CFA - quantitative methods - The future value of a series of cash flows

Present values indexed at times other than t=0 Example

£100 payments in year 2 Year 2 = PV1 = 100/0.05 =2,000 at a 5% discount rate Todays Pv = PV0 =2,000/1.05 = 1904.76

Present values indexed at times other than t=0 Example 2

£6 per year - begin at end of year 4 Last cash flow year 10 = 7 year ordinary annuity from end of year 3 PV end of year 3 and then discount that PV back to t=0 R = 5% and cash flow of £6 per year starting at the end of the 4th year will be worth £34.72 at the end of t=3 and 29.99 today t=0

Ordinary annuity revised formula Because the annuity payment (A) is a constant in the equation it can be factored out as a common term. Thus the sum of interest factors has a shortcut expression

PV = A[1-1/(1+r)N/r] - In the same way you computed the future value of an ordinary annuity we find the present value by multiplying the annuity by a present value annuity factor eg PV = A[1-1/(1+r)N/r]

The present value of a lump sum example An insurance company has issued a contract that promises to pay £100,000 in six years with an 8% return rate. What amount of money must the insurer invest today at 8% for six years to make the promised payment

PV = FVn (1+r)-N FVn = 100,000 R =8% = 0.08 N = 6 PV = FVn(1+r)-N £100,000 (1/(1.08 to the power 6 ) = 100,000(0.6301696) =63,016,96 (PV) (this amount invested today with an interest of 8% is equivalent to the hundred thousand pound to be received in six years)

The present value of single cash flow

Just as the future value factor links today's present value with tomorrow's future value The present value factor allows us to discount future values to present value

Unequal cash flows method

Find the future value of a series of unequal cash flows by compounding the cash flows one at a time As all the payment shown above are different The most direct approach to Finding the future value at t=5 is to compute the future value of each payment as of t = 5 and then sum the individual future values

example The present value of an ordinary annuity - Suppose you are considering purchasing an asset that promises to pay £1000 per year for five years with the first payment one year from now - The required rate of return is 12% per year - How much should you pay for this asset

Formula to use PV = A[1-1/(1+r)N/r] A = 1000 R = 12% N = 5 PV = 1,000 [1-1/(1.12)5/0.12] = 1,000(3.604776) = 3,604.78 The series of cash flows on the thousand pound per year for five years is currently worth 3,604.78 when discounted at 12%

formula for The present value of single cash flow

Given a future cash flow that is to be received in N periods and interest rate per period of r we can use the formula for future value to solve directly from the present value as follows PV = FVn (1+r)-N This formula shows that the present value factor (1+r) -N is the reciprocal of the future value factor (1+r)N

The present value of a series of equal cash flows (Ordinary annuity)

PV = A/(1+r) + A/(1+r)2+ A/(1+r)3 + .. A/(1+r)N-1 + A/(1+r)N A = the annual annuity R = the interest rate per period corresponding to the frequency of annuity payments (annually quarterly monthly etc) N = the number of annuity payments

A perpetuity =

perpetual annuity or a set of never-ending sequential cash flows with the first cash flow occurring one period from now

The PV of a perpetuity example £100 per year in perpetuity Required rate of return 5%

A=£100 R = 5% PV = A/r 100/0.05 -=2,000 - bond would be worth 2000

The present value of a series of unequal cash flows

When you have unequal cash flows you must find the present value of each individual cash flow and then sum the present values You compute the future values of cash flows Using the single payment future value formula (eg working them out one at a time) If the sum of present values = 15,036.46 N =5 R = 5% FVn=PV(1+r)N 15,036.46(1.05)5= FV of the series of cash flows

An annuity =

finite set of level sequential cash flows

An annuity due =

has a first cash flow that occurs immediately (t=0)

An ordinary annuity =

has a first cash flow that occurs one period from now (t =1)

Solving for rates number of periods, or size of annuity payments

see below

General annuity formula

- Annual annuity amount = A - Number of time periods = N - Interest rate per period = r This simplifies to FVn = A[(1+r)N-1/r] - The term in brackets is the future value annuity factor - The factor gives the future value of an original annuity of $1 per period - X the future value annuity factor by the annuity amount gives the future value of an original annuity - The simplified formula gives the future value annuity factor e.g. [(1.05)5 -1/0.05] = the future value of the five year ordinary annuity

equal cash flow is ordinary annuity

- Consider an ordinary annuity paying 5% annually - Five separate deposit of $1000 occurring at equally spaced intervals of one year so the last payment occurs five years from now - Future value of each $1000 deposit as of t=5 using FVn = PV1(1+r)N - The first deposit made at t=1 Will compound over four periods - the second deposit made at t =2 Will compound over three periods etc - The future value of the first deposit at t= 5 is 1,000(1.05)4 = 1215.51 - As we are finding the future value at t=5 the last deposit does not earn any interest - Add all the future values together to arrive at the future value of the annuity

