Introduction to the Time Value of Money Lecture Outline I. II.

Introduction to the Time Value of Money Lecture Outline I. II. III. IV. Why is there the concept of time value? Single cash flows over multiple periods Groups of cash flows Warnings on doing time value calculations – the “tricks”

I. Why is there a time value to money (TVM)? Which would you prefer: a cheque from me to you for 1,000,000 dated today or a cheque from me to you for 1,000,000 dated one year from now Why? What does the TVM mean?

Time Value Calculations Basics so far: Single cash flow over one period C1 PV0 (1 r ) FV1 C0 (1 r ) Objective: Extend to multiple, or “n”, periods

II. The Basic Time-Value-of-Money Relationships for a Single Cash Flow FVt n Ct (1 r)n C t n PVt (1 r) n where r is the interest rate per period n is the duration of the investment, stated in the time units of the period for r C is the cash flow at period t t C t n is the cash flow at period t n FV t n is the future value at time period t n PV is the present value at time period t t

Intuition for the Future Value formula Future Value of 1 invested for 2 years at 8% per year compounded yearly Year 0 1 2 FV2 C0 (1 r)2 1 (1 .08)2 1.1664 Note: simple interest is just 16 over the two years but compound interest is 16.64 which includes the simple interest plus interest on interest.

The Power of Compounding Given an interest rate of 8% per year and an initial 1,000 investment, compare the compound interest in the 2nd year and the 20th year. What is the total compound interest over the 20 year investment? Explain the power of compounding

Intuition for the Present Value Formula Present value of 1 received in 2 years given a discount rate of 8% per year. Year 0 1 PV0 C2 (1 r)2 1 (1 .08)2 0.8573 2

Practice exercises Find the value in 3 years of 10,000 received in 8 years. Find the value in 10 years of 25,000 invested in 2 years. How many years was the money invested if 10,000 grew to 100,000? If you invested 500 for 7 years and now have 1000, how much did your investment return?

TVM in your HP 10B Calculator Housekeeping functions: 1. Set to 8 decimal places: Yellow DISP 8 2. Clear previous TVM data: Yellow C C ALL 3. Set payment at Beginning/End of Period: Yellow Always keep set on “End” – the display should never say “Begin” MAR BEG/END 3. Set # of times interest is calculated (compounded) per year to 1: 1 Yellow PMT P/YR

Using the HP 10B TVM functions N number of periods or number of payments I/YR the effective interest rate or discount rate per period PV present value or a present cash flow PMT repeating cash flow payments (not yet discussed) FV future value or a future cash flow

III. Groups of Cash Flows Consider the following series of cash flows: Year 0 1 2 3 . 10 1,000 1,050 1,102.50 . 1,551.33 What is PV0; what is FV10? We could apply our PV and FV formulae to each individual cash flow – but that would be too painful!

Mathematics of Perpetuities and Annuities 0 Fortunately, math provides a simplified way . . . Consider a growing perpetuity: 1 2 3 . 1,000 C1 1,050 C2 1,102.5 C3 . C1(1 g)1 C1(1 g)2 t Goes on forever . . Ct C1(1 g)t-1 C

Sum of an infinite series PV0 of the growing perpetuity is mathematically equivalent to the sum of an infinite geometric series. The sum is defined and is finite as long as the PV of each subsequent cash flow is a fraction (less than 1) of the PV of the previous cash flow. I.e., as long as r g

PV of a growing perpetuity This sums the PV’s of each individual cash flow in the growing perpetuity. C1 is the first cash flow C1 PV0 r g PV0 is the PV one period before the first cash flow This formula is only correct when r g. If r g or if r g, then the PV is infinite.

Growing Perpetuity – Example To service the current national debt, the government plans to make the following series of payments beginning in one year and continuing in perpetuity: 8 billion initially and then growing by 4% each year. The interest rate on long-term debt is 6%. What is the PV0 of these payments? Note: the PV0 of all future debt payments is equal to the principal amount currently outstanding.

