Question

Caffieneland 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 3% per year, and the nominal interest rate is 10% per year.

A.) Calculate the real interest rate.

B.) Suppose you wanted 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. The replicator works as follows: at time 0, you insert 1,000 cups of coffee. These cups are consumed by the machine, that is you have to 'invest 1,000 cups now to make the replicator operate. Then, for each of the next five years, the replicator produces 300 cups of coffee. You get 300 cups at time 1, another 300 cups at time 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 replicator in terms of coffee. Hint: Use cups of coffee as your currency. Make a coffee-flow diagram and then calculate the 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 D by the current price of coffee.

Answer 1

P₀ = $1.20 per cup. Inflation, i = 3% per year, and the nominal interest rate, r_nom = 10% per year.

Part (A)

r_real = (1 + r_nom) / (1 + i) - 1 = (1 + 10%) / (1 + 3%) - 1 = 6.80%

Part (B)

N₀ = 1,000 cups of coffee

Amount required to buy 1000 cups of coffee today, A₀ = P₀ x N₀ = 1.20 x 1000 = $ 1,200

Borrow an amount A₀ = $ 1,200 and buy 1,000 cups of coffee today.

Maturity amount after 1 year, A₁ = A₀ x (1 + r_nom) = 1,200 x (1 + 10%) = $ 1,320

Price of coffee per cup in 1 year, P₁ = P₀ x (1 + i) = 1.20 x (1 + 3%) = $ 1.236

Number of cups of coffee owed next year = A₁ / P₁ = 1,320 / 1.236 =  1,068 (rounded off to the nearest integral value)

Part (C)

Please see the table below. Please be guided by the second column titled “Linkage” to understand the mathematics. The last row contains your answer. Figures in parenthesis mean negative values. All financials are in $.

Year, n	Linkage	0	1	2	3	4	5
Number of coffee cups	N	(1,000.00)	300.00	300.00	300.00	300.00	300.00
Price of coffee cups	P*	1.20	1.24	1.27	1.31	1.35	1.39
Cash flows in terms of $	CF = N x P	(1,200.00)	370.80	381.92	393.38	405.18	417.34
Discount rate	R	10%
PV factor	PVF = (1+R)-n	1.0000	0.9091	0.8264	0.7513	0.6830	0.6209
PV of cash flows	PV = CF x PVF	(1,200.00)	337.09	315.64	295.55	276.75	259.13
Total PV	Total of PVs	284.16

*Price, P₁ = P₀ x (1 + i); P₂ = P₁ x (1 + i); P₃ = P₂ x (1 + i) and so on.....

Part (D)

Please see the table below. Please be guided by the second column titled “Linkage” to understand the mathematics. The last row contains your answer. Figures in parenthesis mean negative values.

The coffee interest rate = r_real = 6.80%

Year, n	Linkage	0	1	2	3	4	5
Number of coffee cups	N	(1,000.00)	300.00	300.00	300.00	300.00	300.00
Discount rate	R	6.80%
PV factor	PVF = (1+R)-n	1.0000	0.9364	0.8768	0.8210	0.7687	0.7198
PV of coffee cups	PV = CF x PVF	(1,000.00)	280.91	263.03	246.29	230.62	215.95
Total PV	Total of PVs	236.80

Part (E)

PV in terms of coffee converted to dollars = Answer from D x P₀ = 236.80 x $ 1.20 = $ 284.16