What is emphasised in this reading

- Finding the present or future value of any cash flow series - Recognising the equivalent of present value and appropriately discounted future value - Keeping track of the actual calendar time in a problem involving the time value of money

Suppose you own a perpetuity and issue a perpetuity obligating you to make payments that are the same size as the perpetuity you own

- First payment of perpetuity you issue = t = 5 - Payments on second perpetuity exactly offset the payments received from the perpetuity we own at t=5 and all future dates - = nonzero net cash flows at t =1,2,3,4 - This outcomes fits the def on an annuity with 4 payments - We can construct an annuity as the difference between two perpetuities with equal level payments but different starting dates

Present values relates to the discount rate and the number of periods in the following ways

- For a given discount rate the further in the future the amount to be received the smaller the amounts present value - Holding time constant, the larger the discount rate, the smaller the present value of future amount

The future value of an annuity (example)

- Invest up to £20,000 a year - You plan to do this for the next 30 years - Historically this has earnt 9% per year on average - How much money will you have available after making the last payment FV annuity factor = (1+r)N-1/r = (1.09)30 - 1 /0.09 = 136.307539 FVn = 20,000(136.307539) =2,726,150.77 Steps to take 1) Work out the annuity factor 2) X yearly investment by annuity factor

The present value of an infinite series of equal cash flows - perpetuity

- Perpetuity annuity - To derive a formula for the present value of a perpetuity = ∞ - PV = AΣ [1/(1+r)t - t=1 As long as interest rates are positive the sum of present value factors converges PV = A/r ^ is used to value div from stock as they have no predefined life span ^ is only valid for a perpetuity with level payments

Solving for interest rates and growth rates (growth rate comparison)

- This example shows an interest rate could also be considered a growthrate. The particular application will usually dictate whether we use IR or GR Solving FVn=PV(1+r)N for r and replacing the interest rate r with the growth rate g produces the following expression for determining growthrates G = (FVn/PV)1/N-1

example of The present value of single cash flow

- With a 5% interest rate generating a future payoff of £105 in one year - What current amount invested at 5% for one year will grow to 105 - The answer is 100 - therefore 100 is the present value of 105 to be received in one year at a discount rate of 5%

An amenity due as the present value of an immediate cash flow Plus any ordinary annuity example - You are retiring and you can either have a £2 million lump sum - Or 200,000 for 20 years - The interest rate at your bank is 7% per year compounded annually - Which option has the greatest present value

1) Find the present value of each at t=0 2) 2,000,000 = t=0 3) First 200k = t=0 4) 200,000 for 19 years 5) To value this option you need to find a present value of the ordinary annuity using PV = A[1-1/(1+r)N/r] 6) And then add 200,000 A = 200,000 N = 19 R -7% 200,000 = 1-1/(1.07)19/0.07 =200,000(10.335595) = 2,067,119.05 = 19 payments +200k

The projected present value of an ordinary annuity example - A pension fund anticipate the benefits of £1 million per year must be paid - Payments will be made At 10 years from now t=10 - The payments will extend until t=39 (for a total of 30 payments) - What is the present value of the liability if the appropriate annual discount rate is 5% compounded annually

Answer - Annuity with the first payment at T =10 - At t=9 you have an ordinary annuity with 30 payments - Present value of the annuity = PV = A[1-1/(1+r)N/r] A = 1,000,000 R =5% N =30 = 1,000,000 (1-1/(1.05)30/0.05) = 1,000,000(15.372451) = 15,372,451.03 = PV at t=9 Now we need to find the present value at t=0 Rely on the equivalence of present value and future value T=9 = a future value from the vantage point of t=0 FNv =15,372,451.03 (T=9) N9 R =5% PV = FVn(1+r)-N = 15,372,451.03(1.05)-9 = 9,909,219.00

