Question

5. Assume the following interest rates

Current Rate on a 1-year bond due in 2019: 4%

Expected Rate on a 1-year bond due in 2020: 5%

Expected Rate on a 1-year bond due in 2021: 6%

Expected Rate on a 1-year bond due in 2022: 4%

Expected Rate on a 1-year bond due in 2023: 2%

a. According to the expectations theory for the yield curve, what would be the current rate on a 3-year bond due in 2021? Show work.

b. According to the expectations theory for the yield curve, what would be the current rate on a 5-year bond due in 2023? Show work.

c. Graph and explain the yield curve. Explain how and why it might be upward sloping and when it might be downward sloping. Explain.

d. How might the liquidity preference theory change your results? Explain.

e. How might risk premiums change your results? Explain.

Could you answer e?

Answer 1

The question relates with derivation of yield curve using spot rates and forward rates.

The timeline in the question is as follows:-

	1	2	3	4	5
	2019	2020	2021	2022	2023
S01	4%
F12		5%
F23			6%
F34				4%
F45					2%

• Current rate on 1-year bond due in 2019 = spot rate for 1 year = S01 = 4%
• Expected rate on 1-year bond due in 2020 = forward rate for 1 year after 1 year= F12 = 5% (F21 is read as : 1 year forward rate, 2 years from today)
• Expected rate on 1-year bond due in 2021 = forward rate for 1 year after 2 years= F23 = 6% (F23 is read as : 1 year forward rate, 3 years from today)
• Expected rate on 1-year bond due in 2022 = forward rate for 1 year after 3 years= F34 = 4% (F34 is read as 1 year forward rate, 4 years from today)
• Expected rate on 1-year bond due in 2023 = forward rate for 1 year after 4 years= F45 = 2% (F45 is read as : 1 year forward rate, 5 years from today)

Now to derive spot rates from forward rates, the formulae are as follows:-

Sqaured (Spot rate for two years) = Spot rate of one year * 1 year forward rate, 2 years from today
(S02)^2 = S01*F12
(S02)^2 = 4%*5%
S02 =	0.20%	4.47%

Similarly, for S03
	(S03)^3 = (S02)^2* F23
	(S03)^3= 4.47%*4.47%*6%
	S03=	0.01%	4.64%		Answer (a)

.

Similarly, for S04
	(S04)^4 = (S03)^3* F34
	(S04)^4= (4.64%)^3*4%
	S03=	0.0004%	4.47%

Similarly, for S05
	(S05)^5 = (S04)^4* F45
	(S05)^5= (4.47%)^4*2%
	S03=	0.000008%	3.81%		Answer (b)

Answer (c)
Year	Spot rates	Change	forward rates
2019	4%	-
2020	4.47%	0.47%	5%
2021	4.64%	0.17%	6%
2022	4.47%	-0.17%	4%
2023	3.81%	-0.66%	2%

Clearly, from above the yield curve is rising from 2019 to 2020 to 2021 and then falls from 2021 to 2022 to 2023.

In a raw sense we can say for example, that spot rate of 2 years is a very approximate average of spot rate of 1 year and 1 forward rate, 2 years from today

year 1	year 2		year 1	year 2		year 2	year 3		year 3	year 4		year 4	year 5
S01	F12		4%	5%		4.47%	6%		4.64%	4%		4.47%	2%
S02			4.47%			4.64%			4.47%			3.81%

.Hence, if S02 i.e. spot rate for 2 years is to be high, then F12 should be higher to increase the average.

This is what we see happening in the table above.

S02 (spot rate of 2020) is higher than S01 (spot rate of 2019). This is because the forward rate F12 (5%) is higher than S01 (4%). Hence, the approximate average is higher.

The graph is upward sloping till 2021 because of above reasons and downward sloping till 2022 because of above reasons.

Answer (d)

Spot rates are the Yield to Maturity (YTM) of a zero coupon bond.

In a sense, we can say that interest rates are the price of the bond

For example, there is a bond of par value of $100 with a YTM of 10%. Now, if you want to buy this bond at a cheaper price, the seller will increase its YTM and vice versa.

This is because the price of the bond is the present value of cash flows discounted at the yield to maturity.

And there is an inverse relationship between the price of bond and its YTM

Hence, by simple calculation, we see that if the market price is high, the YTM or discount rate has to be low and vice versa.

Hence, we say the interest rate (spot rate in the question) is the price for money.

Higher the interest rates, more expensive is the market and vice versa.

Higher the interest rates, lower the liquidity.

Currently, the we see the yield curve rising first and then falling.

But from the interpretation of liquidity preference, theory, yield curve rising is market being more expensive and less liquid and yield curve falling is market being cheaper and more liquid.

.

Answer (e)

Risk premiums are added to risk free rate to arrive at appropriate discount rate.

They are the additional return we expect from our investment due to taking additional risk

The additional risk could be in terms of longer maturity, more volatility, etc.

In our question, the time period ranges from 2019-2023

Hence, a bond of a duration of 5 years should have an interest rate higher than that of a bond of a duration of 1 year to compensate for the additional risk of time.

Hence, 5 year spot rates = YTM of a 5-year zero coupon bond should be higher than the YTM of 1 year bond

Which is not true as per above findings.