ECO362: Discounting

Flashcards covering riskless assets, no-arbitrage, present value, zero-coupon and coupon bonds, annuities, STRIPS, par yields, and forward rates as presented in the lecture notes.

What are examples of riskless assets?

Savings deposits and certificates of deposit; US Treasury bills, notes, and bonds; and some municipal bonds.

What does the term 'riskless' mean in this context?

They are the safest assets with negligible default risk, but promises are in nominal value and subject to inflation risk; nothing is truly riskless.

Why can different riskless assets trade at different prices?

Differences in maturity and liquidity (how easy it is to resell in the secondary market) can lead to different prices.

What is the general behavior of interest rates and their effect on present value?

Interest rates are usually positive; a $1 today is worth more than $1 in the future; negative rates can occur in extraordinary circumstances.

What is compounding frequency?

The number of times interest accrues per year (e.g., semi-annual, monthly, daily).

How much is $1 growing to after T years with annual compounding at rate Y?

(1 + Y)^{T}

How is the future value computed with semi-annual compounding?

(1 + Y/2)^{2T}

How is the future value computed with monthly compounding?

(1 + Y/12)^{12T}

What is continuous compounding?

Growth is exp(y T) as the number of compounding periods per year grows without bound.

What is the relationship between annual rate Y and continuous rate y?

1 + Y = exp(y); equivalently Y = exp(y) − 1 and y = ln(1 + Y).

At a 5% annual rate, how long to double with annual, semi-annual, and continuous compounding?

Annual: 14.21 years; semi-annual: 14.04 years; continuous: 13.86 years.

What is the discounting formula for P when aiming for $1 after T years with annual rate Y and n compounding per year?

P = 1 / (1 + Y/n)^{nT}; with continuous compounding, P = exp(-yT).

What is a zero-coupon bond?

A bond that pays $1 at maturity; Pb(T) = 1 / (1 + Yb(T))^T; Yb(T) is the yield to maturity.

Is a zero-coupon bond risky if held to maturity?

No, it is riskless if the investment horizon matches the maturity; the price path between now and maturity can be uncertain for shorter horizons.

How is the continuously compounded return from year 0 to 1 defined in terms of prices?

exp(r1) = P1(T−1) / P0(T); generally exp(rt) = Pt(T−t) / Pt−1(T−t+1).

In Example 2, given P0 = 0.90, P1 = 0.96, P2 = 1, what is the continuously compounded yield today?

y = (1/2) ln(1/0.90) ≈ 5.27%.

What are the year-by-year continuously compounded returns r1 and r2 in Example 2?

r1 = ln(0.96/0.90) ≈ 6.45%; r2 = ln(1/0.96) ≈ 4.08%.

What is the interpretation of yield for a zero-coupon bond?

Yield is the average return if the bond is held to maturity.

How does horizon affect the risklessness of a zero-coupon bond?

A 20-year zero-coupon bond delivers a riskless return over 20 years, but the annual returns before maturity are random, so it is not riskless for shorter horizons.

What does a short selling bike example illustrate?

Arbitrage: borrow and sell an asset now to profit from a price spike, then buy back later to return the asset.

What does the short selling a bond example illustrate?

Borrow and sell a bond now, then repay $1 at maturity; cash flows are price today and a future liability, akin to a loan.

What is arbitrage in the context of bonds with identical cash flows but different prices?

Buy the cheaper bond and short the more expensive one to lock in an immediate profit with identical future cash flows.

What does the Theory of No Arbitrage state about assets with identical cash flows?

They must have the same price; if not, arbitrageurs would exploit the price difference until they converge.

What is an annuity and its price Pa(T)?

Annuity pays $1 each year from 1 to T; Pa(T) = ∑_{t=1}^T 1/(1 + Ya(T))^t.

How are the cash flows of an annuity related to zero-coupon bonds?

They are equivalent to owning 1 unit of each zero-coupon bond with maturities 1 through T.

How do you price an annuity by no arbitrage using zero-coupon yields?

Pa(T) = ∑{t=1}^T Pb(t) = ∑{t=1}^T 1/(1 + Yb(t))^t.

What is the difference between yield and present value for annuities, according to the notes?

Yield (Ya(T)) quotes the price; PV is computed from the yield; for a 2-year annuity with Ya = 0.65%, PV ≈ 1.98.

What is the price formula for a T-year coupon bond Pc(T) given coupon rate c and yield Yc(T)?

Pc(T) = ∑_{t=1}^T c /(1 + Yc(T))^t + 1 /(1 + Yc(T))^T.

What are the two ways to price a coupon bond?

(1) Pc(T) = c Pa(T) + Pb(T); (2) Pc(T) = ∑_{t=1}^{T-1} c Pb(t) + (1 + c) Pb(T).

What is STRIPS in US Treasuries?

Coupons are separated from principal; from a newly issued note, STRIPS create multiple zero-coupon bonds (e.g., 20 from a 10-year note).

What is a par bond and when is its price equal to face value?

A T-year par bond has coupon rate c(T) equal to yield; price equals face value; for $1 face value, 1 = ∑_{t=1}^T c(T)/(1 + c(T))^t + 1/(1 + c(T))^T.

What is par yield and how is c(T) determined?

Par yield is the coupon rate that makes a T-year bond trade at face value; c(T) = [1 − Pb(T)] / ∑_{t=1}^T Pb(t).

How is the forward rate between two horizons derived from the zero-coupon yield curve?

From the relationship exp(10 f(1,11)) = exp(11 yb(11)) / exp(yb(1)); the continuous forward rate is f_cont(1,11) = [11 yb(11) − yb(1)] / 10.

What is the continuously compounded forward rate in the example?

yf(1,11) ≈ [11 yb(11) − yb(1)] / 10 ≈ 1.76%.

What is the main takeaway about the present-value formula for riskless cash flows?

Discount any stream of riskless cash flows using the zero-coupon yield curve; riskless depends on the investment horizon.