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1. 18 December, 2023 Land Management Department 1 vAluation mathematics & valuation tables

2. 18 December, 2023 Land Management Department 2 I. VALUATION MATHEMATICS Purpose is to study the mathematics needed as a preliminary requisite to the solving of valuation problems that we encounter later on.

3. 18 December, 2023 Land Management Department 3 I. VALUATION MATHEMATICS There are several formulae that are vital in valuation calculations and assume yearly interests. Other formulae for payments in arrears or in advance for periods of less than a year and for aspects of growth are also included. In each case, the unit is 1. The applicable rate of interest (‘i’) is shown as a decimal (for example 5 per cent becomes 0.05, 11.5 per cent becomes 0.115). The period (‘n’) between the receipt of one rent payment and the next is in years.

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5. 18 December, 2023 Land Management Department 5 I. VALUATION MATHEMATICS Purpose is to study the mathematics needed as a preliminary requisite to the solving of valuation problems that we encounter later on. Simple Interest P = Principal amount i = interest rate per annum expressed as a decimal n = term of years I = total simple interest A = total amount

6. 18 December, 2023 Land Management Department 6 I. VALUATION MATHEMATICS The present value of £1 The single amount to be invested now to accumulate to £1 in ‘n’ years at ‘i’ inter est. It is self-evident that the prospect of £1 at some future date is worth less than £1 available today. The precise amount of the present value will be determined by the period of deferment and the rate of interest. Example: A legacy of £5000 is payable in four years’ time when the recipient reaches the age of 21. Calculate the present value of the sum where the appropriate rate of interest is 6 per cent.

7. Legacy payable in 4 years’ time £5000.00 PV £1 in 4 years @ 6.0% 0.7921 Current value of legacy £3 960.47 Step-by-step workings Legacy due in 4 years’ time £5 000.00 PV £1 in 1 year @ 6.0% 0.9434 Value in 3 years’ time £4 716.98 PV £1 in 1 year @ 6.0% 0.9434 Value in 2 years’ time £4 449.98 PV £1 in 1 year @ 6.0% 0.9434 Value in 1 year’s time £4 198.11 PV £1 in 1 year @ 6.0% 0.9434 Current value £3 950.50 Slight difference due to rounding. 18 December, 2023 Land Management Department 7 I. VALUATION MATHEMATICS

8. 18 December, 2023 Land Management Department 8 I. VALUATION MATHEMATICS The amount of £1 (Or Future value of £1) The formula is designed to show the total amount of compound interest added to the initial and only deposit of £1 for a specified time and at a particular rate of interest. It assumes that interest is added at the end of each year and is left in the account to itself attract interest in subsequent years (the principle of compound interest). Example: A retailer has entered into a new lease, which includes a commitment to spend £20,000 on internal rearrangements to the premises. A loan has been arranged at an interest rate of 7.5 per cent to be repaid at the end of three years. Calculate the amount due at that time.

9. Loan £20000.00 Amt of £1 in 3 years @ 7.5% 1.2423 Repayment due in 3 years £24 846.00 Check through yearly calculations Loan £20 000.00 Amt of £1 in 1 yr @ 7.5% (i.e. × 0.075) £ 1 500.00 £21 500.00 Amount of £1 in 1 yr @ 7.5% £ 1 612.50 £23 112.50 Amount of £1 in 1 yr @ 7.5% £ 1 733.44 Amount for repayment in 3 years £24 845.94 18 December, 2023 Land Management Department 9 I. VALUATION MATHEMATICS

10. 18 December, 2023 Land Management Department 10 I. VALUATION MATHEMATICS COMPOUND INTEREST Formulas: A = P( 1+i ) n Example: To what amount will K7,500,000 accumulate at 8.5% compound interest for 5 years?

11. 18 December, 2023 Land Management Department 11 I. VALUATION MATHEMATICS Years’ purchase (or the present value of £1 per annum) The present value of the right to receive £1 at the end of each year for ‘n’ years at ‘i’ compound interest. This formula is used to find the capital value of the right to receive a stream of income, typically but not necessarily a rent. The same result could be found by adding together a series of present values (PVs) of £1, which is shown as an alternative. Again, this would be practicable only for short periods. Example: Your client owns a freehold office building recently let at a market rent of £30 000 per annum. The rent will be reviewed at the end of five years. You are asked to indicate the present value of the first five years’ income at a discount rate of 6.5 per cent.

