Discounting. Capital Budgeting and Corporate Objectives. Professor Ron Kaniel. Simon School of Business University of Rochester.

Size: px

Start display at page:

Download "Discounting. Capital Budgeting and Corporate Objectives. Professor Ron Kaniel. Simon School of Business University of Rochester."

Matilda Davidson
6 years ago
Views:

1 Discounting Capital Budgeting and Corporate Objectives Professor Ron Kaniel Simon School of Business University of Rochester 1 Topic Overview The Timeline Compounding & Future Value Discounting & Present Value Multiple Cash Flows Special Streams of Cash Flows» Perpetuities» Annuities Interest Rates 2 1 1

2 The Timeline Timeline: a linear representation of the timing of potential cash flows. Two types of cash flows: 1. Inflows (i.e., money we get) are represented by positive numbers 2. Outflows (i.e., money we give) are represented by negative numbers Example:» Assume that you are lending $10,000 today and that the loan will be repaid in two annual $6,000 payments. 3 Money s Time Units Think of money as having a time unit denoting when it is received (or paid)» Just like currency We can only compare money in the same time units:» It doesn t make sense to add $50 US to 50; and» It doesn t make sense to add $50 received today with $50 received next year. Discounting and Compounding are the tools to manipulate money s time units» Discounting converts money s time units back in time» Compounding converts money s time units forward in time 4 2 2

3 Compounding The Future Value (FV n ) of a cash flow T-years from today is:» C = Cash Flow (or CF )» r = discount rate Example:» Would you rather receive $1,000 today or $1,210 in two years if you can earn 10% per year on the $1,000? Timeline and Future Value =? FV C 1r T 5 Discounting The Present Value (PV) of a cash flow T-years from today is: C PV C 1r T 1 r Example:» What is the price of a savings bond that will pay $15,000 in ten years if the annual interest rate is 6%? Timeline =? T Present Value =? 6 3 3

4 Multiple Cash Flows Present Value (PV) and Future Value (FV) are linear operators» PV(C 1 + C 2 ) = PV(C 1 )+PV(C 2 )» FV(C 1 + C 2 ) = FV(C 1 )+FV(C 2 ) Example: If we can earn a 10% annual interest rate and save $1000 today, and $1000 at the end of each of the next two years how much will we have in 3 years? Timeline =? FV =? 7 General Stream of Cash Flows Present Value PV N PV ( C ) n n 0 n 0 N Cn (1 r) n The PV of a stream of cash flows is just the sum of the PVs. Future Value (same idea): N n n 0 n 0 N N n N FV FV ( C ) C 1 r PV 1 r n 8 4 4

5 Perpetuities A perpetuity is a stream of cash flows with no end: Cash Flows 0 C 1 C 2 C 3 C 4 C 5 C 6 Periods Examples:» Cencus Agreements issued in 12 th century in Italy, France, and Spain to circumvent usury laws of Catholic Church (no principal = no loan)» Hoogheemraadschap Lekdijk Bovendams 17 th century Dutch Water Board to upkeep local dikes (they still pay interest!)» British consol bonds» Panama Canal perpetuities How do we compute PV? 9 Valuing Perpetuities Step 1: Write out the PV of the perpetuity PV C 1r C 1r C 1 r... Step 2: Pull out the cash flow, C PV C 1r 1r 1 r... Step 3: Multiply both sides by 1/(1+r) PV 1r C 1r 1r 1 r... Step 4: Subtract (3) from (2) 1 1 PV 1 1 r C 1r Step 5: Do some algebra C PV r

6 Perpetuity Example What does the timeline look like? The stream of cash flows is a? with a PV =? 11 Growing Perpetuities A growing perpetuity is a stream of cash flows that grow at a constant periodic rate, g, with no end. Cash Flows 0 C C(1+g) C(1+g) 2 C(1+g) 3 C(1+g) 4 C(1+g) 5 Periods Again, infeasible to calculate by brute force so is there a shortcut? PV C r g

7 Annuities An annuity is a level stream of regular payments that last for a fixed number of periods Cash Flows 0 C C C C C Periods N-1 N Examples:» Mortgages» Lottery prizes (sometimes )» Retirement savings plans How do we compute PV? 13 Valuing Annuities Part I An annuity is just the difference in two perpetuities starting at different times!» Perpetuity #1 starts today: Cash Flows 0 CF CF CF CF CF CF Periods N-1 N N+1 It has present value at time 0 equal to C/r.» Perpetuity #2 starts in period N: Cash Flows CF Periods N-1 N N+1 It has present value at time N equal to C/r and at time 0 equal to (C/r)(1+r) -N

8 Valuing Annuities Part II Subtracting the cash flow streams of the two perpetuities gives us the cash flow stream for our annuity Cash Flows 0 CF CF CF CF CF 0 Periods N-1 N N+1 Therefore, difference in present values for the two perpetuities must equal the present value of our annuity Perpetuity #1 Perpetuity #2 PV PV PV What s the future value of an annuity?? 15 Annuity Example PV of Option A:» What is the timeline?» What is the present value of all the cash flows? PV of Option B =?

