## Calculating the Net Present Value (NPV)

Financial problems such as valuing an asset can be solved by calculating its net present value. The underlying principle in the calculation is that a dollar today is worth more than a dollar a year from now. This principle was well known to our ancestors thousands of year ago.

A bird in hand is worth two in the bush.

— Aesop’s Fables

So a stream of cash flows that we will receive in the future have to be discounted to calculate the net present value. We will see some methods to discount this cash flow in the following sections.

``````# We will be using the folowing libraries for this chapter

library(tidyquant) # For financial data
library(DT) # to create pretty tables

options(digits = 3, scipen = 100)``````

Let us take the following example. If we receive \$100 per year for the next five years what is the net present value if the current interest rates are 5%. Let us set up our cash flow table.

``````# Creating a variable r for interest rates and the cash flow table

r <- 0.05

# cf is the table for our data

cf <- tribble(~Year,~cash_flow,
1,100,
2,100,
3,100,
4,100,
5,100)
cf %>%
datatable() %>%
formatRound(columns = "cash_flow",digits = 2)``````
``````r <- 0.05

cf <- tribble(~Year,~cash_flow,
1,100,
2,100,
3,100,
4,100,
5,100)

# Calculating the Net present value for each year

cf %>%
mutate(pv = cash_flow/(1 + r)^Year) %>%
datatable() %>%
formatRound(columns = "pv",digits = 2)``````
``````# Calculating the total Net present value

npv <- cf %>%
mutate(pv = cash_flow/(1 + r)^Year) %>%
select(pv) %>%
sum()

npv``````
``## [1] 433``

### Calculating net present value of uneven cash flows

We can use the same method to calculate the net present value of uneven cash flows as shown below.

``````# We create a table of cash flows

cf <- tribble(~Year,~cf,
1,100,
2,200,
3,300,
4,400,
5,500)
datatable(cf)``````
``````npv <- cf %>%
mutate(pv = cf/(1+r)^Year) %>%
select(pv) %>%
sum()

print(npv)``````
``## [1] 1257``

### Calculating the PV with negative cash flow

In many cases we will have an initial cash flow that is negative. Think of this as buying a bond, where we buy a bond for \$250 and it pays an interest of \$100 each year. We can solve the present value of such a bond as below.

``````r <- 0.05
cf <- tribble(~Year,~cf,
0,-250,
1,100,
2,100,
3,100,
4,100,
5,250)

datatable(cf)``````
``````npv <- cf %>%
mutate(pv = cf/(1+r)^Year) %>%
select(pv) %>%
sum()``````

The present value of this bond is 300.48.

### Calculating the present value of a finite annuity

In some cases we have to calculate the present value of stream of equal payments. Many times a lottery winner is given the option to get the money in equal payments or a lump sum. Let us suppose you win a \$50000 lottery and you are given two options

1. Receive five equal payments of \$10000
2. Lump sum \$45000.

Let us suppose the bank pays 5% interest rate on the deposit.

At first glance it may appear that five equal payments of \$10000 is better than \$45000. After all \$50000 is more than \$45000. To solve this problem we can calculate the net present value of those payments as below.

``````pmt <- 10000                #payments
n <- 5                      # Years
r <- 0.05                   # interest rates

cf <- tibble(period = 1:5,
pmt = pmt)

npv <- cf %>%
mutate(pv = pmt/(1+r)^period) %>%
select(pv) %>%
sum()``````

The net present value of five equal payments is 43294.767. This payment is less that \$45000 offered today. Therefore is better to take the lump sum payment of \$45000 and deposit it in the bank than five equal payments of \$10000.

### Calculating the present value infinite annuity

Sometimes one can buy an infinite payment stream. Bank of England issued such bonds that paid interest forever. How can we calculate the net present value of such bonds. Let us suppose we are given an offer to buy a bond that pays \$1000 per year in interest forever. If the current interest rate is 5%, how much should we pay for this bond? We can do the calculation as below.

``````cf <- 1000 # Interest paid each year
r <- 0.05 # Interest rate 5%

value <- cf/r``````

The value of this bond is 20000

### Calculating present value of finite growing annuity

There are cases when when an investor may be offered a stream of payments that grow at a certain rate. Suppose you own an apartment complex. The tenants pay you a fixed rent every month. You can increase the rent to tackle higher inflation. This apartment building may be described as an annuity payment that grows at the rate of inflation. We can calculate the present value of those payments as below.

``````payment <- 1000
growth <- 0.03 # Growth rate of payments
n <- 5 # years
rate <- 0.06 # Interest rate

# Creating the tibble

cf <- tibble(period = 1:5,
cf = 1000)

# making the cash flow grow at 3% per year

cf <- cf %>%
mutate(cf = payment*(1+growth)^(period - 1))

cf <- cf %>%
mutate(pv = cf/(1+rate)^period)

datatable(cf, caption = "Growing Annuity payments")``````
``````npv <- cf %>%
select(pv) %>%
sum()``````

The present value of this payment stream is 4457.

### Calculating present value of infinite growing annuity

In case of infinite payment stream that grows at a certain rate, we simply subtract the growth rate from the current interest rate as shown below.

Suppose you get paid \$100 per year forever, and this payment grows at 3% each year. What is the net present value of this payment stream.

``````payment <- 100
growth <- 0.03
rate <- 0.06

annuity <- payment/(rate - growth)``````

Here the present value of annuity is 3333.333, for \$100 payment that grows at 3% forever.

## Calculating Future Value (FV)

We can use the above formula to calculate the calculate the Future value of a stream of payments. For example if we are promised \$100 per year for 5 years at 5% interest rate, what is the future value of such a payment.

``````# Lets create our table

r <- 0.05

cf <- tibble(year = 1:5,
cf = 100)

fv <- cf %>%
mutate(fv = cf * (1 + r) ^ year) %>%
select(fv) %>%
sum()``````

The future value of our cash flow is 580.19.