Present value calc

8/10/2019 Present value calc

1/18

Present value Calculations

Introduction.

The following two decision criteria are used for investments:

NPV I Ct

(1r) t 0

Invest until the return on the investment is equal to the return infinancial markets.

The fundamental vaue of an asset is given by:

= price.PV Ct

1r t

1


2/18

Here we look at how to perform the calculations:

if there is only one cash flow involved, but this cash flow may occurseveral periods into the future:

NPV I C

(1r)n 0

if there is more than one cash flow in the future:

NPV I C

1r C

(1r)2 ..

C

(1r)n

clearly we can just calculate it as a sum of individual cash flows but wewill show how one can, under certain circumstances, use annuities to

simplify the calculations.

Calculate the return on investments if there are:

One future payment in n periods

Several payments in the future

2


3/18

Calculation of present and future values of single cash

flows.

Below we show have to calculate present value for cash flows severalperiods into the future.

Key assumption in all calculations:

Interest rates are the same in each period .

Future value.

How much will an initial amount grow to after n years at an interest of r%.

?

n10 2 Time

Kr X

Consider the case of investing 100 for four years at a rate of 5%:

121.555.79115.764115.765.51110.253110.255.25105210551001

Amount atthen end

of the year

Interestearned

Amount atbeginning of

the year

Year

Note:It is assumed that the interest earned is reinvested in an asset earningthe same rate of return, i.e. here the interest earnings are reinvested at

5%.

3


4/18

In symbolic notation we have that the value after n years is:

X(1+r)n-1+rX(1+r)n-1= X(1+r)nrX(1+r)n-1X(1+r)n-1n....

X(1+r)3+rX(1+r)3= X(1+r)4rX(1+r)3X(1+r)34X(1+r)2+rX(1+r)2= X(1+r)3rX(1+r)2X(1+r)23X(1+r)+rX(1+r) = X(1+r)2rX(1+r)X(1+r)2X+rX = X(1+r)rXX1

Amount at then endof the year

Interestearned

Amount atbeginning of

the year

Year

So the future value of X dollars compound n periods from now earning r% ininterest is:

FV X1 rn

4


5/18

Example.Consider the case where you can invest kr. 100 at 6% for five years.

Thus after 5 years you have 133.82.FV 1001.065 133.82

Excel function:

FVReturns the future value of an investment based on periodic, constant payments and a constantinterest rate.

Syntax

FV(rate,nper,pmt,pv,type)

For a more complete description of the arguments in FV and for more information on annuityfunctions, see PV.

Rate is the interest rate per period.

Nper is the total number of payment periods in an annuity.

Pmt is the payment made each period; it cannot change over the life of the annuity. Typically,pmt contains principal and interest but no other fees or taxes. If pmt is omitted, you must includethe pv argument.

Pv is the present value, or the lump-sum amount that a series of future payments is worth rightnow. If pv is omitted, it is assumed to be 0 (zero), and you must include the pmt argument.

Type is the number 0 or 1 and indicates when payments are due. If type is omitted, it is assumedto be 0.

At the beginning of the period1

At the end of the period0

If payments are dueSet type equal to

Remarks

Make sure that you are consistent about the units you use for specifying rate and nper. Ifyou make monthly payments on a four-year loan at 12 percent annual interest, use12%/12 for rate and 4*12 for nper. If you make annual payments on the same loan, use12% for rate and 4 for nper.

For all the arguments, cash you pay out, such as deposits to savings, is represented bynegative numbers; cash you receive, such as dividend checks, is represented by positive

numbers

5


6/18

Excel:

Method 1:

Formula in B4: FV(B3;B2;0;B1)

133.82Future value46%Interest rate35Number of periods2

100Amount of initial cash:1 BARow/columns

Notice the 0 for the amounts paid each period. Notice that the initial amount is given by B1.

Method 2:

Formula B4: B1*(1+B3)^B25

133.82Future value46%Interest rate35Number of periods2

100Amount of initial cash:1BARow/columns

6


7/18

Finding the interest rate or return.

Sometimes the future value, FV, and the initial value, X, is known and wewant to find the rate of return:

FV X1 rn 1 rn FV

X r n

FV

X 1 r (

FV

X)

1n 1

Consider the case where we know that we receive 133,82 in five years time

from an investment of 100 today.

Formula in B4: (B3/B1)^(1/B2)

6%Interest rate4133.82Future value3

5Number of periods2100Amount of initial cash:1BARow/columns

An alternative method is to use the spreadsheet function RATE.

