Interest, Present Value, and Yield Curves



Math 53 – Financial Mathematics

January 24, 2007

Interest, Present Value, and Yield Curves

I. Interest on deposits

Notation:

t time in years; time t = 0 represents “now”

D(t) “future value function” – value of the account at time t,

relative to initial deposit

B initial deposit, $ [ so actual value of the account at time t is B D(t) ]

Simple interest:

D(t) = 1 + Rt ( R = interest rate )

Compound interest, compounding period ( > 0 (( in years)

[pic] ( R = interest rate )

Example: An initial deposit of $100 draws interest at 10% compounded twice per year. How much is in the account after 4 years?

Solution: Here B = $100; R = 0.10; t = 4; ( = 1/2 (for the half-year compounding period). So

Amount of deposit after 4 years

= B D(4)

= ($100) (1 + (0.10)(1/2))4 / (1/2)

= $147.75.

Compound interest, annual compounding:

[pic] ( R = “effective rate” or “annual percentage rate”

or “APR” or “annually compounded rate” …

The text uses re for R.)

Continuous compounding:

[pic] ( r = continuously compounded rate )

Example: $100 draws continuously compounded interest at 10%.

How much is in the account after 4 years?

Solution:

$100 e(0.10) 4 = $149.18.

Notes on Interest Rates:

(1) Units: Times t and ( are in years. Interest rates R and r are in inverse years (e.g., 0.05 per year).

Other time units can be used. But the convention that interest rates are in units of inverse years is so strong that analysts will express interest rates in units of “per year” even if other parts of a problem use different time periods. For example, someone who says “6% compounded monthly” usually means 6% per year, or ½ % per month, even if all of the time variables in the problem are in months.

In this class, “%” is a synonym for “times 0.01,” so, for example, “5 %” means “0.05 per year.”

(2) R and r: We’ll usually use r for a continuously compounded rate, and R for an annually compounded rate or any other kind of rate.

(3) What’s real and what’s a construct? V(t) is real. Interest rates are constructs. When things get confusing, retreat to explicit formulas for V(t).

All compounding rules give the same family of functions V(t). Assuming compound interest with compounding period (,

[pic]

So V(t) is an exponential function, regardless of the compounding period. The compounding period just determines which interest rate we associate with which curve.

Example: An annual rate of 6% (( = 1, R = 0.06) gives the same function V(t) as a continuous rate of [pic]. A rate of 5.8411% compounded monthly ( ( = 1/12, R = 0.058411 ) also gives r = 0.058269. All of these terms describe the same function V(t), but using different interest rates.

Relation between r (continuous rate) and R (annually compounded rate)

Suppose R is an annually compounded rate, and r is related to R by

ert = (1+R)t

or equivalently,

r = ln (1 + R),

or

R = er – 1.

Then R (annually compounded) and r (continuously compounded) represent the same future value function and the same actual transactions.

Example: If r = 0.18, then R = er – 1 = 0.19722. A continuously compounded rate of 18% is exactly the same as an annually compounded rate of 19.722%.

Try expanding these expressions using Taylor series. Verify that:

r is approximately R – (1/2) R2 + … and

R is approximately r + (1/2) r2 + ….

That is, to first order r and R are the same, but to second order R is larger

by about (1/2) r2.

Example: r = 0.18 corresponds to R = 0.19722. Note that

r + (1/2) r2 = (0.18) + (1/2) (0.18)2 = 0.1962.

II. Economic Theory

Actual interest rates vary because of…

Term of loan

Repayment risk

Transaction cost

In this class we will always disregard the repayment risk. That means that we are studying what are called “risk-free” interest rates. (That means free of repayment risk, not necessarily free of all risk.) Government bond interest rates are generally assumed to be risk-free rates. So are market rates in certain private markets, such as the Eurodollar market and the “repo” market.

We’ll mention transaction costs occasionally, but usually just to assume them away again.

When we also assume away dependence on the term of the loan, we’re assuming that the

interest rate is constant. The extreme form of this is that the interest rate is a known

constant, applicable at all times to everyone.

Constant Interest Rate Assumption: There is a market interest rate r, such that anyone can borrow or lend any amount for any length of time at this rate.

At other times we allow the interest rate to vary with the duration of the loan. This is the more realistic “variable interest rate assumption.”

III. How bonds work

A typical corporate bond has…

A face value, F, usually $25,000;

A coupon rate — an interest rate, R, such as 8% or 6%

An term, T, such as 10 years.

“Coupons” (interest payments) are usually paid at six-month intervals. As usual, we quote R as a per-year rate, even though all the interest payments are for half-year periods. The first coupon is paid at time t = ½, and the last coupon is paid at time t = T. An additional payment of F is made at time T.