9 Valuing Growing Annuities A growing annuity is a constant growing stream of regular payments that last for a fixed number of periods Cash Flows 0 CF (1+g)CF (1+g) 2 CF (1+g) N-2 CF (1+g) N-1 CF 0 Periods N-1 N N+1 The present value of this stream is T C 1 g PV 1 rg 1r 17 Internal Rate of Return (IRR) The Internal Rate of Return (IRR) is the one interest rate that sets the net present value of the cash flows equal to zero N Cn Initial Cost = 0 n n 0 (1 IRR) Example 1:» The IRR of a security (e.g., bond, stock, CD, etc.) is just the one interest rate that sets the present value of all the cash flows equal to the price (a.k.a. PV) of the security: N Cn Price 0 n n 0 (1 IRR) Example 2:» The IRR of an investment project (e.g., acquisition, merger, capital expenditure, etc.) is just the one interest rate that sets the present value of all the cash flows equal to the initial outlay (a.k.a. PV) of the investment: N Cn Initial Outlay 0 n (1 IRR) n

10 Computing the Internal Rate of Return Example Jessica takes out a $1 million loan today. Her bank offers her the following repayment option: pay $100,000 at the end of the first year, afterwhich the repayment amount will increase by 4% per annum forever. What is the IRR? The Timeline = Present Value =?? IRR =? 19 Effective Annual Rate (EAR) The Effective Annual Rate (EAR) indicates the total amount of interest that will be earned at the end of one year» Considers the effect of compounding» Also referred to as the effective annual yield (EAY) or annual percentage yield (APY)» We can use this to discount cash flows, as long as we express time in annual units (i.e., years) So far everything was on an annual basis» Cash flows were every year» Interest was on an annual bases (i.e., compounded once a year)» Therefore, distinction was irrelevant: EAR = r

11 Adjusting the Discount Rate to Different Time Periods Earning 5% annually is not the same as earnings 2.5% every six months because of compounding So, if the EAR is 5% but we have semi-annual discounting the Equivalent Periodic Rate (EPR) is EPR EPR More generally, $1 $1.025 $ / EPR 1/ m (1 EAR) 1» where m = # of compounding periods per year (e.g., semi-annual m = 2, quarterly m = 4, monthly m = 12, )» EPR is just an n-period discount rate 21 EAR and EPR Examples If the EAR is 10% and we have quarterly compounding, what is the EPR? If the EPR is 0.6% and we have monthly compounding, what is the EAR?

12 Valuing Monthly Cash Flows Example Timeline:? Monthly EPR = Periodic Cash Flow =?? 23 Annual Percentage Rate (APR) The Annual Percentage Rate (APR), indicates the amount of simple interest earned in one year.» Simple interest is the amount of interest earned without the effect of compounding.» The APR is typically less than the effective annual rate (EAR) which incorporates the effect of compounding Counterexample? The APR itself cannot be used as a discount rate.» The APR with m compounding periods is a way of quoting the actual interest earned each compounding period: APR Interest Rate per Compounding Period i m periods / year

13 EAR vs APR How do I convert an APR (not a discount rate) to an EAR (a discount rate)? m APR 1 EAR 1 m» EAR increases with the frequency of?» If compounding is once per year (m=1) then EAR=?» Continuous Compounding: In limit as m, (1+APR/m) m exp(apr) Some notation» R = APR (not a discount rate!)» i = APR/m = interest rate per compounding period 25 Valuing Monthly Cash Flows Revisited Example Recall the problem on slide 22:» Monthly interest with an EAR of 6% What is the APR (R) on this account? How much interest is earned each period? How much do you have to save at the end of each month to accumulate $100,000 in 10 years?

Converting the APR to a Discount Rate Example Strategy: Compute the PV of the lease and compare it with the $150,000 Timeline:? This cash flow stream is an? with? periodicity 27 Converting the APR to a Discount Rate Example (Cont.

14 Converting the APR to a Discount Rate Example Strategy: Compute the PV of the lease and compare it with the $150,000 Timeline:? This cash flow stream is an? with? periodicity 27 Converting the APR to a Discount Rate Example (Cont.) Computing the monthly discount rate:» Method 1: We re given an APR of 5% with semiannual compounding, which implies the EAR =? Convert annual discount rate into monthly discount rate?» Method 2: Compute an effective periodic interest rate from the APR,? Convert six-month discount rate into monthly periodic rate:?

15 Converting the APR to a Discount Rate Example (Cont.) With the monthly discount rate in hand, the PV of the annuity is? The PV of the lease is greater than the upfront payment of $150,000 so purchase the system outright 29 Nominal Versus Real Interest Rates Nominal Interest Rate: The rates quoted by financial institutions and used for discounting or compounding cash flows, r Real Interest Rate: The rate of growth of your purchasing power, after adjusting for inflation, rr 1 r Growth of Money Growth in Purchasing Power 1 rr 1 Growth of Prices rr r r

16 US Interest Rates and Inflation 31 Summary Money has a time unit» Can only compare money in same units!» Compound to get future values» Discount to get present values Future and Present Values are linear» Use them on streams of cash flows Special streams of cash flows» Perpetuity» Annuity Interest Rates» APR vs. EAR» Real vs. Nominal

Financial Management I

Financial Management I Workshop on Time Value of Money MBA 2016 2017 Slide 2 Finance & Valuation Capital Budgeting Decisions Long-term Investment decisions Investments in Net Working Capital Financing