3. Time value of money


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1 1 Simple interest 2 3. Time value of money With simple interest, the amount earned each period is always the same: i = rp o We will review some tools for discounting cash flows. where i = interest earned r = interest rate (per period) p o = principal Therefore, at the end of n periods, we will have (principal plus interest) p t = p o + trp o = p o(1 + tr) Example simple interest 3 Compound interest 4 If we invest $100 at 10% simple interest for 7 years, how much will we have? With compound interest, we earn interest not only on the principal but also on the interest earned in previous periods: = 170 p 1 = p o + rp o = p o(1 + r) p 2 = p 1(1 + r) = p o(1 + r) 2 p 3 = p 2(1 + r) = p o(1 + r) 3. p t = p t 1(1 + r) = p o(1 + r) t
2 Example compound interest 5 Simple vs compound interest 6 If we invest $100 at 10% compound interest for 7 years, how much will we have? How much of the interest earned in the previous example was from the principal, and how much was earned on previous periods interest? The total will be = , of which = is interest on interest. Compounding over many periods 7 Present value and future value 8 The value of an investment at present is often referred to as the present value (PV). Its value in the future is often referred to as its future value (FV). Thus, one might also write the formula for compound interest as FV t = PV (1 + r) t Because of compounding, small differences in interest rate can make a large difference after many periods.
3 Discounting 9 Four variables 10 Computing the present value of a future cash flow is often referred to as discounting the cash flow. By rearranging the previous formula, we get PV = FVt (1 + r) t There are four variables in the equation FV t = PV (1 + r) t. Given values for any three, we can solve for the fourth. It is not hard to do this algebraically. But, it is easier to use our financial calculators. Calculators 11 Example compound interest 12 There are a couple of things to be careful about when using your calculators: Be sure that you are in end mode rather than begin mode. This means that payments are at the end rather than the beginning of each period. This is the standard convention unless noted otherwise. Be sure that the number of payments per period is set to 1. Suppose I invest $100 initially. After 8 years, the investment is worth $190. What interest rate did I earn (assume annual compounding)? n = 8 pmt = 0 pv = 100 fv = 190 rate = 8.35%
4 Example compound interest 13 Example compound interest 14 Suppose I invest $100 initially. The investment earns 8% compounded annually. How long until the investment is worth $200? Suppose I have an investment that will pay off $1000 after 20 years. What is the present value of this investment if I discount at 11% per year? r = 8 pmt = 0 pv = 100 fv = 200 n = fv=1000 n=20 r=11% pmt=0 pv = Example compound interest 15 Rule of I invest $300 initially. The investment earns 10% compounded annually. How much is it worth in 15 years? pv=300 r=10 pmt=0 n=15 A good rule of thumb is that time required for an investment to double multiplied by the rate is about 72. Example: About how long is needed for an investment at a 6% annual rate (compounded annually) to double? fv=
5 Time periods 17 Example time periods 18 You have to be sure when doing these problems that the time units are consistent between I invest $200 at an annual rate of 8% compounded weekly. How much do I have after 3 years? compounding period interest rate Note: Be careful about rounding! period of investment If not, it is usually best to convert everything to the same units as the compounding frequency. Note: For the purposes of this course, always assume a year is comprised of 12 months (each of equal length), 52 weeks, and 365 days. pv=200 r=8/52 n=3*52 pmt=0 fv= Effective annual rate (EAR) 19 Annual percentage rate (APR) 20 There are many different ways to quote rates: Rate: annual, monthly, weekly,... Compounding: annual, monthly, weekly,... Truthinlending laws in the US require that lenders disclose the APR. The APR is just the nominal rate quoted on an annual basis. It says nothing about the compounding interval. To compare different rates, it is convenient to standardize them. The effective annual rate (EAR) is the equivalent annual rate based on annual compounding.
