Business Finance Lecture 4

Lecture 4: Introduction to Valuation: The Time Value of Money 4.1 Future value and compounding Investing for a single period ! ! Future value (FV) Ñ the amount an investment is worth after one or more periods. ! Year%0 • You%invest%£100%at%10% Year%1 • Your%money%grows% to%£100%+%10%%of% £100%=%£110 So: ! V1 = V0 (1 + r)! Where V1 is the value after 1 period; r is the interest/discount rate for one period. ! *One period = one year! Example What is the future value for an investment for more than one period at a compound interest rate of 10%? ! £100 Ñ> £110 Ñ> £121 Ñ> £133.10 ! + £10 + £11 + £12.10 ! Where: ! £100 x 1.10 = £110! £110 Ñ £100 = £10 (interest you earn in period) ! £110 x 1.10 = £100 x 1.102 = £121! £121 Ñ £110 = £11 (interest you earn in period) ! £121 x 1.10 = £100 x 1.103 = £133.10! £133.10 Ñ £121 = £12.10 (interest you earn in period) ! Investing for more than one period ! Compounding Ñ the process of accumulating interest on an investment over time to earn more interest. ! Interest on interest Ñ interest earned on the reinvestment of previous interest payments.! Simple interest Ñ interest earned only on the original principal amount invested.!   Compound interest Ñ interest earned on both the initial principal and the interest reinvested from prior periods. ! In general: ! Vt = V0 x (1 + r)t! Where Vt is the value after t periods; r is the interest/discount rate for one period.! We call Vt the FUTURE VALUE at the end of t periods of compound interest r. ! We call V0 the PRESENT VALUE at the start of the time period being considered. ! Note: PV may not be TODAY. ! The formula on your mid-term test formula sheet is written:! FV = PV x (1+r)t! Example 2a What was the interest on interest in period 2 in example 1? ! £10 (interest you earn in period 2) x 10% (interest rate paid)! = £1! Interest on interest in period 2 was £1. Without compounding, period 2 interest would have been £10. The extra £1 is interest on interest, being 10% x the original interest of £10.! Example 2b What was the interest on interest in period 3 in example 1? ! £21 (interest earned up to the end of period 2: £10 + £11) x 10% (interest rate paid) ! = £2.10 ! Without compounding, period 3 interest would have been £10. The extra £2.10 is interest on interest, being 10% x the interest earned up to the end of period 2, being £21 (where £21 = £10 + £11). Example 3 What would the interest payment have been in period 3 in example 1 if simple interest had been paid? ! Interest is simply 10% of the original amount invested. That is, interest is £10 every period. ! £100 Ñ> £110 Ñ> £120 Ñ> £130, etc! + £10 + £10 + £10 !   The interest on interest is growing every period: e.g. period 1 = 0, period 2 = 1, period 3 = 2.10, period 4 = 3.31, etc. ! Future value of £1 for different periods and rates Figure'4.2: Future&value&of&£1&for&different&periods&and&rates ! Growth of £100 original amount at 10% per year. The orange shaded area shows the simple interest. The green shaded area represents the portion of the total that results from compounding of interest. Future values depend critically on the assumed interest rate, particularly for long-lived investments. Figure 4.2 illustrates this relationship by plotting the growth of £1 for different rates and lengths of time. Notice that the future value of £1 after 10 years is about £6.20 at a 20% rate but it is only about £2.60 at 10%. In this case, doubling the interest rate more than doubles the future value. To solve future value problems, you can use a table that contains future value factors for some common interest rates and time periods. ! ! 