12. Rent per annum £ 30 000.00 YP £1, 5 years @ 6.5% 4.1557 Value of first 5 years’ rent £124 671.00 Check through yearly calculations Rent per annum £ 30 000.00 PV of £1 in 1 year @ 6.5% 0.93897 PV of £1 in 1 year @ 6.5% 0.88166 PV of £1 in 1 year @ 6.5% 0.82785 PV of £1 in 1 year @ 6.5% 0.77732 PV of £1 in 1 year @ 6.5% 0.72988 4.1557 Value of first 5 years’ rent £124 670.40 18 December, 2023 Land Management Department 12 I. VALUATION MATHEMATICS

13. 18 December, 2023 Land Management Department 13 I. VALUATION MATHEMATICS Amount of £1 per annum The formula shows the amount to which £1 invested at the end of each year would accumulate at a given rate of interest. Example Janet intends to save £3000 each year for the next three years. Calculate the value at the end of the period if the savings are invested in a tax-free ISA account at an interest rate of 5 per cent.

14. Savings each year £3 000.00 Amt of £1 p.a , 3 years @ 5% 3.1525 Amt available at end of 3 yrs £9 457.50 Check through yearly calculations Amount invested at end of year 1 £3 000.00 Interest added at end of year 2 £ 150.00 Balance carried forward £3 150.00 Amount invested at end of year 2 £3 000.00 Balance carried forward £6 150.00 Interest added at end of year 3 £ 307.50 Balance carried forward £6 457.50 Amount invested at end of year 3 £3 000.00 Total return at end of 3 years £9 457.50 18 December, 2023 Land Management Department 14 I. VALUATION MATHEMATICS

15. 18 December, 2023 Land Management Department 15 I. VALUATION MATHEMATICS Annual sinking fund The annual premium required to be invested at the end of each year to accumulate to £1 at a specified compound rate of interest over a period of years. One of its principal uses is to provide for the replacement of capital in investments that have a limited life, referred to as wasting assets, in which case it will be linked to the years’ purchase formula. A feature of compound interest is that the interest payments include not only interest on the capital sum invested, but an additional amount, which, if reinvested at the same rate of interest, will return the original amount expended at the end of the period.

16. 18 December, 2023 Land Management Department 16 I. VALUATION MATHEMATICS Example The tenant of a small flat has agreed to purchase it in five years’ time when a deposit of £25 000 will be required. How much should be invested each year if a rate of interest of 5.25 per cent can be obtained from a five-year fixed interest bond underwritten by the government? Computation Deposit required in 5 years’ time £25 000.00 Annual sinking fund, 5 years @ 5.25% 0.1819 £ 4 547.50

17. 18 December, 2023 Land Management Department 17 I. VALUATION MATHEMATICS The annuity purchased by £1 The formula envisages investing a single amount now to provide an income for a specified period commencing at the end of the first year. The capital and interest accruing will be returned over the life of the annuity. The whole of the fund will have been used by the end of the period.

18. 18 December, 2023 Land Management Department 18 I. VALUATION MATHEMATICS Example Your client is taking early retirement and wishes to purchase an annuity to supplement investment income pending receipt of a pension in 10 years’ time. Advise on the annuity available from a capital investment of £15 000 at a discount rate of 5.5 per cent. Working Sum available to purchase annuity £15 000.00 Annuity, 10 years @ 5.5% 0.1327 Annuity £ 1 990.50

19. 18 December, 2023 Land Management Department 19 I. VALUATION MATHEMATICS Years’ purchase in perpetuity (the present value of £1 per annum in perpetuity) If it can be assumed that the income will be received indefinitely on a yearly in arrears basis, the formula is of the simplest form. The years’ purchase is found by dividing the decimal equivalent of the appropriate yield into unity.

