: INFLATION Suppose inflation is running at ƒ% per annum. Then the purchasing power of X received in one year is X(+ ƒ) - or X / ( + ƒ). So 00 received one year hence with inflation running at 3.5% is worth 00 / (.035) = 96.62. Now, if an investment pays j% per annum compound interest, with inflation running at ƒ% per annum, then the real accumulated values for years,2,n will be, respectively: ( j) ( j) 2 ( j) n ( f ) ( f ) ( f )
INVESTMENT APPRAISAL: INFLATION There must, thus, be a real interest rate i% such that: ( j) ( i) ( f ) Hence: i ( ( j) f ) And NOT simply j-f... With inflation at 8% and interest at 5%, the real rate = j f.5.08 i.065 ( f ).08 Notice it is NOT 5%-8% = 7%. This becomes important whenever inflation is high. j ( f f )
Investment Appraisal Warm Up You deposit 200 in an account paying 6% interest. How much will you have in the account in five years time? An investment pays 5,000 in four years time. If the appropriate discount rate is 8%, what is the present value of the investment? An investment pays interest at.5% per month. What is the nominal annual rate? What is the effective annual rate?
Investment Appraisal Warm Up Future value: 200 (.06) 5 = 200(.3382) = 267.65 Present value: 5,000 (.08) -4 = 5,000(0.7350) = 3,675.5 Nominal rate is 8% Effective rate is (.05)^2 - = 9.56%
INVESTMENT APPRAISAL: ANNUITIES
INVESTMENT APPRAISAL ANNUITIES Annuities are regular payments made (or received). We may want to know: How much a series of deposits earning interest will be worth in n years (a future value). What is the cash equivalent in today s money of n payments at X (a present value). There are obvious applications here in investment and in loans. Annuities, as the name implies, are traditionally annual payments. An ordinary annuity is received at year end, an annuity due is received at the start of each year. The basic calculations can be applied to other time periods.
INVESTMENT APPRAISAL ACCUMULATION: ORDINARY ANNUITIES Invest at the end of each of next 0 years, earn compound interest of 0%. How much will be in account at end? 0 2 8 9 0 First pound worth (.) 9 ; second pound (.) 8 and so on. The last pound earns no interest and is worth : In total (.) 9 + (.) 8 + Or [(.) 9 + (.) 8 + (.) + ] More generally, with n years and i% interest: [( +i) n- + ( + i) n-2 + ( + i) + ] This can be used as multiplier for any amount invested
INVESTMENT APPRAISAL ACCUMULATION: ORDINARY ANNUITIES The term in the square bracket is the accumulation accumulated value or future value. Valuation terminology calls this the amount of p.a. written A pa s Actuarial notation: n i% Now the term is a geometric series and so we can reduce it to a simple formula. See the handout for the n algebraic derivation. ( i) n % = i s i In original question n = 0 and i = 0% So accumulated value = [(.) 0 -] / 0. = 5.94 Note that this is 0 of capital invested and 5.94 accrued interest.
INVESTMENT APPRAISAL PRESENT VALUES, ORDINARY ANNUITIES What is value today of 000 received at end of each of the next 8 years if your discount rate is 0%? 000 000 000 000 000 The first payment is worth 000 0% 2 The second payment is worth 000 0% and so on The year eight payment 000 In total, we have: Which is: 0 2 6 7 8 000 000[ 8 0% 2 0% 000 0%... 000 0% 2 0%... 8 0% ] 8 0%
INVESTMENT APPRAISAL PRESENT VALUES, ORDINARY ANNUITIES Generalising the square bracket for n years and i% a n i% This is the Present Value of an Ordinary Annuity or, in UK valuer speak, the Years Purchase As before, this is a geometric progression, which simplifies to the most important valuation formula you will meet: In our example, n = 8 and i = 0% S0: present value = 000 2 n [ i% i%... i% ] a v n i% i a 8 0% 000 8. = 000 (5.3349) = 5,335 0. n i
INVESTMENT APPRAISAL PRESENT VALUES, ORDINARY ANNUITIES Note that the total payments received are 8,000 but the present value to us is only 5,300. The lower value reflects the impact of anticipated inflation, time impatience, alternative investment opportunities, borrowing costs and the risk associated with this type of cashflow You can acquire a property lease, making an annual profit of 000. The lease has six years to run and alternative investments have returns of 2%. How much should you pay? a,000 6 2% =,000 (4.4) = 4,.4 Valuation: Profit Rent,000 YP 6 years @ 2% 4.4 Value 4,.4
INVESTMENT APPRAISAL PRESENT VALUES, ORDINARY ANNUITIES We can show that we get our capital back and receive 2% on the outstanding capital: YEAR CAPITAL O/S AT START INTEREST @2% CAPITAL RETURNED 4.4 493.37 506.63 2 3604.78 432.57 567.43 3 3037.35 364.48 635.52 4 240.84 288.22 7.78 5 690.06 202.8 797.9 6 892.86 07.4 892.86 Why is this just like a residential mortgage?