4.2 Present value and discounting Present value Ñ the current value of future cash ßows discounted at the appropriate discount rate. ! Discount Ñ calculate the present value of some future amount.! Present value x 1.1 = £1 ! Present value = £1 / 1.1! = £0.909! Year%1 • Receive%£1,% interest% rate%is%10% Year%0  • What%is%the% Year%0% equivalent% value%of%£1% received%in% Year%1?  The present value of future cash ßows is equal to: ! PV (V0) = Vt ! (1+r)t! Where Vt is the same value at time t; r is the interest rate used, sometimes called the DISCOUNT RATE. ! Example 4a What is £100 payable in one yearÕs time worth today at a n interest rate of 10%?! Note: interest rates should be assumed to be compound rates unless the question says Ôsimple interest rateÕ.! PV = 100 / (1+10%)! = 100/1.10 ! = £90.91 ! This time, DIVIDE by (1+r), that is, divide by 1.10 in this example as r = 10% = 0.10.! So £100 in one yearÕs time is worth 100/1.10 = £90.91 today. ! Example 4b What is £100 payable in two yearÕs time worth today at an interest rate of 10%? ! PV = 100 / (1+10%)2! = 100/1.102 ! = £82.64 ! Divide by 1.102 in one step. So £100 in two yearÕs time is worth 100/1.102 = £82.64 today. ! 4.3 More about present and future values Discount rate Ñ the rate used to calculate the present value of future cash ßows. ! Discounted cash ßow (DFC) valuation Ñ calculating the present value of a future cash ßow to determine its value today. ! General rule: ! PV = FVt ! (1+r)t! That is, present values are smaller than future values (as long as r is a positive number). In order to get to this formula, you have to derive the PV formula from the FV formula: ! PV x (1+r)t = FVt! PV = FVt / (1+r)t ! = FVt x [1/(1+r)t]! = FVt x (1+r)-t!   Calculating t (Þnding the number of periods) ! So far, we have just calculated PV or FV from values we have been given for r and t. You may also be asked to calculate r or t. ! Example 5 You've been saving up to buy Godot Ltd. The total cost will be £10 million. You currently have about £2.3 million. ! (a) If you can earn 5% on your money, how long will you have to wait? ! (b) At 16%, how long must you wait?! (a) From the basic present value equation: ! Use PV = FV / (1+r)t ! PV = £2.3 million ! FV = £10 million ! r = 5% ! £2.3 million = £10 million / 1.05t ! £10 million / £2.3 million = 1.05t ! 1.05t = 4.35! Now we need to solve for t. Using trial and error: Try t = 10, 1.05^10 = 1.629 ! Try t = 20, 1.05^20 = 2.653! Try t = 30, 1.05^30 = 4.322! Try t = 31, 1.05^31 = 4.533! So the best answer is 30 years. ! (b) Now assume r = 16%, what is t? How many years before you have £10 million?! Use PV = FV / (1+r)t ! PV = £2.3 million ! FV = £10 million ! r = 16% ! £2.3 million = £10 million / 1.16t ! £10 million / £2.3 million = 1.16t ! So, 1.16t = 4.35 ! Now solve for t using trial and error: Try t = 10, 1.16^10 = 4.41 ! Try t = 9, 1.16^9 = 3.80! So, the best answer is 9 years. !   Calculating r! We often need to calculate the return that the investment has generated. That is, calculate the implicit interest rate, r.! Use the following formula, which is on your formula sheet:! r = (FV/PV)(1/t) - 1! Proof: PV = FV! (1+r)t ! Solve for r: ! (1 + r)t x PV = FV! So, (1 + r)t = FV/PV! So, 1 + r = (FV/PV)(1/t) ! So, r = (FV/PV)(1/t) - 1! Example 6 You invested £10,000 Þve years ago and your investment is now worth £15,000. What was the annual return (interest) earned on your investment? ! Use r = (FV/PV)(1/t) - 1! Where FV = value at the end of the investment: £15,000! PV = value at the start of the investment: £10,000 ! t = 5 years ! Solve for r: ! r = (15,000/10,000)(1/5) - 1 ! r = 1.5^0.2 - 1! r = 1.08447 - 1! r = 0.08447! = 8.447%! Answer, to 2 decimal places = the annual return earned on the investment was 8.45%!  