Whether you are investing your own money or making decisions as an investment fiduciary, it is essential that you understand how to calculate the fair market value of future cash flows. Depending on the nature of the cash flow and the degree of accuracy you need, these calculations can become quite complex. However, you can go a long way towards protecting yourself and your clients with just a basic understanding of the interrelationship between spot, forward and zero interest rates.

Spot Rates

A Spot interest rate is the interest rate at which a person can enter into a contract today to borrow or lend money for a fixed period of time. Usually the longer the contract period, the higher the rate of interest. Spot interest rates are usually quoted as an annual rate but with the assumption that interest will be paid every six-months in arrears. To simplify the explanation, however, we will assume that all interest rates are paid annually and use rates that make it easier to understand the calculations rather than accurately reflecting current market levels. We will also ignore bid-offer spreads and credit spreads and assume that all transactions happen at a mid-market rate.

Let us assume that the spot market interest rate for a period of one year is 2%. This means that if you borrow $100,000 today from the market for one year you would need to pay back a total of USD102,000 one year from today, consisting of $2,000 in annual interest plus $100,000 principal. Similarly, let us assume that the market interest rate for a 2-year period is 3% per annum. This means that if you invest $100,000 for two years (i.e. the market borrows $100,000 from you for two years), you would receive $3,000 interest one year from today and $103,000 (another $3,000 annual interest plus $100,000 principal) two years from today.

Forward Rates

Imagine that a customer comes to you and says that they want to borrow $100,000 from you exactly one year from today and pay it back one year later i.e. two years from today. They also want you to fix with them today the interest rate you will charge on the loan that starts one year from now. The interest rate that you are being asked to quote is called a 1-year forward rate, because it is payable for a 1-year period that starts in the future. You need to quote a fixed rate but you also want to make sure that you will not lose money if the borrower pays you this rate. There is a simple mechanism to accomplish both tasks. Imagine that today you borrow $100,000 from the market for two years at our assumed spot rate of 3% per annum (payable annually) and at the same time you lend $100,000 to the market for one year at the spot rate of 2% per annum. Table A summarized the actual cash flows you would experience over the next two years.

Table A: Net Cash Flow From Combined Borrowing and Lending

Action Today Year 1 Year 2
Borrow For 2 Years: $100,000 -$3,000 -$103,000
Lend For 1 Year: -$100,000 $102,000 $0
Net Cash Flow: $0 $99,000 -$103,000

The bottom row of Table A shows that, in combination, entering into these two market transactions today results in no net cash being paid or received today. Instead, at the end of year 1 the market will repay you $102,000 on your 1-year loan and you will need to pay interest of $3,000 to the market on the money that you borrowed for two years. This leaves you with a net amount of $99,000 cash in hand. At the end of year 2 you will then need to repay $103,000 to the market, consisting of $3,000 in interest for the second year plus the original principal of $100,000. The above transactions result in you having $99,000 in hand at the end of year 1 but this is less than the $100,000 that your customer wants you to lend to them on that date. However, if instead of doing the two spot transactions in the amount of $100,000, you increased the amount by a factor of 100/99 = 1.01, Table B shows the results.

Table B: Adjusting Principal to Hedge Customer Transaction

Action Today Year 1 Year 2
Borrow For 2 Years: $101,010 -$3,030 -$104,040
Lend For 1 Year: -$101,010 $103,030 $0
Net Cash Flow: $0 $100,000 -$104,040

Now at the end of year 1 you have exactly the $100,000 cash in hand that you need to lend to your customer . Note, however, that the amount you must now repay at the end of year 2 on the 2-year loan you obtained from the market has also risen to USD104,040. To ensure that you do not lose any money, the forward rate you would therefore need to quote your customer is 4.04% per annum. Table C below shows the final cash flows, in which your two transactions today, combined with the forward starting loan to your customer, results in a net zero cash flow for you.

Table C: Combining Customer Transaction with Hedging Transaction

Action Today Year 1 Year 2
Borrow For 2 Years: $101,010 -$3,030 -$104,040
Lend For 1 Year: -$101,010 $103,030 $0
Net Cash Flow: $0 $100,000 -$104,040
Loan To Customer: $0 -$100,000 $104,040
Final Cash Flow: $0 $0 $0

This 4.04% forward rate is not only the rate at which you would be willing to lend money to somebody else, starting one year from today. It is also the rate at which the market in general would be willing to contract today to borrow or lend money for one year starting 12 months from today. Why is this? Because if the market in general was willing to pay a higher forward starting interest rate of, say, 5%, you would receive $105,000 at the end of year 2 and make a windfall profit of $960. Similarly, if the market’s forward rate was only 3%, anybody who tried to fund such a transaction using the current spot rates, would inevitably lose $1,040 in year 2 on every transaction. Neither of these outcomes is sustainable because eventually those market participants who were paying 5% would offer a lower forward rate, while those who were receiving 3% would demand a higher rate, moving the market until the rate would stabilize at 4.04%. This is the only rate at which nobody can make a free windfall profit and is called the “arbitrage-free rate”.

Zero Rates

Up to now we have assumed that interest is paid annually on the principal amount. We assumed that the market rate for a 2-year loan under these conditions is 3%. When we receive $3,000 in interest at the end of year one we could simply spend the money or, alternatively, we could invest it again for one year and today we could lock in our forward rate of 4.04% for that second year. Table D shows the resulting cash flows.

Table D: Effect of Reinvesting Annual Interest Until Maturity

Action Today Year 1 Year 2
Lend For 2 Years: $100,000 -$3,000 -$103,000
Reinvest Year 1 Interest: $0 -$3,000 $3,121
Net Cash Flow: -$100,000 $0 -$106,121

The initial $100,000 has grown to $106,121 by the end of year 2, a total increase of 6.121% The equivalent annualized compound rate of return is 3.02%, which is calculated using the following formula:

PV * (1+r) n = FV

(1+r) n = FV/PV

(1+r) = (FV/PV) (1/n)

r = [ (FV/PV) (1/n) ] – 1

where:
FV = future value ($106,121 in our example)
PV = present value ($100,000 in our example)
n = number of compounding periods (2 in our example)
r = compound rate of return (an annual rate in our example)

From Zero Rates to Zero Values

We have seen above how to use spot and forward rates to calculate that $100,000 would grow to $106,121 over two years and we can express that growth as a zero rate (i.e. an annualized compound rate of growth). An alternative and useful way to express this relationship is to ask, given the zero rate, how much would I need to invest today for my principal to grow to $1 within some time period, such as 2 years in our example. The answer is obtained by dividing $100,000 by $106,121 and expressing the result as a decimal. In our example, the result in 0.9423. This value is called the 2-year “zero value” – not be confused with the two-year “zero rate”. For a one-year period, we assumed a market interest rate of 2%, so $100,000 would grow to $102,000 and the one-year zero value is 100,000 divided by 102,000 or 0.9804.

What is interesting and very useful to know is that we can easily calculate the forward rate between years 1 and 2 using the formula:

Forward = [(1-year zero) / (2-year zero) ] – 1

Using our example, the results is (0.9804/0.9423) – 1 = 0.0404 or 4.04%

In a later post we will see how we can take zero yields on US Treasury bonds, apply this simple formula and construct an accurate set of forward investment returns that will be the cornerstone or planning for retirement security.

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