MEASURING WEALTH: TIME VALUE OF MONEY CHAPTER 3 N.Alsaleh-Fin421-Chapter 3 4/12/2019

A. future value of a single amount FVn = PV (1+ K)n , FV = PV (1+K/q)qn q = the number of times interest is compounded (1+ K)n , (1+K/q)qn , FVIF1 k,n,nq Find the future value after 20 years of $100 deposited in an account paying 10% rate of interest , compounded annually Solution: FV20 = $100 ( 1+ .10)20 = $100 x 6.7275 = $672.75 If any three of the four variables (FV, PV, k, n) are known, the fourth can be calculated. N.Alsaleh-Fin421-Chapter 3 4/12/2019

B. Present value of a single amount PV = FVn [1/(1+k)n ] , 1/(1+k)n = PVIF1k10%,n Find the present value of $20,000 to be received 5 years from today, if the opportunity cost of capital is 10 percent Solution: PV5 = $20,000 [ 1/(1+.10 )5 ] = $20,000 x 0.6209 = $12,418 Relationship between Present value, rate of retune, and time Figure 3-1, P70) N.Alsaleh-Fin421-Chapter 3 4/12/2019

C. Annuity Problems- Future value of ordinary annuity: payments are the same and the first payment occurs at the end of the year FVAn = PMT X FVIFA1n,k% , FVAn = PMT X [(1+k)n -1]/k Find the future value at the end of year 3 of 3 equal payment of $1,000 each at 10 percent rate of return. The first payment occurs one year from today. N.Alsaleh-Fin421-Chapter 3 4/12/2019

Solution : FVA3y = $1,000 X FVIFA1 3y,.10% = $1,000 X 3.310 = $ 3,310 If any three of the four variables ( PMT, k, n, FVA) are known the fourth variable can be calculated. N.Alsaleh-Fin421-Chapter 3 4/12/2019

D. Annuity Due (annuity in advance) AD has payments at the beginning of each period, and the future value is found one period after the last payment. Year 0 1 2 3 OAN $1000 $1000 $1000 AND $1000 $1000 $1000 N.Alsaleh-Fin421-Chapter 3 4/12/2019

Find the future value of $1,000 , 20 payment annuity due at 8 percent FVAD3 = PMT X FVIFA120y,8% x (1+.08) , = $1,000 X 45.762 X 1.08 =$ 9,423. E. Present value of an annuity E1. Ordinary annuity : PVOAN = PMT x PVIFA1 n,k% E2: Annuity due: PVAND = PMT X PVIFA1 n.k% x (1+k) N.Alsaleh-Fin421-Chapter 3 4/12/2019

F. Complex Cash Flow Problems: A complex cash flow is one where the cash flow amounts are not equal and consistent but instead either miss periods or change in amount. Problem. Find the present value of a stream of cash flows such that cash flow will be $1,000 at the end of each year for years 4 through 20. = $1,000(PVIFA1 20y,10% – PVIFA1 3y,10% ) N.Alsaleh-Fin421-Chapter 3 4/12/2019

Problem: Find the present value of the following cash flow stream at 10 percent cost of capital Year zero = -100,000, Year 1 ($50,000), Year 2 through 10 $25,000, Year 10 $80,000. Solution: NPV = [($100,000) x 1 + ($50,000) x .9091 + $25,000 (6.1446 -.9091) + $80,000 x 0.3855] = $16,273. N.Alsaleh-Fin421-Chapter 3 4/12/2019

Multiple inflows and outflows net present value example (P.81) A rental property can be purchased for $40,000. The property will generate cash flows of $8,000 at t he end of each year for 7 years, before equipment replacement. A new $6,000 furnace will be needed at the end of 5 years, and the building will be worth an estimated $50,000 at the end of 7 years. If the opportunity cost of capital is 10%, what is the project’s NPV?. N.Alsaleh-Fin421-Chapter 3 4/12/2019

G. Effective Interest Rate : K = (1+ K` / q)q -1 3-6, K` = nominal annual rate of return rate, q = the number of times interest is compounded Find the future value of $100 after 3 years at 10 percent rate of interest, if the interest is compounded semiannually Solution: FV3 = $100 (1+ .10/2)3x2 = $134 , k = (1+.10/2)2 -1 =10.25% N.Alsaleh-Fin421-Chapter 3 4/12/2019

Quarterly compounding example (P.83-84) Find the present value of $10,000 to be received in 10 years at an nominal required return of 16% a year, compounded quarterly. PV = $10,000 [1/(1+.16/4)10x4] = $10,000 x .2083 = $2,083 What is the effective rate of interest? N.Alsaleh-Fin421-Chapter 3 4/12/2019

H. Alternate Payment Patterns Fractional time period. Payments occur at some date within a period. Find the future value of $100 deposit in a savings account that has semiannual compounding and pays 5 percent each 6-months period and the money is left for 3.5 years. Solution: FV3.5y = $100 (1.05)7 = $140.71 or Alternatively the problem can be solved by finding effective rate of interest. N.Alsaleh-Fin421-Chapter 3 4/12/2019