INVESTMENT APPRAISAL PERPETUITIES A freehold property gives a (theoretical) income forever ; so does owning a share in a company. How can you value this? a Well n v n i% i = i n If n is infinitely large then i% must tend to zero since (+i) -n = /(+i) n So, in perpetuity the present value becomes simply i,000 received in perpetuity, discounted at 0% has a present value of,000 / 0.0 = 0,000. In property valuation, i is the yield, initial yield or cap rate and /i is the Years Purchase. In equity markets, i might be the dividend yield.
INVESTMENT APPRAISAL REVERSIONS You are to receive 000 per annum from the end of years 6 to 0 inclusive, but nothing from years to 5. If you discount at 0%, what is this worth? 000 000 000 000 000 0 5 6 7 8 9 0 You know how to value the income stream from Yr a 6 to 0 it s the PV of an annuity 000 5 0% That is 000 (3.7908) = 3,79 But that is a LUMP SUM at the start of Year Six, which is the end of Year 5 So we must PV that lump sum back to today: 5 v 3,79 0% = 3,79(0.6209) = 2,354
INVESTMENT APPRAISAL REVERSIONS - 2 The same principle applies to perpetuities Assume that you are going to receive 000 in perpetuity, starting at end Year Six. Calculation is essentially the same. The present value of a perpetuity values the income to start i year 6, end year 5.,000 / 0.0 = 0,000 Now we must PV it back to today, five years at 0%: 0,000 (.) -5 = 0,000 (.6209) = 6,209 Reversions are a very common feature of the UK property market. You will meet Term and Reversion where you receive one income for a set period and then a higher income thereafter
CALCULATORS AT THE READY If you invest,000 in an account paying compound interest of 4% per annum, how much will be in the account after 4 years? If the appropriate discount rate is 8%, what is the present value of,000 received exactly six years from now? Again assuming a discount rate of 8%, what is the present value of a cashflow of,000 per annum received at the end of each of the next six years? Why do you use different discount rates to discount different types of project and investment opportunity?
How Did You Do? 000 (.04) 4 = 000 (.699) =,69.86,000 (.08) -6 = 000 (0.6302) = 630.7 a,000 6 8% = 000(4.6229) = 4,622.88 Since [-.08-6 ]/0.08 = 4.6229 If they are valued in the same country and at the same time, it s differential risk, init? Impatience, anticipated inflation will be the same for all projects.
INVESTMENT APPRAISAL SINKING FUNDS You need 2,000 in five years time. You can invest, from your income and earn 0%. How much should you invest each year? Well X = 2,000 so X = 2,000 ( / ) That is / s i s 5 0% s 5 0%. 2000 5 (.) 2000(.638) 327.59 n % is the Annual Sinking Fund or ASF in valuation terminology. It is the amount of annual investment at i% required to produce after n years. You can then multiply it by whatever the required end amount is. In UK traditional valuation, this is used extensively in the leasehold valuation process
INVESTMENT APPRAISAL ANNUITIES DUE: PAYMENT IN ADVANCE Invest at the start of each of the next n years: 0 2 n-2 n- n The first will be worth (+i) n The second will be worth (+i) n- and so on The final pound will be worth (+i) That is: (+i) n + (+i) n- + (+i) 2 + (+i) s n i % Now was (+i) n- + (+i) n-2 + (+i) + So each separate term is (+i) higher than for an ordinary annuity So accumulated (future) value of an annuity due: s n i % = (+i) s n i %
INVESTMENT APPRAISAL ANNUITIES DUE 200 invested at the start of each of the next ten years earning 8% interest will be worth: 200 s 0 8% that is 200 or 200 (.08) 4.4866 = 3,29.0 (.08) (.08).08 0 Exactly the same principle applies to discounting / calculating present values. Each payment is received one year earlier, so it is discounted one year less. Therefore the Present Value of an Annuity Due is: a n i% = (+i) a n i%
INVESTMENT APPRAISAL ANNUITIES DUE The present value of 0,000 received at the start of the next 5 years discounted at 8% is: a 5 8% (.08) a 8% 0,000 = 0,000 5 0,000 (.08) (8.5595) = 92,442 Exactly the same principle applies to a perpetuity. a n i % = i so a n i % = (+i) i = ( ) i i An investment that pays 2,000 at the start of each year in perpetuity, discounted at 8% has a present value of 2000 (.08)/.08 = 2000 (3.50) = 27,000
