Question

Caffeineland is a small country in Central America. The main product of the country is coffee, which is currently priced at $1.20 per cup. Inflation is expected to be 10% per year, and the nominal interest rate is 10% per year.

a) Calculate the real interest rate.

b) Suppose you want to borrow 1000 cups of coffee now and pay back the loan next year. How many cups of coffee would you owe? What is the interest rate for this coffee loan?

HINT: Borrow money now and use it to buy coffee at the current price. Then calculate how much you owe in dollars next year. Use next year's coffee price to determine how much you owed in terms of "cups of coffee."

c) Juan Valdez, a brilliant local scientist, has invented a coffee replicator. It works as follows: At time zero you insert 1000 cups of coffee. These cups are consumed by the machine - you have to 'invest' 1000 cups now to make the replicator work. The for the next 5 years, the replicator produces 300 cups of coffee per year. you get 300 cups at year 1, year 2 and so on. after five years the replicator is used up. CALCULATE THE PV OF THE REPLICATOR IN TERMS OF DOLLARS.

d) Now calculate the PV of the replcator in terms of coffee. HINT: Use cups of coffee as your currency. Make a coffee-flow diagram and then calculate rhe PV. Be sure to use the coffee interest rate in your PV calculations.

e) Convert the PV in terms of coffee from part d into dollars. Multiply the answer from part d by the current price of coffee.

Answer 1

Answer: Qa- 0; Qb- 1000 cups, 0% coffee rate; Qc- $600; Qd- 500 cups, Qe- $600

Solution -

Qa) Real interest rate = nominal interest rate - Inflation = 10% - 10% = 0 % per year

Qb) Current price of coffee = P₀= $1.2 per cup,

Inflation = 10% or 0.1 per year

We know that Price of coffee at time t = current price * (1+ inflation rate)^t

Price of coffee next year = P_f = current price * (1+ inflation rate) =

                                         = 1.2*(1+10%) = 1.2*1.1 = $1.32 per cup

To borrow 1000 cups of coffee, we need borrow 1000*1.2 = $1200 (P = principal) currently

Then we need to pay 10% (r = interest rate) on it.

So total amount to be paid next year (i.e. time 1 year) = principal + interest

A = P + P*r*t/100 = 1200 + (1200*10*1/100) = 1200+120 = $1320

So, number of coffee cups owed = amount to be paid next year / Price of coffee next year = A/P_f = 1320/1.32 = 1000 (answer for first part)

Note we borrowed 1000 coffee cups and paying 1000 coffee cups, so in effect coffee interest rate = 0 as coffee cup value didn’t change at all. (answer for second part)

Note coffee interest rate = 0 is because real interest rate is 0 as we calculated already in Qa.

Qc) Given Cash flow at time 0 (now) = CF0 = -1000 coffee cups = -1000*1.2 = -$1200 [ -ve sign as cash/cup spent and +ve sign for cash/cup receiving]

Cash flow at time 1 = CF1 =+300 coffee cups = 300* Price of coffee at time 1

                                 = 300* current price * (1+ inflation rate)¹ = 300*1.2*(1+.1)¹ = $396

Cash flow at time 2 = CF2 =+300 coffee cups= 300* Price of coffee at time 2

                                 = 300* current price * (1+ inflation rate)² = 300*1.2*(1+.1)² = $435.6

Cash flow at time 3 = CF3 =+300 coffee cups= 300* Price of coffee at time 3

                                 = 300* current price * (1+ inflation rate)³ = 300*1.2*(1+.1)³ = $479.16

Cash flow at time 4 = CF4 =+300 coffee cups= 300* Price of coffee at time 4

                                 = 300* current price * (1+ inflation rate)⁴ = 300*1.2*(1+.1)⁴ = $527.08

Cash flow at time 5 = CF5 =+300 coffee cups= 300* Price of coffee at time 5

                                 = 300* current price * (1+ inflation rate)⁵ = 300*1.2*(1+.1)⁵ = $579.78

To calculate Present value, we need to discount all cash flows at respective interest rates

PV = CF0 + CF1/(1+r)¹ + CF2/(1+r)² + CF3/(1+r)³+ CF4/(1+r)⁴ + CF5/(1+r)⁵

          = -1200 + 396/(1+0.1)¹ +435.6/(1+0.1)² + 479.1/(1+0.1)³ + 527.08/(1+0.1)⁴ + 579.78/(1+0.1)⁵

= -1200+360+360+360+360+360 = $600

So, required PV = $600

Qd) To calculate Present value, we need to discount all cash flows in terms of coffee cups at respective interest rates [ we already calculated such cash flows above]. Also coffee interest rate = 0 as calculated in Qb.

PV = CF0 + CF1/(1+r)¹ + CF2/(1+r)² + CF3/(1+r)³+ CF4/(1+r)⁴ + CF5/(1+r)⁵

          = -1000 + 300/(1+0)¹ +300/(1+0.0)² + 300/(1+0.0)³ + 300/(1+0.0)⁴ + 300/(1+0.0)⁵

= -1000+300+300+300+300+300 = 500

So, required PV = 500 coffee cups

Qe) PV = 500 coffee cups* current price of coffee = 500* 1.2 = $600 (same as in Qc)