Example: If F = $25000, R = 8%, and T = 10 (years), then the payments are…

20 payments of $1000 each (four percent of $25000) at times ½, 1, 1½, …, 10; and

1 additional payment of $25000 at time 10.

Government bonds follow the same pattern, except that the face value may vary.

Note: The initial sale price of the bond may be different from the face value. This is

because the coupon rate is usually a round number, chosen a few weeks before the bond is issued. The market interest rate at the time of sale is likely to be different from the coupon rate, so the bond will be sold at a price slightly above or below F.

So, while we might think of the final bond payment as “paying back the principal,” there was probably never a principal amount of exactly F at the outset.

If the bond is sold from one person to another, the sale price is usually different from the face value, because market interest rates have changed.

Zero-coupon bonds. For some bonds, the coupon rate is R = 0. These are called “zero-coupon bonds.” They involve only one payment, equal to the face value F at time  T. Of course, before time T these bonds are usually traded for much less than the face value. Because they involve only one payment, they are much easier to analyze than other bonds.

Treasury strips are like zero-coupon government bonds. (An intermediary separates the coupons from the bonds and sells all rights separately.) Their prices are reported daily in financial newspapers, so we can see what people are willing to pay now for guaranteed payments at various future dates.

IV. Present value and the present value function

Assume that everyone can borrow or lend any amount at the rate r

(continuously compounded).

Then anyone can trade $1 at time 0 for ert at time t, or vice versa. So,

$1 at time 0 is worth exactly the same amount as a guaranteed ert at

time t. Similarly, a guaranteed $1 at time t is worth exactly as much as e-rt at

time 0. (That’s the same as 1/ert.)

In general, we define the present value function, P(t), as follows:

P(t) = “present value” of $1 delivered at time t; that is,

the value in dollars at time 0 that is equivalent to a payment

of $1 at time t.

With a constant interest rate r,

P(t) = e-rt .

V. Present value of a cash flow stream

Suppose we are entitled to receive a sequence of payments. A cash flow stream is a sequence of payments indexed by i = 1, 2, …, n, where the i-th payment is ci dollars at time ti.

Then the “present value” of the cash flow is the sum of the present values of the payments:

[pic]

or, since we’re assuming a constant interest rate and that forces P(t) = e-rt,

[pic].

VI. Variable interest rates

Now allow future interest rates to vary. We assume that everyone agrees about future interest rates, but that the rates may depend on time.

Write:

r(s) = interest rate that will apply at time s.

Then the function r( ) is called the “forward interest rate curve.” One interpretation: If we were to agree now to a short term loan at time s (that is, borrowed at time s and repaid soon after) then we would agree on the interest rate r(s). Everyone can borrow or lend at these rates.

Then the present value function becomes:

[pic].

If we’re given P(t), we can reconstruct r(s) as follows:

r(t) = - P’(t)/P(t). (Verify! This uses the fundamental theorem of calculus.)

VII. Yield curves

[pic]

[pic]

VII. Yields of bonds and Treasury strips

VIII. Internal rate of return

IX. Summary: Relationships among D(t), P(t), r(s), and the yield curve.

The pattern of future interest rates (called the “term structure” of interest rates) can be described by any one of the following functions:

P(t) = present value (at time 0) of a guaranteed payment at time t

D(t) = amount on deposit at time t, assuming deposit of 1 at time 0

r(t) = (instantaneous) interest rate that will apply at time t.

(These are also called forward interest rates. Sometimes we denote time by s when discussing this function, so that we write it as r(s). That makes the notation easier in those integrals.)

[pic](t) = yield on a zero-coupon bond due at time t.

Typically P(t) can be observed in the market (treasury strips, for example) and it is all we need for computing present values of cash flow streams. The forward interest rates r(t) are important in economic theory. Yields of bonds are the common language of actual traders.

From any one of these curves, you can determine the others. They are related as follows:

P(t) = 1 / D(t)

D(t) = 1 / P(t)

[pic].

r(t) = – P’(t) / P(t).

[pic].

r(t) = + D’(t) / D(t).

P(t) = exp ( –[pic](t) t )

D(t) = exp ( +[pic](t) t )

[pic](t) = – (1/t) ln P(t) = + (1/t) ln D(t)

[pic]

X. “Flat dollars”

A flat dollar, delivered at time t, is the same as (1/P(t)) ordinary dollars.

This means that anyone should be willing to exchange one flat dollar payable at any time, for one flat dollar payable at any other time. In terms of flat dollars, all interest rates are zero.

Analyzing other financial problems (such as option values) is much easier if we can assume zero interest rates. So, a typical approach is to translate the problem into flat dollars, do the analysis in the presence of zero interest rates, and then translate the results back into ordinary dollars.

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