THINGS GET GRIM QUARTERLY IN ADVANCE UK commercial property rents are paid quarterly in advance even though quoted as annual figures We could estimate a quarterly discount rate and estimate the present value as an annuity due However, it is convenient to have a formula which we can use as a multiplier for the annual rent. Effectively, we take the quarterly rent elements and convert them into an annual equivalent rent received at year end The algebraic transformation to do this is relatively straightforward:
I Really CAN Do This, You Know R Let the annual rent = R. Then the quarterly payments are 4 The effective rental value must take into account the early arrival of income. Suppose our annual discount rate is i%. Then the quarterly rate is j = (+i) ¼ -. Then the effective annual rent is: R s s 4 j = R 4 (+j) 4 j% 4 % s 4 j% = [(+j) 4 ] / j But j = (+i) ¼ - so (+j) = (+i) ¼ Hence (+j) 4 = [(+i) ¼ ] 4 = (+i) And so s 4 j% = i / [(+i) ¼ -] s j % 4 s = (+j) 4 j% = (+i) ¼ {i / [(+i) ¼ -]} This now produces an effective annual in arrears rent, which we can present value using a n i %. s a n i% R 4 4 (+j) j% = R 4 (+i) ¼ {i / [(+i) ¼ -]} * [ (+i) -n ] / i] Cross multiplying removes i from denominator and numerator. 4 n ( i) [ ( i) ] R4 n ( i) Now divide top and bottom by (+i) ¼ R4 ( i) n 4 [ ( i) ]
INVESTMENT APPRAISAL QUARTERLY IN ADVANCE So, the Present Value of an Income Stream received quarterly in advance is: 4 a ( ) i% n = For example, 2,000 received QinA for six years, discounted at 2% has a present value of: 2000 4( (.2) (.2) 6 4 R ) 4[ For a perpetuity, the numerator becomes ( ( For other periods, change the 4s to the appropriate number e.g. monthly, change the 4s to 2s i) i) n 4 ] 2000(4.454) 8,830.79
IS THERE NO END TO THIS SUFFERING? CONTINUOUS INCOME Suppose income is received continuously. Then the present value: n vi a n i % = log ( i) What is the value of turnover at 0,000p.a. received continuously for six years, discounting at 0%? 0,000 ( 0. ) 6 log ( 0. ) e Log e (+i) is the force of interest, an important parameter in serious investment mathematics e = 0,000. 4355. 0953 = 0,000 (4.5693) = 45,693
NO, REALLY, IT IS USEFUL. INCREASING ANNUITIES Consider an annuity payable at year end, starting at but increasing by (+g) per annum. What is its present value discounting at i%? Year is worth: ( i) Year 2 is worth: ( g ) 2 ( i) Year n: ( g ) ( i) n n and so on until Thus Ia n i% = ( i ) + ( g ) 2 ( i ) +. + ( g ) ( i) This is yet another geometric series. Avoiding the algebra, this reduces to: n n Ia n i% = ( g i ) i g n
IF ONLY TO BAFFLE YOU. INCREASING ANNUITIES Now where n tends to infinity (that is, for a perpetuity), the numerator tends to (assuming that i > g), leaving: Ia n i % = i g In equity markets this is the Gordon Dividend Growth model: i is the required return, g is the expected growth in dividends and d = i - g is the dividend yield. The same principle applies to property. The initial yield k = e - g where k is the initial yield, e is the equated yield (required return) and g is the expected rental growth. However, we need to correct for the rent review period.
INVESTMENT APPRAISAL INCREASING ANNUITIES An investment pays 0,000 per annum in perpetuity: the income is expected to increase at 3% per annum and the investor wishes to make a return of 9% from this type of investment. What should she pay? 0,000.09.03 66,667. 0,000(6.6667) An investment property is sold at an initial yield of 6.5%. Investors require a return of 9%. What rental growth is required to achieve this? k = e - g, so e - k = g => 9 6.5 = 2.5% per annum
Investment Appraisal Warm Up You deposit 200 in an account paying 6% interest. How much will you have in the account in five years time? An investment pays 5,000 in four years time. If the appropriate discount rate is 8%, what is the present value of the investment? Why might the appropriate discount rate be 8%? A property investment provides an annual nominal return of 5%. What is the real return, if inflation is expected to run at 8% per annum?
Investment Appraisal Warm Up Future value: 200 (.06) 5 = 200(.3382) = 267.65 Present value: 5,000 (.08) -4 = 5,000(0.7350) = 3,675.5 Anticipated inflation, risk (of what?), returns on alternative investments, cost of borrowing, target returns in investment strategy I = (+j)/(+f)- = (.5)/(.08)- or (.5-.08)/.08 =.0648 = 6.48% and NOT 5%-8% = 7% except as a quick approximation