20. 18 December, 2023 Land Management Department 20 I. VALUATION MATHEMATICS Example Calculate the capital value of the right to an income of £5000 per annum in perpetuity where the appropriate return by reference to the sales of other similar properties is 8 per cent. Annual rent £ 5 000.00 Years’ purchase in perpetuity @ 8% 12.50 Capital value £62 500.00

21. 18 December, 2023 Land Management Department 21 I. VALUATION MATHEMATICS Years’ purchase (YP) in perpetuity deferred ‘n’ years If receipt of payments is deferred for an initial period, the effect can be calculated in one of two ways, both of which give the same result: ● YP in perpetuity minus YP for initial period of delay ● YP in perpetuity × present value of £1 for initial period of delay. The following example will be used to demonstrate both approaches.

22. 18 December, 2023 Land Management Department 22 I. VALUATION MATHEMATICS Example Assume the right to receive £1000 per annum in perpetuity after the expiration of three years from now. The current interest rate may be taken as 5 per cent. Income in perpetuity commencing in 3 years’ time £ 1 000.00 YP perpetuity @ 5% 20.0000 Less YP £1, 3 years @ 5% 2.7232 17.2768 Capital value £17 276.80

23. 18 December, 2023 Land Management Department 23 I. VALUATION MATHEMATICS Example Also Income in perpetuity commencing in 3 years time £ 1 000.00 YP perpetuity @ 5% 20.0000 × PV £1, 3 years @ 5% 0.8638 17.2760 Capital value £17 276.00 Slight differences due to rounding of components.

24. 18 December, 2023 Land Management Department 24 I. VALUATION MATHEMATICS Mortgage Repayments Repayment formula: P = M( 1+i )n i ( 1+i )n – 1 Example: What would be the annual repayments on a mortgage of K4,500,000 borrowed for 25 years at 11% compound interest?

25. 18 December, 2023 Land Management Department 25 I. VALUATION MATHEMATICS Depreciation Loss in value of any asset such as buildings, machinery and equipment. Formula: D = P(1-i) n Example: What is the value of an asset depreciating at 8% per annum, valued at K3,000,000 after 7 years?

26. CONSTRUCTION & ANALYSIS OF VALUATION TABLES

27. 18 December, 2023 Land Management Department 27 I. GENERAL Valuer’s business to make a carefully considered estimate of property value; Valuation tables enable valuers to express the estimate value; Valuation tables objective: To save valuer time; To reduce risk of error in calculations; As a mathematical tool for use by the valuer; Parry’s Valuation & Investment Tables; & All tables based on Principle of Compound Interest.

28. 18 December, 2023 Land Management Department 28 ii. SINGLE RATE TABLES Amount of K1 (A) Amount to which K1 will accumulate at i interest rate for n years invested at the beginning of each year Tables assume interest added annually Amount = (1+i) n

29. 18 December, 2023 Land Management Department 29 EXAMPLE 1 A building estate was purchased for K5,000,000 & K1,000,000 was spent on roads & other development costs @ once. For 5 years no return was received from the property. What was the total cost of this property to the purchaser at the end of 5 years assuming interest of 9%? Answer: Total invest ( K5,000,000 + K1,000,000 ) = K6,000,000 Amount of K1 in 5 yrs @ 9% = 1.5386 Cost to purchaser at the end of 5 yrs = K9,231,600

30. 18 December, 2023 Land Management Department 30 EXAMPLE 1: USING TABLES

31. 18 December, 2023 Land Management Department 31 EXAMPLE 2 A development was purchased 3 yrs ago & a loan of K2,000,000 was obtained for this purpose at a fixed annual interest rate of 8% rolled up until the development is completed. The development will be completed in 4 yr’s time. Calculate the sum due for payment at that time. Answer: Capital sum invested = K2,000,000 Amount of K1 in 4 yrs @ 8% = 1.360 Sum due for repayment = K2,720,000

32. 18 December, 2023 Land Management Department 32 EXAMPLE 2

33. 18 December, 2023 Land Management Department 33 EXAMPLE 3 Interest compounded more than once a yr Formula: A = (1+ i / m ) mn Example: To what amount will K1,000,000 invested @ 15% accumulate in 8 yrs if interest is payable monthly? Answer: Amount = K1,000,000 (1+ 0.15 / 12 ) 12*8 = K3,227,000

