2. I =interest (in dollars and cents, accumulated over some period)

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Lionel Singleton
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1 A. Recap of the Variables 1. P = principal (as designated at some point in time) a. we shall use PV for present value. Your text and others use P for PV (We shall do it sometimes too!) 2. I =interest (in dollars and cents, accumulated over some period) 3. i = conversion rate of interest 4. j m = nominal rate of interest to be compounded over m periods 5. m = number of conversion periods annually 6. n= number of periods in total 7. S = an accumulated value a. face value of a discounted loan instrument at maturity b. an accumulation of the principal over time 8. j eff = annual effective rate of interest

2 2. A comparison of compounding: take 9.5% as the nominal rate of interest; i.e. j m =.095 Effective Annual Rate of Interest annually semi-annually quarterly daily % % % % Note: j eff = ( /1) 1 1 = =.0950 annually j eff = ( /2) 2 1 = =.0973 semi-annually j eff = ( /4) 4 1 = =.0984 quarterly j eff = ( /365) = =.0996 daily Notice that, as expected, the effective rate of interest, j eff increases as the number of compounding periods m increase.

3 Def. 1: A set of dated values is some collection of points (dates) on the time line (cashflow) where money flows in or ou (cash flows in or out). Def. 2: A focal date is a point (date) on the time line which we use to compute the net value at that date, of a set of dated values. That net value is the dated value of the set of dated values at the given focal date. A focal date (focal time) may be used to compare two different sets of dated values. Example 1 (of a set of dated values):

4 CAUTION: Two sets of dated values cannot be compared without bringing them to a common focal date. Why? Answer (using 2 simple sets of dated values:

5 Example 2: Lawless and Partners, Chartered Accountants, the accountants for BGE, charge monthly interest of "2.5% monthly". The terms of the statement state: "Payable on receipt." (a) What is the nominal interest rate? (b) What is the effective interest rate? (c) If BGE owed $ 5, for one year, how much interest would BGE have to pay Lawless? (d) If BGE owed $ 5, for six months, how much interest would BGE have to pay Lawless?

6 Solution with cashflow explanation: (a) By definition, j 12 = 12 i = =.30 The nominal rate of interest is 30% (100% x.30), (Recall that j 12 is nominal interest.) which means that the nominal interest rate is 30%.

7 For (b): Choose the focal date to be t = 12. Move the initial $1 to t = 12 using the Fundamental Formula For Compound Interest (FFCI). S (1.025) 12 = 0 so S = (1.025) 12 = so j eff = = The effective rate of interest is 34.49%

8 For (c): Choose the focal date t = 12. Move $5,000 from t = 0 to t = 12 using the Fundamental Formula for Compound Interest (FFCI) At t = 12 S (1.025) = 0

9 so S = (1.025) = 6, I = S _ P = 6, _ 5,000 = 1, The interest payable by BGE to Lawless is $1, For (d): Take the focal date as t = 6 Move all flows to the focal point.

10 S (1.025) = 0 (cashflow equation) so here the accumulation S = (1.025) = 5, I = S P so I = 5, = Hence BGE would have to pay Lawless interest.

11 Example 3: BGE invests $10,000 of its capital in a one year 9.5% certificate of deposit which is compounded quarterly. What will BGE be paid when the certificate matures? Solution: P = 10,000 S =? n = 4 m = 4 j m =.095 i =.095/4 =.0238

12 Cashflow Diagram: S = (1+i) n P = ( /4) 4 10,000 The actual payment is made at the end of the year so at that point S = ( /4) 4 10,000 = 10, At the end of the year, BGE will be paid 10,

13 B. Discounted Value or Present Value Example 4: Suppose that a $10,000 certificate of deposit (CD) is sold at a discount (that is for some number less than $10,000 and that its value at maturity is $10,000). What shoul BGE pay for the certificate in order that the interest be the same as a 9.5% certificate compounded quarterly? Def. 3: The present value or discount value of an amount S available to us n periods from now is the amount P that would be required now so that after n periods compounded using the nominal interest rate, j m, the accumulated amount would be S S = (1+i) n P so P = S (1i)+ n = (1+i) n S The process of calculating P from S is called discounting.

