Interest Rate Risk

Money Market Interest Rates in HK & US

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HK: HK Exchange Fund Bills: Yield: 12 Months

US: Treasury Bills Yield: Constant Maturity: Nominal: MA: 1 Year

HK Interest Rates

• Two factors determine short-term interest rates in Hong Kong

• US Monetary Policy• Market Expectations about exchange

rates.– If market expects a devaluation there will be

high HK interest rates– If market expects a revaluation there will be

relatively low HK interest rates.

US Monetary Policy• US Monetary policy is decided by Federal

Open Market Committee headed by Fed chair.

• US Monetary policy implemented using setting of Fed Funds rate (interbank rate).

• Some scholars suggest that US monetary can be described by a rule of thumb

Taylor rule.5 ( ) .5 ( )

% %

;

CPI CPIt t t t

t

i r Output Gap

Output Gap Deviation of GDP from Trend

Long term Inflation Target r Natural Real Interest Rate

In USA, 2%; 2%r

Discount Bonds• Most money market debt securities are usually zero-

coupon/discount bonds.• Bond has a face/par value (usually normalized to 100)

and is sold at a price below Par. Interest rate is equivalent to the capital gain on the bond.

• Yield on 1 Year bill is the discount factor that sets present value of cash flow generated by the bond equal to the price.,

• Note: Yield is the - PricePrice

Price

Par Pari

1+i

Example Par = 100Price 90 i 0.1i 0.111111 Price 90.90909

Definitions

• Present Value – A concept used to evaluate credit market instruments by placing all payments in terms of today’s dollars so that they can be added together. The present value of a payment is less than its nominal value because a dollar today can be used to earn interest.

• Discount Factor: The rate used to discount the value of future payments. Should be equivalent to the interest that could be earned over the time until the future payment is made.

Discount Bonds with Greater than 1 year maturity.

• The yield on a bond which has a payoff at some future date, T years from now is the continuous return achieved from the capital gain

Equivalently, the price is the present value of the face or par value.

00

1P (1 )

TT

PAR PARi

i

Price

Coupon Bonds

• Most bonds with a maturity of 1 year will pay an annual (or more frequent) coupon.

• Yield to maturity is the discount factor which sets present value of cash flow equal to price.

2 3

1...

1 1 1 1 1T T

c c c c cPrice Par

i i i i i

;Coupon Coupon

Coupon Rate c Current RatePar Current Price

Simple Math Trick:PV of Stream of Constant Payments

• You receive $C dollars next year, and every following year until year T. The present value of your payments would be given by:

2 3 T

C C C C+ + +....+ =

( 1+i) ( 1+i) ( 1+i) ( 1+i) 1

11

Ti

Ci

Example (5-9-05)

• The Kowloon-Canton Railroad has a bond with maturity date of about 8 years. – The bond pays an annual coupon payment of 4.65 and has a yield

to maturity of 4.29.

The price is equal to

• The coupon rate is 4.65%. • The current rate is 100%*4.65/102.40 = 100%* 0.0454. =

4.54%

Coupon Maturity YTMKCRC 4.65 A 10-Jun-13 4.29%

1[1 ]

100(1.0429)4.65 = 102.40

.0429 (1.0429)

T

T

YTM vs. Ex Post Return

• YTM is the future return you get on the asset if you hold it till maturity (and you are abIe to reinvest coupons at the YTM).

• Single period ex post return is the return that the investor will get if they bought yesterday and sell today: current rate and capital gain.

1 0

0 0

P P cPar

P P

YTM vs. Coupon Rate vs. Current Yield

• If the price of the bond is equal to the face value, then the yield to maturity is equal to the coupon rate and the current yield.

2 3

1....

(1 ) (1 ) (1 ) (1 ) (1 )

11

111

(1 )

T T

T

T

c c c cPar Par

i i i i i

ic i c

i i

• If the price of the bond is lower than the face value, the yield to maturity is greater the current yield which is greater than the coupon rate face value.

2 3

1....

(1 ) (1 ) (1 ) (1 ) (1 )

11

111

(1 ) (1 )

T T

T

T T

c c c cPRICE Par

i i i i i

PRICE Par

iParPRICE PRICE c Par

i i i

c ParPRICE i c

i

• If the price of the bond is higher than the face value, the yield to maturity is less than the current rate which is less than the coupon rate.