34. 18 December, 2023 Land Management Department 34 ii. SINGLE RATE TABLES Present Value of K1 Table This is the inverse of amount of K1 table Current value of the right to receive K1 @ a known date in future PV is termed the deferred value of a future sum PV = 1 / (1+i)n

35. 18 December, 2023 Land Management Department 35 EXAMPLE 4 What sum invested now will accumulate to K1 in 6 yrs’ time @ 5% compound interest p.a? Answer: Using formula: 1/(1+0.05) 6 and Tables: 1/1.3400956 = K 0.75

36. 18 December, 2023 Land Management Department 36 EXAMPLE 5 M/s Jombo has the right to receive K100,000 in 3 years time assuming that capital can invested at 5% annual return, compound interest? Using formula: K100,000 /(1+0.05) 3 = K100,000 /1.1576 = K86,383 Using tables: Sum receivable = K100,000 PV K1 in 3 yrs @ 5% p.a. = 0.8638376 = K86,383 Therefore PV of K100,000 deferred 3 yrs @ 5% = K86,383

37. 18 December, 2023 Land Management Department 37 EXAMPLE 4

38. 18 December, 2023 Land Management Department 38 ii. SINGLE RATE TABLES Amount of K1 per annum Amount to which K1 invested at the end of each year will accumulate @ i compound interest in n years. Formula: ((1+i) n -1)/i or = (A-1)/i

39. 18 December, 2023 Land Management Department 39 EXAMPLE 8 A new timber tree plantation will mature in 80 yrs time. The initial planting cost was K200,000 per ha & annual expenses average K20,000 per ha. What will be the total cost per ha by the time the trees mature ignoring any increase in value of the land & assuming that interest required on the other outstanding capital @ 5%?

40. 18 December, 2023 Land Management Department 40 EXAMPLE 8 Answer: Using formula: 1 st part: amount of K200,000 @ 5% in 80 years = K200,000 (1+0.05) 80 = K200,000 *1.05 80 = K200,000 *49.5614 = K9,912,288.21 2 nd part: amount of K20,000 @ 5% in 80 years = K20,000 ((1+0.05) 80 – 1)/0.05 = K20,000 *(1.05 80 – 1)/0.05 = K20,000 * 48.5614/0.05 = K20,000 * 971.2280 = K19,424,560.00 Total cost per hectare in 80 years = K29,336,848.21

41. 18 December, 2023 Land Management Department 41 EXAMPLE 8 Using tables: Initial capital outlay p.ha = K200,000 Amount of K1 in 80 yrs @ 5% = 49.5614 = K9,912,288.21 Annual cost per hectare = K20,000 Amount of K1 p.a. in 80 yrs @ 5% = 971.229 = K19,424,560.00 Total cost p.ha in 80 yrs = K29,336,848.21

42. 18 December, 2023 Land Management Department 42 ii. SINGLE RATE TABLES Annual Sinking Fund This is an annual sum, s, required to be invested at the end of each year to accumulate to K1 in n years @ i compound interest. Formula for end of yr payments: s = i/(1+i) n -1 or i/(A-1) Formula for start of yr = s = i/(1+i) n+1 -1

43. 18 December, 2023 Land Management Department 43 EXAMPLE 9 The owner of a house anticipates that he will need to provide a new staircase in 10 years’ time @ an estimated cost of K70,000 . If capital can be invested @ 8% CI, what amount should be invested annually to meet his future liability? a) At the end of the yr & b) at the beginning of the yr.

44. 18 December, 2023 Land Management Department 44 EXAMPLE 9 0.08/(1.08 10 -1) = 0.08/2.1589-1 = 0.08/1.1589 = 0.0690 = 0.0690*K70,000 = K4,832.06 = 0.08/(1.08 10+1 -1) = 0.08/(1.08 11 -1) = 0.08/2.3316-1 = 0.08/1.3316 = 0.0601 = 0.601*K70,000 = K4,205.48

45. 18 December, 2023 Land Management Department 45 EXAMPLE 9

46. 18 December, 2023 Land Management Department 46 ii. SINGLE RATE TABLES Years’ Purchase or PV of K1 per Annum This is the PV of the right to receive K1 @ the end of each year for n years @ i CI. Formula: (1- (1/( 1+i ) n ))/i Example 10 : A landlord will receive K100,000 p.a rent from his tenant for the next 20 yrs. Assuming 8% CI, what is the capital value of the income?