PV of a Growing Annuity 0 An annuity is a finite series of cash flows – i.e., a series that has an end – assume the end as at time “n”. We can determine the PV of the growing annuity by subtracting off the latter part from a growing perpetuity. 1 2 . n n 1 . C1 C2 . Cn Cn 1 . C C1(1 g)n-1 C1(1 g)n C1(1 g)1

PV of a Growing Annuity so the PV of the growing annuity is just the PV of the whole growing perpetuity minus the PV of the latter part of the growing perpetuity. The latter part of the growing perpetuity is just another growing perpetuity that starts at a later time with a different initial cash flow.

PV of a Growing Annuity Cn 1 C1 1 PV0 r g r g (1 r ) n n C1 C1 (1 g ) 1 PV0 n r g r g (1 r ) n C1 (1 g ) PV0 1 n r g (1 r ) Subtract off the PV of the latter part of the growing perpetuity PV of the whole growing perpetuity PV0 is the PV one period before the first cash flow

Growing Annuity – Example In 20 years you plan on retiring and you would like income each year that grows at the expected inflation rate of 3%. You desire your year 20 income to be 105,000 and you expect to need 30 years of retirement income. If you are confident your savings will earn 8% per year, how much do you need saved by year 20?

FV of a Growing Annuity If PV0 discounts all the cash flows to time zero and sums up the discounted amounts . . . then FVn, the future value of all the cash flows taken to time n, . . . is just PV0 (1 r)n C1 (1 g ) n n FVn 1 ( 1 r ) r g (1 r ) n C1 FVn (1 r ) n (1 g ) n r g In effect, this takes all cash flows of the growing annuity, including the last cash flow, forward to the time period of the last cash flow

FV of Growing Annuity – Example Given your retirement plans of the previous example, how much do you need to save each year beginning in one year and ending with year 19? Assume your savings will earn 8% and you increase your contributions by 6% each year.

Simple (non-growing) series of cash flows For constant C regular perpetuity PV 0 annuities and r constant perpetuities, the C 1 PV0 1 time value r (1 r) n formulas are regular annuity simplified by setting g 0. C FVn (1 r) n 1 r We can use the PMT button on the financial calculator for the annuity cash flows, C

Regular Annuities & Perpetuities – Examples Your father “loaned” you 20,000 as your down payment on your new house. If you repay him in equal amounts of 2,600 each of the next 10 years, what rate of interest are you, in effect, paying him?

Regular Annuities & Perpetuities – Examples You have won the lottery and are offered cash payments of 1 million per year for the next 20 years (first payment is one year from today). If you could invest at a rate of 10%, how much as a single lump sum would you be willing to receive today in exchange for the 20 yearly cash flows?

Regular Annuities & Perpetuities – Examples You have just donated to the University of Manitoba and your donation stipulates that the University must spend the income earned from your donation each year. If your donation is 10 million and it earns a 6% rate of return, how much can be spent each year and for how long can this continue?

IV. Some final warnings Even though the time value calculations look easy there are many potential pitfalls you may experience Be careful of the following: PV0 of annuities or perpetuities that do not begin in period 1; remember the PV formulas given always discount to exactly one period before the first cash flow. If the cash flows begin at period t, then you must divide the PV from our formula by (1 r)t-1 to get PV0. Note: this works even if t is a fraction.

Be careful of annuity payments Count the number of payments in an annuity. If the first payment is in period 1 and the last is in period 2, there are obviously 2 payments. How many payments are there if the 1st payment is in period 12 and the last payment is in period 21 (answer is 10 – use your fingers). How about if the 1st payment is now (period 0) and the last payment is in period 15 (answer is 16 payments). If the first cash flow is at period t and the last cash flow is at period T, then there are T-t 1 cash flows in the annuity.

Be careful of wording A cash flow occurs at the end of the third period. A cash flow occurs at time period three. A cash flow occurs at the beginning of the fourth period. Each of the above statements refers to the same point in time! 0 1 2 3 4 C If in doubt, draw a time line.