14 Discounting at an interest rate for simple interest: S j (= j eff ) P S PSj ==+ (1t) + j (1t) 1 Discounting at an interest rate for compound interest: S i = j m /m n periods P PSj ==+=+ (1/m + j S m ) n ( 1/m)(1) m nn Si

15 The difference S P is called compound discount on S at an interest rate i because we use (1+i) as the factor. For simple interest, S P is called simple discount on S at an interest rat i. (For PS, S P is interest. For SP, S P is discount.) Solution to Example 4: Cash Flow Diagram: Let the focal date be t = 0. Move S to the focal date.

16 P = (1+.095/4) 4 (10,000) = 9, Hence BGE should pay $9, for the certificate. Example 5: The premium for BGE's property and liability polic from its insurance carrier, Great White Whale Insurance, is $7,500. BGE is offered the choice of paying the premium in fu immediately or making an instalment payment of $2, now and an additional payment of $5, six months from now. Find the nominal rate j 2. Solution: Method 1 informal: $2,500 of the $7,500 is paid immediately, so the amount outstanding (still owing) is $7,500 $2,500 = $5,000 After 6 months, the total owing is discharged (paid so no amount left owing). However, the amount paid exceeds the $5,000 owing by $(5,150 5,000), so interest of $150 was paid for six months. So the interest rate over the six months is

17 i = 150 / 5,000 =.03 Now j 2 = 2 x i = 2 x.03 =.06 Hence the nominal interest rate charged is 6%. Method 2 a cash flow approach: Pmt. Schedule 1 7,500 j 2 =? i = j 2 / Pmt. Schedule 2 5,150 j 2 =? i = j 2 /

18 Note: The payment schedules are considered cash flows even though they each have net present value (NPV) different from 0. It is understood that the NPV could be put in at time 0 to offset the NPV but that this flow is often suppressed. Divide by (1+i) Focal date: t = (1+i) 2 = 2500 (1+i) (1+i) Equation of Value of two sets of dated values so Therefore 7500 (1+i) = 2500 (1+i) (1+i) = 5150 i = =.03 so j 2 = 2 x.03 =.06 Hence the nominal rate of interest is 6%.

19 Method 3 discounting: Pmt. Schedule 1 7,500 j 2 =? i = j 2 / Pmt. Schedule 2 5,150 i = j 2 / 2 =? Take the focal date to be t = 0: NPV I = 7500 NPV II = (1+i) 1 Set NPV I = NPV II 7500 = (1+i) 1

20 so 5150 (1+i) 1 = 5000 Hence (1+i) = 5150 / 5000 so i =.03 j 2 = 2 x i =.06 The nominal interest rate is 6%. A note on the two different Equations of Value for Example 5 Take the focal date as t = 2. (It is preferable to use t as the variable for the focal date rather than n so as not to confuse the number of periods n with the focal date.) Equation of Value (Cashflow Equation) focal point t = 2: (1) 7500 (1+i) 2 = 2500 (1+i) (1+i)

21 Now take the focal date as t = 0. Equation of Value (Cashflow Equation): (2) which reduces to = (1i) (1 + i) = 7500 (1 + i) Dividing through (1) by (1 + i) yields this same result. In this case both equations of value can be solved easily. However, in general, the choice of the focal point can make the calculations easy or hard so keep this fact in mind.