7


8/18

RATE

Returns the interest rate per period of an annuity. RATE is calculated by iteration and can havezero or more solutions. If the successive results of RATE do not converge to within 0.0000001after 20 iterations, RATE returns the #NUM! error value.

Syntax

RATE(nper,pmt,pv,fv,type,guess)

For a complete description of the arguments nper, pmt, pv, fv, and type, see PV.

Nper is the total number of payment periods in an annuity.

Pmt is the payment made each period and cannot change over the life of the annuity. Typically,pmt includes principal and interest but no other fees or taxes. If pmt is omitted, you must includethe fv argument.

Pv is the present value the total amount that a series of future payments is worth now.

Fv is the future value, or a cash balance you want to attain after the last payment is made. If fv isomitted, it is assumed to be 0 (the future value of a loan, for example, is 0).

Type is the number 0 or 1 and indicates when payments are due.


At the end of the period0 or omitted


Guess is your guess for what the rate will be.

If you omit guess, it is assumed to be 10 percent.

If RATE does not converge, try different values for guess. RATE usually converges ifguess is between 0 and 1.

Remark

Make sure that you are consistent about the units you use for specifying guess and nper. If youmake monthly payments on a four-year loan at 12 percent annual interest, use 12%/12 for guessand 4*12 for nper. If you make annual payments on the same loan, use 12% for guess and 4 fornper.

8


9/18

Formula in B4: RATE(B2;0;B1;B3)6%Interest rate4

133.82Future value35Number of periods2

-100Amount of initial cash:1BARow/columns

Note that the second parameter is 0, i.e. there are no intermediate cash

flows. Note that the initial cash flow is set to -100, i.e. a cash outflow, either

the initial cash inflow or the cash outflow should be coded as negative.

Finding the number of periods.

Recall that:FV X1 rn

FV X1 rn lnFV lnX n ln1 r

nlnFV lnX

ln1 r

ln(FV

X)

ln(1 r)

Using the above figures we have:

Formula in B4: ln(B2/B1)/ln(1+B3)

5Number of periods46%Interest rate3

133.82Future value2100Amount of initial cash:1BARow/columns

Again one can alternatively use one of the build-in-functions:

9


10/18

NPER

Returns the number of periods for an investment based on periodic, constant payments and aconstant interest rate.

Syntax

NPER(rate, pmt, pv, fv, type)

For a more complete description of the arguments in NPER and for more information aboutannuity functions, see PV.

Rate is the interest rate per period.

Pmt is the payment made each period; it cannot change over the life of the annuity. Typically,pmt contains principal and interest but no other fees or taxes.

Pv is the present value, or the lump-sum amount that a series of future payments is worth rightnow.

Fv is the future value, or a cash balance you want to attain after the last payment is made. If fv isomitted, it is assumed to be 0 (the future value of a loan, for example, is 0).



At the end of the period0 or omittedIf payments are dueSet type equal to

Formula in B4: nper(B3;0;B1;B2)

5Number of periods46%Interest rate3

133.82Future value2-100Amount of initial cash:1

BARow/columns

10


11/18

Present value.

Definition of present value:The present value of an amount due n years in the future is the amount

which, if it was on hand today, would grow to equal the future amount wheninvested at the opportunity rate.

FV

n10 2 Time

Kr ?

But future value is what X, in the FV value function, will grow to in n years(compound periods), so by the above definition we have that X is the presentvalue. So:

FV PV1 rn PV FV

1rn

11


12/18

Let us calculate the present value of 100 received in four years time:

82.270386,3838/(1.05)86.3838186.383890,703/(1,05)90.703290.70395,24/(1,05)95.2381395.2381100/(1,05)1004

Amount at the

beginning ofthe year

Discount

factor

Amount at end

of the year

Year

100

410 2Time

3

95,2490,7086,3882,27

PV at different times

Notice that 82, 2703 1,054 100

FV/(1+r)40FV/(1+r)4(FV/(1+r)3)/(1+r)FV/(1+r)31FV/(1+r)3(FV/(1+r)2/(1+r)FV/(1+r)22FV/(1+r)2(FV/(1+r))/(1+r)FV/(1+r)3 FV/(1+r)FV/(1+r)FV4

Amount at thebeginning of theyear

DiscountAmount at end ofthe year

Year

12


13/18

Consider the case, as above, where we want to calculate the Present valueof 133.82 received in five years with an interest rate of 6%.