2 3

2 3

1....

(1 ) (1 ) (1 ) (1 ) (1 )

....(1 ) (1 ) (1 ) (1 ) (1 )

11

111

(1 )

T T

T T

T

T

c c c cPRICE Par

i i i i i

Par PRICE

c c c c PRICEPRICE Par

i i i i i

iPRICE c Par

i i

i c

Expectations Theory of Maturity Structure

• Portfolio holders are indifferent between long and short-term bonds.

• Yield to maturity over the life of a long-term bond must be equal to average yields on repeated rollovers of short-term bond holdings during the same period.

• Arbitrage between markets implies equal returns on equal assets.

• In general, if the pay-off for investing in a T period bond should be the same as the pay-off from rolling over 1 year bonds for T periods:

• Then a T period bond yield is (approximately) equal to the average expected yield on 1 period bonds between today and date T.

, 1, 1 1, 1 1, 2 1, 1(1 ) (1 ) (1 ) (1 ) ... (1 )T e e eT t t t t t Ty y y y y

1, 1, 1 1, 2 1, 1...e e et t t t T

n

y y y yy

T

Average Yield Curve

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% p.a.

1 Year 2 Year 3 Year 5 Year 7 Year 10 Year

Yield Curve HK Exchange Fund Bonds

Liquidity Premium Theory

• Short-term bonds are inherently more liquid than long-term bonds which have much smaller turnover.

• People who believe that there is some chance that they will need funds will pay lower prices for long-term funds and interest rates will be greater.

Calculate Liquidity Premium• For T year returns, we can calculate

average liquidity premium over time:

• Assume that the liquidity premium stays constant over time. Take time series averages

1, 1, 1 1, 2 1, 1 1 2 1... ...e e e

t t t t T t t t Tn

y y y yy

T T

1 2 11,

...t t t TT ty y

T

Exchamge Fund BillsT y Σχ/T

1 4.21027 02 4.623595 0.4133243 4.984468 0.7741985 5.45036 1.240097 5.769973 1.559703

10 6.078234 1.867964

EX. T = 2, 1 2 0.413=0.826t

Uses of the Term Structure • The term structure represents the market’s

forecast of future costs of funds. These forecasts can be used to:– forecast future costs of funds, – set interest rates for longer term lending, – Estimate net interest margins.

• In the USA, the yield curve also helps predict future business cycle movements. Flatter or inverted yield curves are associated with business cycle downturns in the immediate future.

Present Value of a Stream of Income is equal to the sum of the present value of

each component• Examples

1. Coupon Bond

2. Fixed Payment Loan

3. General

2 3....

(1 ) (1 ) (1 ) (1 ) (1 )T TC BOND

C C C C FACEPRICE

y y y y y

2 3....

(1 ) (1 ) (1 ) (1 )TPAY PAY PAY PAY

y y y y

1 1

Tt

t tt

XPV

y

Changes in interest rates, change in bond prices

• Bond prices are inverse to interest rates which determine bond yields. A rise in interest rates reduces bond prices. A fall in interest rates increases bond prices.

• % effect of a permanent change in interest rates results in a change in price/present value proportional to T.

0

0

1log(1 ) log( )

1log

1

Pary

T P

yP

y T

How does a change in the discount factor affect present value?

1

1

1

1

1

1

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1 1

T

t ttt

Tt

tt

Tt t

t tt

PVX y

y y

Xt

yy

PVX XPV yt PVX

PV PV y y

1

1

Tt

t

PVXduration t

PV

PVPV duration

yy

A way to measure the maturity structure of an income stream is to calculate what percentage of the present value of an income stream comes from different maturity dates

To sum up this measure, calculate a weighted sum of the years until final maturity using these percentages as weights.

This measure is called DURATION of the income stream.

Duration also measures the sensitivity of the income stream to changes in the interest rate.

Duration Gap• Market Value of equity should be the gap

between the present value of assets and the present value of liabilities

NWMV = APV – LPV

• Use the duration of assets to calculate the effect of a change in interest rates

A LNW Lduration duration DGAP

A A

1

y

y

NW A L A L LNW A L

A A A A L A

Immunization

• Banks may deal with interest rate risk by structuring assets and liabilities so as to close the duration gap as much as possible.

• Firms conduct simulation series to calculate the effect of different interest rate scenarios on balance sheets.