I. Payments Spread Evenly across the Year: If payment series are spread over the year, with a payment received at the end of each period of 1/q of a year: Then effective rate per payment period is: Kq = (1 + k)1/q – 1 Problem: if you receive $1,000 a year for 10 years, divided into equal daily payments, With an effective annual interest rate of 10%. Find the present value of the payments. N.Alsaleh-Fin421-Chapter 3 4/12/2019

To solve for PVA1n,k use equation 3-4b, page 77 Solution: K365 = (1+ 0.10)1/365 – 1 =.000261158 PVA 3650 days = ($1,000/365) PVA 13650days,.0261158% = 6,446.07 To solve for PVA1n,k use equation 3-4b, page 77 PVA1n,k = [1 -1/(1+k)n ] /k, N.Alsaleh-Fin421-Chapter 3 4/12/2019

Example (page 86) number in million $ Year 0 1 2 3 4 5 ets. Year-end 30 30 30 30 30 Mid-year 30 30 30 30 30 Daily*----------------------82.19 per day for 10 Years, *30,000,000/365 =$82.19 If payments received at year-end, the present value = PVA 10y,10% = $30m PVIFA110y,10% = $30m x 6.1446 = $184,338,000 N.Alsaleh-Fin421-Chapter 3 4/12/2019

b. If payments are received at mid-year, present value = PVA 10y = = $30m PVA1 10y,10% x (1+.10).5 = $193,335,325 c. If payments are spread evenly over the year on a daily basis. Kq = ( 1+.10)1/365 -1 = .0261158% PV = ($30m/365) PVA13650,.0261158% = 193,382,189 Difference between daily cash flow and mid-year cash flow is only .02%! Difference between daily cash flow and year-end cash flow is 4.68%! N.Alsaleh-Fin421-Chapter 3 4/12/2019

Using midyear cash flow to approximate daily cash flow- example on page 87 Y0 ½ y1 1.5y y2 2.5y y3 $1m/3 $1m/3 $1m/3 $1m/3 $1m/3 $1m/3 FVA3y The compound cost of Boeing’s investment at the end of years 3 is $1m/3 x FVIFA13y,10% x (1+.10)1/2 = $115,718,576 N.Alsaleh-Fin421-Chapter 3 4/12/2019

J. Present value of a No-Growth Perpetuity PVp = PMT/k K. Present value of a Constant- Growth Perpetuity, PVgp = PMT1 / k-g L. Contentious Compounding A situation in which interest is added continuously rather than at discreet points in Time. FVn = PV (ekxn), PV = FVn/ (ekxn), N.Alsaleh-Fin421-Chapter 3 4/12/2019

The effective interest rate with Continuous compounding: K = ek` – 1, if k`= 10% then k = e.10 -1 =10.52 K/ = ln (1+k) An investment of $1,000 at a 10% nominal interest rate with continuous compounding will grow in 5 years to: FV5 = $1,000 e.10x5 = $1,000 x 1.648.7 = $1,648.70 Table on page 97 shows the amount to which $1,000 will grow with various compounding schemes if invested at 10% a year for 5 year. N.Alsaleh-Fin421-Chapter 3 4/12/2019

Future value of an annuity FVAn = PMT [ek` n -1 ]/ k` Find the future value of an annuity of $10,000 a year f or 20 years, with payments spread continuously over the year. The money I s expected to earn an effective annual rate of 10.517percent. N.Alsaleh-Fin421-Chapter 3 4/12/2019

The future value of the annuity FVA20 = FVA20 = $10,000 [e.10x20 -1] /.10 = $10,000[7.3891-1]/.10 = $638,910 With continuous compounding, K` = ln (1 + .10517) = 10 percent N.Alsaleh-Fin421-Chapter 3 4/12/2019

Present value of an annuity PVAn = PMT [1-1/ek`n]/k` Example: ( page 98) Searcy hat co. is considering a Building expansion that will provide cash flows of $100,000 a year, arriving continuously over the years, for 20 years. At the end of the 20- years period, the investment can be sold For $200,000. The effective required return is 10.517 percent a year. N.Alsaleh-Fin421-Chapter 3 4/12/2019

the present value of the benefits is: PVAn = PMT [1-1/ek`n]/k` K` = ln(1+ .10517) = 10 percent, e.10x20 =7.3891 the present value of the benefits is: PV = $100,000[1 -1/7.3891 ] /.10 + $200,000/e.10x20 = $100,000 [1 -1/7.3891]/.10 + $200,000/7.3891 = $891,732 N.Alsaleh-Fin421-Chapter 3 4/12/2019

N.Alsaleh-Fin421-Chapter 3 4/12/2019

N.Alsaleh-Fin421-Chapter 3 4/12/2019

N.Alsaleh-Fin421-Chapter 3 4/12/2019

N.Alsaleh-Fin421-Chapter 3 4/12/2019

N.Alsaleh-Fin421-Chapter 3 4/12/2019

N.Alsaleh-Fin421-Chapter 3 4/12/2019