Solving for interest rates and growth rates - Suppose a bank deposit of £100 is known to generate the payoff of £111 in one year - To work out the interest rate use

FVn=PV(1+r)N (N=1) - With PV FV and N known we can solve r directly - 1+r = FV/PV - 1+r = 111/100 = 1.11 - R = 0.11 = 11% - This example shows an interest rate could also be considered a growthrate

Example the present value of a lump sum with monthly compounding The fund make a lump sum payment of 5 million 10 years from now Want to invest an amount today to grow to that required amount Current interest rate is 6% a year compounded monthly How much should she invest today

PV = FVn(1+rs/m)-mN FVn = 5,000,000 Rs = 6% M = 12 Rs/m = 0.06/12=0.005 N = 10 mN=12(10) =120 PV = FVn(1+rs/m)-mN 5,000,000(1.005)-120 5,000,000(0.549633) = 2,748,163.67 In applying this equation we use the periodic rate (monthly rate in this case) and the appropriate number of periods with monthly compounding (in this case 10 years of monthly compounding or 120 periods)

The frequency of compounding: With more than one compounding period a year =

PV = FVn(1+rs/m)-mN Rs = quoted interest rate and equals the periodic interest rate x the number of compelling periods in each year M = Number of compounding periods per year N = Number of years - The only difference in the above formula is the use of periodic interest rate and the corresponding number of compounding periods

Annuity or perpetuity beginning sometime in the future can be expressed in present value terms one period prior to the first payment. That present value can be discounted back to today's present value Example the present value of the a projected perpetuity £100 per year Beginning t =5 What is the present value today at t=0 given a 5% discount rate

Solution 1) Find the present value of the perpetuity at t=4 2) Discount that back to t=0 3) Recall that a perpetuity or an ordinary annuity has its first payment one period away explaining that t=4 for present value calculations T = 4 = A =100 R =5% PV =A/r 100/0.05 = 2,000 2000 is a FV FVn = 2,000 (PV at t=4) R = 5% N= 4 PV = FVn(1+r)-N = 2,000(1.05)-4 = 1,645.40 = PV of the perpetuity Annuity = payment of a fixed amount for a specified number of periods

Example calculating a growth rate Sales increased from 10,503.0 billion in 2008 to 14,146.4 billion in 2012 Net decline from 822.5 billion in 2008 to 796.4 billion in 2012 Calculate the following growth rates fall the four year period from 2008-2012 1) Sales growth rate 2) Net profit grocery

Solution 1) Use G = (FVn/PV)1/N-1 we denote sales in 2008 as PV and sales in 2012 to FV4 g=4√14,146.4/10,503.0-1 =4√1.346891-1 = 1.077291-1 = 0.077291 or 7.7% The calculator growth rate of about 7.7% a year shows that sales grew substantially during 2008 - 2012

The present value of an ordinary annuity as the PV of a current minus projected perpetuity example - 5% DR - Find the PV of a 4 year ordinary annuity of £100 per year starting in year 1 as the difference between P1 £100 per year starting year 1 (first payment = t=1) P2 £100 per year starting in year 5 (first payment = t=5)

Solution 1) Subtract P2-P1 = ordinary annuity of £100 per period for 4 years (payments 1,2,3,4) 2) - PV of P2 from P1 = PV of 4 year ordinary annuity P1 PV0 = 100/0.05 = 2k P2 PV4 = 100/0.05 =2k PV0 = 2,000/(1.05)4 =1,654,40 Annuity PV0 = PV0(P1)-PV0(P2) = 2k - 1645.40 = 354.60 = 4 year ordinary annuity's PV

Example of unequal cash flow metods

Suppose you have the five cash flows below indexed relative to the present (t=0) Time Cash flow Future value 5 years T = 1 1,000 1,000(1.05)4 T = 2 2,000 2,000 (1.05)3 = T=3 4,000 4,000(1.05)2 = T=4 5,000 5,000(1.05)1 = T=5 6,000 6,000(1.05)0 = Sum = add above together =19,190.76

The projected present value of a more distant future lump sum example - You own an asset that will pay you $100,000 in 10 years - You want to know the value in four years - 8% discount rate

The value of the asset is the present value of the assets promised payment. At t=4 the cash payment will be received 6 years later FVn = 100,000 R=8% N=6 PV = FVn(1+r)-N = 100,000 1/(1.08)6 = 100,000(0.6301696) = 63,016.96 (PV 4 years)