47. 18 December, 2023 Land Management Department 47 EXAMPLE 9 Answer: Using formula = 1- {1/(1+i) n }/i = [1- {1/(1+0.08) 20 }]/0.08 = {1-(1/1.08 20 )}/0.08 = {1-(1/4.661)} /0.08 = (1-0.2145)/0.08 = 0.7855/0.08 = 9.818 = Capital value = K100,000*9.818 = K981,800.00

48. 18 December, 2023 Land Management Department 48 EXAMPLE 9 Answer: using tables of Years’ Purchase Rent receive p.a. = K100,000 YP for 20 yrs @ 8% = 9.818 Capital value = K981,800

49. 18 December, 2023 Land Management Department 49 EXAMPLE 9

50. 18 December, 2023 Land Management Department 50 ii. SINGLE RATE TABLES Years’ Purchase in Perpetuity (YP) This is the PV of the right to receive K1 @ the end of each year in perpetuity @ i CI. This is for an endless period of time YP in perpetuity formula: 1/ i

51. 18 December, 2023 Land Management Department 51 EXAMPLE 11 What is the capital value of a freehold interest with an annual income of K25,000 assuming a 7% CI? Answer: Annual net income per annum = K25,000 YP in perpetuity @ 7% = 1/0.07 = 14.2857 Capital value = K357,142.86 Work out the same using tables

52. 18 December, 2023 Land Management Department 52 ii. SINGLE RATE TABLES YP of a Reversion to a Perpetuity The PV of the right to receive K1 @ the end of each year in perpetuity @ i CI, but receivable after the expiration of n years. Formula: 1/i(1+i) n Example 12: What is the capital value of the right to receive K1 p.a. in perpetuity starting in 7 yrs’ time at 7% CI?

53. 18 December, 2023 Land Management Department 53 EXAMPLE 12 Using formula: = 1/(0.07(1+0.07) 7 ) = 1/(0.07*1+0.07 7 ) = 1/(0.07*1.6058) =1/0.1124 = K8.8964 Using tables - Net income p.a. = K1 - YP in perpetuity @ 7% for 7 yrs = 8.89642 Capital value = K8.89642

54. 18 December, 2023 Land Management Department 54 EXAMPLE 13 The owner of a freehold property will receive net income of K275,000 p.a. commencing in 4 yrs’ time assuming a return of 8%. Value his interest. Use the formula. Answer = K2,524,000

55. 18 December, 2023 Land Management Department 55 ii. SINGLE RATE TABLES Income to be received @ intervals of more than 1 yr Example 14: What is the capital value of the right to receive 5 payments each of K100,000 at 5 year intervals @ 3% CI?

56. 18 December, 2023 Land Management Department 56 EXAMPLE 14 Using tables Each payment = K100,000 PV of K1 for 5 yrs @ 3% = 0.8626 PV of K1 for 10 yrs @ 3% = 0.7441 15 yrs @ 3% = 0.6419 20 yrs @ 3% = 0.5537 25 yrs @ 3% = 0.4776 = 3.2799 = K327,990

57. 18 December, 2023 Land Management Department 57 iii. DUAL RATE TABLES This is the capital value of the right to receive K1 @ the end of each yr for n yrs @ i CI while allowing for a sinking fund s to recoup the initial capital after ‘n’ yrs. Example 15: What is the capital value of K100,000 p.a. receivable in perpetuity? The owner requires 8% return.

58. 18 December, 2023 Land Management Department 58 iii. DUAL RATE TABLES Answer: Net income p.a. = K100,000 YP in perpetuity @ 8% (1/0.08) = 12.5 Capital value = K1,250,000 If the income stream is receivable for 25 yrs only, then the capital value could be calculated as follows.