Formula in B4: (B1/(1+B2)^B3

100Present value45Number of periods3


BARow/columns

Again we can also use the a build-in-function PV ( p dansk NV):

13


14/18

PV

Returns the present value of an investment. The present value is the total amount that a series offuture payments is worth now. For example, when you borrow money, the loan amount is the

present value to the lender.

Syntax

PV(rate,nper,pmt,fv,type)

Rate is the interest rate per period. For example, if you obtain an automobile loan at a 10 percentannual interest rate and make monthly payments, your interest rate per month is 10%/12, or0.83%. You would enter 10%/12, or 0.83%, or 0.0083, into the formula as the rate.

Nper is the total number of payment periods in an annuity. For example, if you get a four-year

car loan and make monthly payments, your loan has 4*12 (or 48) periods. You would enter 48into the formula for nper.

Pmt is the payment made each period and cannot change over the life of the annuity. Typically,pmt includes principal and interest but no other fees or taxes. For example, the monthlypayments on a $10,000, four-year car loan at 12 percent are $263.33. You would enter -263.33into the formula as the pmt. If pmt is omitted, you must include the fv argument.

Fv is the future value, or a cash balance you want to attain after the last payment is made. If fv isomitted, it is assumed to be 0 (the future value of a loan, for example, is 0). For example, if youwant to save $50,000 to pay for a special project in 18 years, then $50,000 is the future value.You could then make a conservative guess at an interest rate and determine how much you mustsave each month. If fv is omitted, you must include the pmt argument.



At the end of the period0 or omitted


Formula in B4: PV(B2;B3;0;B1)

100Present value45Number of periods3


BARow/columns

14


15/18

More frequent compound periods.

So far we have assumed that the returns are received once a year at the endof the year. What happens if you receive interest more than once a year, e.g.

once a month. First some important terms:

Nominal interest rate:This rate does not include compounding, for example a bank may state thatyou earn 12% per annum paid monthly. This is equivalent to being paid 1%per month, and it does not reflect the amount of interest earned over theyear!

Effective interest:Is the rate that would have produced the future value under annualcompounding. In the above example the effective rate is:

(1.01)12-1 = 12.68%

If you invest $1 at the beginning of the year and receive 1% per month ininterest which is reinvestedat the rate of 1% per month then you have

1 + .1268 = 1.1268 after one year.

Note that the effective interest rate is always at least as large as the nominalinterest rate since you earn interest on the interest during the year.

15


16/18

Example - semiannual compounding.

Nominal rate = 5% Earn 2.5% for half a year

After half a year you receive interest which is reinvested "Period" refers to compound period

121.84118.87x2.5=121.8404118.87118.87115.97x2.5=2.899115.974115.97113.41x2.5=2.8285113.41113.14110.38x2.5=2.7595110.383 110.38107.69x2.5=2.6922107.69

107.69105.06x2.5=2.6265105.062105.06102.5x2.5=2.5625102.5102.5100x2.51001

Amount at theend of theperiod

Interest earnedAmount at thebeginning ofthe year

Year

The future value from annual compounding is 121.55.

From above we have that the formula for the future value is:

rnom = Annual nominal interest ratem = number of compound periods per yearn = number of years

FV PV 1 rnomm

nm

PV FV

[1 rnomm ]

nm

Note that is equal to the effective annual rate[1 rnomm ]

m 1

From above we have:100 (1

0.62

)42 121.8403

16


17/18

Continuous Compounding.

From above we have that the effective rate of interest is:

[1 rnom

m ]m

1

where m is the number of compound periods. We can rewrite the expressionin the following way:

[1 rnomm ]

mrnom

rnom

1

define then we have:w mrnom

([1 1w ]w

)rnom

1

Note that as m w .

Recall that e is defined as:

e llim 1

1l

l

2.71828

Letting m (and therefore w) go to infinity we get the effective annual rate with

continuous compounding:

wlim ([1

1w ]

w )rnom 1 e rnom 1

And the future present values with continuos compounding are:

FV PVenrnom

PV FVenrnom

17


18/18

Summary.

Assumptions: All cash f lows are reinvested at the rate r.

Present and future values of single cash flows.

Future value: FV X 1 rn

Present value: PV X1rN FV

1rn

Frequent compound periods.

Key words: Nominal and effective interest rates.

Future value: FV X 1 rnomm mn

Present value: PV X

1rnomm nm

Continuos compounding.

Effective annual rate: ernom

Future value: PVernomn

Present value: FVernomn

18

Present value calc

Documents

Transcript of Present value calc