6 Examples EAR/APR 21 IMPORTANT 22 What is the APR for a loan with a daily rate of 0.03% compounded daily? What is the EAR? What is the EAR for an annual rate of 11% compounded weekly? Unless otherwise noted, interest rates (and other kinds or rates) will be quoted as nominal annualized rates (i.e., APR). Note: Be careful about rounding!! APR = = = 10.95% If I start with $1, then the future value after 1 year will be FV = = so EAR = 11.6% The weekly rate must be 11/52, so if I start with $1, after one year I will have FV = (1 +.11/52) 52 = so, EAR = 11.61%. Continuous compounding 23 Continuous compounding formula 24 You could think of compounding over shorter and shorter intervals. In the limit, as the compounding interval becomes infinitely short, we refer to this as continuous compounding: FV t = lim PV (1 + r/m)mt m = PV e rt As with the standard compounding problems, there are four variables. Given any three, we can solve for the fourth. The formula is obtained using L Hopitals rule: ( lim x [ ( = exp lim x x) x log )] x x [ log ( )] x = exp lim x 1 x 1/x 2 1+1/x = exp lim x 1/x 2 [ ] 1 = exp lim x x = e
7 Example continuous compounding 25 Example continuous compounding 26 Suppose we have $100 invested at 12% annual interest. How much do we have at the end of two years if the interest is compounded continuously? Suppose I invest some sum of money with 8% interest compounded continuously. At the end of 5 years, I have $200. How much did I invest? FV = 100 exp(0.24) = = P V exp(5.08) P V = 200/ exp(5.08) = Example continuous compounding 27 Present value with unequal cash flows 28 Suppose I invest $100 with continuously compounded interest. At the end of three years, I have $185. What is the interest rate? Suppose I am to receive cash flows of $300 after one year, $500 after two years, and $700 per year for the next three years. What is the present value of these cash flows if I discount at 8% per year? 185 = 100 exp(3 r) 185/100 = exp(3 r) log(185/100) = 3 r r = 1 log(185/100) = 20.5% 3 Discount the cash flows individually and add them up: Or, use cash flow function on our calculators. cf = 0, 300, 500, 700, 700, 700 r=8 PV = npv = Note: You don t have to reenter 700 three times. Just hit the cash flow button three times in a row.
8 Future value with unequal cash flows 29 Interest rate with unequal cash flows 30 Suppose I am to receive cash flows of $300 after one year, $500 after two years, and $700 per year for the next three years. What is the future value of these cash flows at the end of the fifth year 8% annual interest? Suppose I am to receive cash flows of $300 after one year, $500 after two years, and $700 per year for the next three years. If the present value of these cash flows is $2400, what is the interest rate? (Use cash flow function and then compute IRR using calculator.) Compound the cash flows individually and add them up: FV = = Or, use cash flow function on our calculators (use NFV button if there is one, otherwise, use NPV and then multiply by ). cf = 2400, 300, 500, 700, 700, 700 irr = 5.91% Perpetuity 31 Deriving the perpetuity formula 32 A perpetuity is a stream of cash flows that continues forever. The perpetuity formula is based on the identity P V = C R Example: What is the present value of a perpetuity that pays $100 per quarter (use a discount rate of 12%)? PV = 100/.03 = This is referred to as a geometric series. With a little algebra, we can get C 1 + r + C (1 + r) x + x 2 + = 1 1 x = C 1 + r ( 1 1 ) 1 + r + (1 + r) 2 + = C 1 + r 1 1 1/(1 + r) = C r
9 Annuities 33 Deriving the annuity formula 34 A common situation is where a fixed number of equal cash flows are paid out. This is referred to as an annuity. The present value of an annuity can be calculated using the following formula: (Or, use calculator). PV = C [ r 1 ] 1 (1 + r) n We can derive the formula using the perpetuity formula: An annuity with payments of size C at the end of years 1 through n is equal to: the present value of a perpetuity with payments of size C minus the present value of a perpetuity with payments of size C beginning in year n+1. I.e., P V = C R C R 1 (1 + r) n = C [ ] R 1 1 (1 + r) n Annuity with final lump sum payment 35 Annuities continued 36 A common problem involves There are five variables involved in a standard problem: an annuity with payments of size C at the end of years 1 through n present value plus a final lump sum payment, F V n. interest rate The present value of this stream of cash flows is: P V = C [ R 1 1 (1 + r) n ] + F Vn (1 + r) n number of periods periodic payments final lump sum payment Given any four, you should be able to solve for the fifth.
