Week 4 - Valuing government bonds

Question 1

2 definitions of arbitrage

Answer 1

1. A strategy where one side you go long and another side you go short. Get cash inflow today and 0 CFs in future. No negative CFs / no cash outflow in the future [usually use this definition 1]
   > same as 0 CF today & guaranteed +ve CF in the future
2. 0 cash flows today & definitely no -ve CF in future, w/ at least 1 +ve CF in future

Question 2

Spot rate, rt

Answer 2

Interest rate fixed today on a loan that is made TODAY

Question 3

Macaulay duration, Dmac & its weight

Answer 3

A weighted average of the years in which the bond pays its cash flows

weight = PV of cash flow as a % of the PV bond price

Question 4

Modified duration, Dmod (aka volatility)

Also try to derive it

Answer 4

1. Duration is a measure based on computing the SLOPE/gradient of the price/yield relationship
2. Use it to approximate the change in bond price when the yield changes
3. Main disadvantage: it is a LINEAR APPROXIMATION to a CONVEX FUNCTION

Question 5

How to make the approximation for changes in bond price using Dmod more accurate?

Answer 5

Dmod is a linear approximation to a convex function, thus only good for small changes in x.

Include a CONVEXITY TERM.

Question 6

Why are prices of long term bonds more interest rate sensitive (and have higher Dmac and Dmod) than short term bonds?

Answer 6

CFs of long term bonds are farther in the future & are more heavily discounted

Question 7

Forward rate, ft

Answer 7

Interest rate fixed today on a loan to be made in the FUTURE over a specified future period agreed today. Quoted on an annual basis

Question 8

Term structure & the 3 theories

Answer 8

A plot of spot rates against maturities (= shape of yield curve)

1. (unbiased) Expectations theory
   - Our best guess of the expected 1-year spot rate in 1 year’s time is equal to the forward rate 1f1
   (1+r2)^2 = (1+r1)(1+ E(1r1) ) then 1r1 = 1f1
   > flat term structure, increasing, upward sloping, downward structures
2. Liquidity premium theory
   - same as above + Savers have a preference for short term securities over long term ones
   ie. investors have a preference for liquidity
   - need to reward investors for long-term deposits through TERM PREMIUM
3. Market segmentation theory
   - Diff. clientele of investors don’t trade across short and long term securities, thus short- and long-term rates are set separately

Question 9

Conditions for bond trading at a premium & discount

Answer 9

YTM < coupon rate => premium
YTM > coupon rate => discount