59. 18 December, 2023 Land Management Department 59 iii. DUAL RATE TABLES Example 15 continues: Let capital value be x Risk-free net rate of s, say 2.5% (accumulative rate of interest) Net income p.a. = K100,000 Less s to recoup K1 in 25 yrs @ 2.5% = 0.025/(1.025 25 -1) = 0.025/0.8539 = 0.0293 S factor = 0.0293

60. 18 December, 2023 Land Management Department 60 iii. DUAL RATE TABLES Example 15 continues s to recoup x in 25 yrs @ 2.5 % = 0.0293x Spendable income p.a. = 100,000- 0.0293x YP in perpetuity @ 8% = 12.5 Capital value = 1,250,000- 0.3663x Capital value is x, hence 1,250,000- 0.3663x = x 1,250,000 = 1.3663x x = 1,250,000/1.3663 C apital value = K914,879.60

61. 18 December, 2023 Land Management Department 61 iii. DUAL RATE TABLES Example 15 continues. Calculation using Dual Rate Tables Net income p.a. = K100,000 YP for 25 yrs @ 8% & 2.5 % = 9.15 Capital value = K915,000

62. 18 December, 2023 Land Management Department 62 iii. DUAL RATE TABLES Example 16: What is the capital value of an income of K500,000 p.a. for 12 yrs @ 8% return & s rate of 2.5%? Net income p.a. = K500,000 YP for 12 yrs @ 8% & 2.5% = 6.5579 Capital value = K3,278,950

63. 18 December, 2023 Land Management Department 63 iii. DUAL RATE TABLES Brain teaser: Calculate the YP using the formulas (single & dual rate) for 10 yrs @ 6%. Please do this on your own!!!!!

64. iii. DUAL RATE TABLES 18 December, 2023 Land Management Department 64

65. iii. DUAL RATE TABLES 18 December, 2023 Land Management Department 65

66. iii. DUAL RATE TABLES Commentary The remaining lease term of 40 years is regarded as a wasting asset requiring replacement of the original capital at the end of the term. The years’ purchase formula is modified to include a sinking fund taken out at a safe and therefore low rate of interest to provide for the original capital to be replaced. As the premiums for the sinking fund will be payable from taxed income, the effect on tax must also be provided for. Selection of the tax rate creates some difficulty but should reflect the rate paid by companies or individuals likely to engage in such activities. It should be noted that some investors, such as charities, enjoy certain tax benefits and would be able to consider the provision of a sinking fund from gross income. The provision of a sinking fund has a limited effect on the capital value, given the remaining term. As a comparison, an income in perpetuity at the same yield would be worth approximately £8000 more. 18 December, 2023 Land Management Department 66

67. 18 December, 2023 Land Management Department 67 iii. DUAL RATE TABLES The Effect of Tax on the Sinking Fund Element of the Dual Rate YP So far no tax effect mentioned or taken into account In dual rate computations, a proportion of income set aside for sinking fund provision (for leasehold) Sinking fund & interest on it subject to tax Dual Rate YP adjusted for the tax using the net adjustment factor, T N

68. iii. DUAL RATE TABLES Where T N = 1-x Where x = Rate of tax (tambala)/100 For example, if gross rate of interest = 4.5% & tax liability = 30 t in K Then Net rate of interest (T N ) = Gross rate * (1-x) = 4.5*(1- 30 / 100 ) = 4.5* 70 / 100 = 4.5*0.7 = 3.15% 18 December, 2023 Land Management Department 68

69. iii. DUAL RATE TABLES Example 17 What is the capital value of an income of K500,000 p.a. receivable for 15 yrs only? The investor requires a 10% return and a sinking fund at 4.25% gross with a typical tax rate of 33%. Try this problem with income of K750,000 for 25 yrs @ 9% & s of 5% gross & tax of 33%. 18 December, 2023 Land Management Department 69

70. iii. DUAL RATE TABLES Example 7 work out: First calculate the net interest for the sinking fund, s => s = Gross interest * (1-x) = 4.25*(1- 33 / 100 ) = 4.25* 67 / 100 = 4.25* 0.67 = 2.8475 => Round off to the nearest 5= Say, 3.0% 18 December, 2023 Land Management Department 70

71. iii. DUAL RATE TABLES Second step is the calculation of capital value: Net income p.a. = K500,000 YP for 15 yrs @ 10% & 3% net (tax 33t in K) = 5.5479 Capital value = K2,773,950 18 December, 2023 Land Management Department 71

72. 18 December, 2023 Land Management Department 72 THANK YOU FOR YOUR ATTENTION