Cash flow additivity and implied forward rates

What is cash flow additivity?

The cash flow additivity principle says that cash flows falling on the same date can be added, and so can their present values: PV(A + B) = PV(A) + PV(B). Break a stream into pieces, value each, and add. Or net two streams and value the difference.

$$PV(A + B) = PV(A) + PV(B)$$

The same idea compares strategies. Two investments that look different can be economically the same.

What does no arbitrage require?

The no-arbitrage condition says no one can earn a riskless profit without investing anything. If two strategies deliver identical cash flows on the same dates, additivity makes their present values equal, so they must cost the same. If they did not, a trader would buy the cheaper, sell the dearer, collect the difference today and owe nothing later. Trading pushes the prices together.

Two strategies with identical cash flows on the same dates

• Same price today Memorize the no-arbitrage condition holds
• Different prices today Memorize an arbitrage exists
  • Buy the cheap one, sell the dear one cash flows cancel, the price gap is a riskless profit
  • Trading pushes the prices together until the profit disappears

How do you find an implied forward rate?

A spot rate is the rate today for money lent from now to a given date. An implied forward rate is the rate agreed today for a loan that starts at one future date and ends at a later one, deduced from spot rates. We write $F_{A,B}$ for a loan from year A to year B, so $F_{1,2}$ is the one-year rate one year from now.

Exam tip Not every source writes it this way. Some put the length of the loan second, so the one-year rate starting in two years can appear as F₂,₁ or F₂,₃. Read the words in the question (when the loan starts and how long it lasts) and the arithmetic is the same.

Rates on the yield curve

• Spot rate S Memorize the rate today for money lent from now until a date
• Forward rate Memorize the rate agreed today for a loan from year A to year B, written $F_{A,B}$
  • Implied by spot rates the rate that makes rolling over equal to investing long

Investing for two years at the 2-year spot rate must give the same money as investing for one year and then rolling over at the forward rate. If it did not, there would be an arbitrage. The forward rate is the breakeven reinvestment rate : the rate at which rolling over one-year deposits exactly matches the two-year deposit.

$$(1 + S_2)^2 = (1 + S_1)(1 + F_{1,2}) \;\Rightarrow\; F_{1,2} = \frac{(1 + S_2)^2}{1 + S_1} - 1$$

A forward loan can last longer than a year. The general form compares the long way round with the short way round plus the forward loan.

$$(1 + S_B)^{B} = (1 + S_A)^{A}\,(1 + F_{A,B})^{B-A}$$

When the spot curve rises, forward rates sit above the spot rates (F₁,₂ is above S₂); when it falls, they sit below.

From zero-coupon prices

A 1-year zero costs P₁ per 100 and a 2-year zero costs P₂. Buying the 2-year zero should give the same as buying the 1-year zero and rolling the proceeds forward.

$$F_{1,2} = \frac{P_1}{P_2} - 1$$

Go deeper An arbitrage, step by step

With spot rates of 3.0% and 3.8%, the implied rate F₁,₂ is 4.61%. Suppose a bank offers to lock a one-year rate, starting in a year, of 5.50%. That is above the implied rate, so lending at it pays too well.

1. Borrow 100 for two years at 3.8%. In two years you owe 100 × 1.038² = 107.7444.
2. Lend 100 for one year at 3.0%, then lend the 103 for a second year at the locked 5.50%. In two years you receive 103 × 1.055 = 108.665.
3. Profit: 108.665 − 107.7444 = 0.92 in two years, from nothing today and no risk.

Everyone would do this, so the 5.50% could not last. The locked rate would be pushed back to 4.61%.

On your calculator

TI BA II Plus HP 12C

1. 103×1.055= 108.665 received
2. −(1.038y^x2×100)= 0.92 profit

y^x power

1. 103ENTER1.055× 108.665 received
2. 1.038ENTER2y^x100× 107.7444 owed
3. − 0.92 profit

ENTER enter (on the HP, puts the number on the stack) · y^x power

Go deeper A forward rate is a break-even rate, not a forecast

The implied forward rate says nothing about what will happen. It is the rate at which a borrower or lender is indifferent between one long loan and two short ones. If rates in a year turn out above it, rolling over short loans beats the long loan; if below, the long loan wins. The Fixed Income module on the term structure uses forward rates to describe what the curve implies, not to predict.