22 Example 6: BGE has a payment coming in one year from now for $10,000. However it needs the money now. A lender has offered to give BGE $8,500 now and to take over (to assume) the $10,000 debt at the end of the year. What is the effective annual rate of interest charged by the lender? Solution: P = 8500 S = 10,000 t =1 (yr) This is all we are given. I = interest payable for 1 yr = S P = 1,500 Hence compound discount = S P = 1,500 Simple Interest earned in a year = j eff x P Amount payable at the end of the year = (1 + j eff ) P Then (1 + j eff ) P = S

23 so Hence j eff = SS = PP 1 j eff = 10,0008,500 1,500 = = ,500 80,50 P The effective annual interest rate is 17.65% Alternate approach: P = 8,500 Discounting (1+i) 1 S = P so (1+j eff ) 1 S = P j 1 = j eff = i (1+j eff ) 1 = 8,500/10,000 =.85 m=n=1 S =10,000 j eff = 1/.85 1 = The effective interest rate is 17.65%.

24 Example 7: The lender in Example 6 uses a nominal rate j 4 (quarterly compounding). Find this nominal rate. Solution: i = interest rate per period (conversion rate) 8,500 focal date t = m = n = 4 i = j4 /4 =? 10,000 Cash Flow equation: so Therefore so that 10,000 (1+i) 4 = 8500 (1+i) 4 = 10,000 / 8,500 = 1 /.85 1+i = (1 /.85) 1/4 = 1 / (.85) 1/4 =

25 i = The nominal interest, j 4, is j 4 = 4 x i = The nominal interest rate is 16.59%.

26 C. Using Net Present Value (NPV) Example 8: BGE must decide which of two machines it should purchase. Machine A costs $25,000 and requires a downpayment of $10,000, while Machine B costs $30,000 and requires a down-payment of $12,000. Both machines have an expected life of 3 years with no disposal value. BGE expects the machines to produce the net cash inflows shown below and will owe nothing more for the machines. If the value of money is j 1 =.16 (= 16%), determine which machine BGE should purchase. Net Cash Flow at the End of Year Machine Investment Now A 10,000 8,500 9,000 4,000 B 12,000 8,000 8,500 9,500 Solution: Draw the cash flows for machines A and B.

27 Cash Flow for Machine A: 8,500 9,000 4,000 focal date: t = 0; j 1 = j eff = ,000 At the focal date t = 0: NPV A = (1+.16) (1+.16) (1+.16) = 6,578.69

28 Cash Flow for Machine B: 8,000 8,500 9,500 focal date: t = 0; j eff = ,000 At the focal date t = 0: NPV B = (1+.16) (1+.16) (1+.16) = 7, Machine B should be bought since NPV B > NPV A.

29 Remark: We could have shortened the calculation by computing NPV B NPV A = ( [ 10000]) + ( )(1+.16) 1 + ( )(1+.16) 2 + ( )(1+.16) = (1+.16) 1 500(1+.16) (1+.16) = > 0 Note that the second calculation results in NPV B NPV A being one cent more than in the 1st calculation as a result of rounding. Example 9: What is the minimum net cash flow at the end of year 3 for Machine B in order that BGE should purchase Machine B? Solution: Let x = cash flow of Machine B at end of year 3.

30 As before NPV B NPV A = ( [ 10000]) + ( )(1+.16) 1 + ( )(1+.16) 2 + (x 4000)(1+.16) 3 = (1.16) 1 500(1.16) 2 + (x 4000)(1.16) 3 = (x 4000)(1.16) 3 = (x 4000)(1.16) 3 At the breakeven point NPV B NPV A = 0 so (x 4000)(1.16) 3 = 0 Solving for x x = (1.16) 3 = 8,374.58

31 Hence, the breakeven point for choosing Machine B is that Machine B provides a flow of $8, This means that as long as Machine B provides a (positive) cash flow of more than $8, in the third year, B is the better choice.

Lecture 3. Chapter 4: Allocating Resources Over Time

Lecture 3. Chapter 4: Allocating Resources Over Time Lecture 3 Chapter 4: Allocating Resources Over Time 1 Introduction: Time Value of Money (TVM) $20 today is worth more than the expectation of $20 tomorrow because: a bank would pay interest on the $20