10 Example annuity 37 Example annuity 38 Consider an investment that pays $100 at the end of each of the next 20 years. What is the present value of these cash flows if I discount at 9% (APR)? Suppose that I have a loan for $100,000 at an annual rate of 9% that I wish to pay off with 5 equal annual payments. What is the required payment? PV = 100/ / / / = 100 (1/ / / ) [ ] 1 1/ = = [ ( 1 1 ) 5 ] , 000 = C 0.09 C = 100, 000/3.98 = 25, 709 or, or, pmt=100 r=9 fv=0 n=20 pv= r=9 n=5 fv=0 pv= pmt = 25,709 Example annuity 39 Example annuity 40 If I borrow $1000 at 12% and make annual payments of $150 for 10 years, what is the balance on the loan after making the payment at the end of year 10? I invest $1000 in some project. The investment pays back $120 per year for 10 years. I then sell the investment for $500. What is the rate of return? pv=1000 pmt=150 n=10 r=12 fv = pv=1000 pmt=120 fv=500 n=10 r=8.65
11 Reminder 41 Example loan 42 Be careful that the time units match up for compounding interval time to maturity Suppose that I buy a house for $300,000. If I put down $50,000 and take out a 30year loan at an annual rate of 9% (compounded monthly) for the remainder. Assume equal monthly payments, and that the loan is paid off after the last payment. What are the monthly payments? payment period. Of the first payment, how much goes toward the principal? What is the remaining balance on the loan after 10 years? What would the payments be for a 15 year loan? Solution 43 Calendar time vs calculator time 44 The monthly rate is r monthly =.09/12 = Interest: i = = 1875 Dave is to receive a perpetuity with annual cash flows of $100 beginning five years from today. If the appropriate discount rate is 12%, what is the value of those cash flows at time t=5? At time t=4? Today? 30 year loan: ( ) r = C r C = r ( ) r = t=5: FV5 = /.12 = t=4: FV4 = 100/.12 = t=0: FV4/(1.12)^4 = Toward principal: = Balance after 10 years: pmt= , n=240, fv=0, r=.75, pv=?=223,575. I.e., after 10 years, you have only paid off 17,000. Alternate solution: pmt= , n=120, pv=250000, r=.75, solve for fv=?=223, year loan: ( ) r = C r C = r ( ) r =
12 45 Jill is thinking of buying an annuity that pays out 10 annual cash flows beginning 5 years from today. If Jill thinks the appropriate discount rate is 8% and is willing to pay $500 today for this annuity, what must the amount of the cash flows be? A note on cash flow timing The timing of the cash flows is very important: 46 p4 = 500*1.08^4 = Now, Plug this into the calculator as PV= n=10 r=8 FV=0 and solve for PMT=?= The usual convention (unless stated otherwise) is that the cash flows occur at the end of the period. Also, we will generally assume that the compounding frequency is the same as that of the cash flows unless otherwise stated. Annuity due 47 Example annuity due 48 Recall that the usual convention is that cash flows occur at the end of each period. An annuity with cash flows at the beginning of each period is called an annuity due. Annuity due value = Ordinary annuity value (1 + r) (Or, you can switch your calculator to begin mode.) I borrowed $1000 which I wish to pay off in five years making equal monthly payments. The annual interest rate (APR) is 12% and the payments are due at the beginning of each month. What are the payments? Two ways to do this: (1) switch to begin mode!! pv=1000 n=60 r=1 fv=0 pmt = Or, (2) in end mode, pv = / 1.01 = n = 60 r = 1 fv = 0 pmt = 22.02
13 Growing perpetuity 49 Example growing perpetuity 50 A growing perpetuity is a series of cash flows which grow at rate g. If C 1 is the size of the first cash flow, then John is to receive a series of quarterly cash flows which are to grow at an annual rate of 6% and continue forever. If the appropriate discount rate is 10%, and the present value of the cash flows is $100,000, what is the first cash flow? P V = C1 r g C 1 = P V (r g) = (.10.06)/4 = 1000 Growing annuity 51 Example growing annuity 52 A growing annuity is like a growing perpetuity, except only n cash flows are paid. The formula can be derived in a manner similar to the annuity formula, as the difference between a growing annuity starting immediately and one starting at time n: John is to receive a series of 10 annual cash flows beginning in one year. The first cash flow will be $100, and the cash flows grow at 10%. If the appropriate discount rate is 15%, what is the present value of the cash flows? P V = C1 r g [ 1 ( ) n ] 1 + g 1 + r [ ( ) ] =
14 Some common types of loans 53 Exercise Loan comparison 54 Pure discount loan The borrower receives the money today and repays the principal plus accumulated interest in a single lump sum at some time in the future. Interestonly loan The borrower repays the interest each period and repays the entire principal in a lump sum at some point in the future. Amortized loans The borrower pays the interest each period plus some amount toward the principal. The loan is paid off when the entire principal has been paid down. The most common structure is for the borrower to make equal payments. Balloon loan The borrower makes a payment each period (usually of equal size) and pays off the balance at some point in the future. Suppose that I buy a house for $300,000, putting $50,000 down and borrowing $250,000. I plan to take out a 30 year loan. The annual rate is 9% (compounded monthly). What are my monthly payments if I take out a pure discount loan? How much will I have paid for the house (principal plus accumulated interest)? What if I take out an interest only loan? How about an amortized loan (equal payments)? Which of these is the better deal? Solution 55 Discounting cash flows using a spreadsheet 56 Pure discount : r monthly = 0.09/12 = See posted examples. FV 360 = 250, = 3, 682, 644 So, 359 payments of zero followed by one of $3,682,644. Total amount paid is that plus $50,000 Interest only : 359 payments of = 1875 plus a final payment of 251,875 for a total of , , 000 = 975, 000. Amortized : ( ) r = C r C = r ( ) r = Total payments: = 774, 162
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