Question

Suppose the term structure of risk-free interest rates is as shown below:

Term (years)	1	2	3	4	5
Rate	1.99%	2.32%	2.74%	3.03%	3.43%

a. Calculate the present value of an investment that pays $1000 in two years and $2000 in five years for certain.

Answer: the present value is $. (round to two decimals)

b. Calculate the present value of an investment that pays $100 at the end of each of year from 1 to 5 for certain.

Answer: the present value is $. (round to two decimals)

c. If you wanted to value the investment in b. correctly using the annuity formula, which discount rate (%) should you use?

Answer: the discount rate is %. (round to three decimals)

(Hint: value correctly means that the present value of payments using the annuity formula equals the PV that has been computed in b.)

d. What is the shape of the yield curve given the term structure in the Table above?

Answer: Yield curve is . (fill in "increasing" or "decreasing")

Answer 1

a) Calculation of Present value of investment that pays $ 1000 in two years, and $ 2000 in five years.

Given interest rate fo a two year periofd = 2.32%

interest rate fo a five year period = 3.43%

Present value = $ 1000*PVIF(2.32%,2) + $ 2000*PVIF(3.43%,5)

=  $ 1000/(1+0.0232)^2 + $ 2000/ ( 1+0.0343)^5

= $ 1000*0.9551+$ 2000*0.8448

= $ 955.1+$ 1689.6

= $2644.7

So present value is $ 2644.7

b) Calculation of Present value of investment that pays $ 100 at the end of each year from 1 to 5

Year	Cash flow	Discountin factor calculation	Discounting factor	Discounted cash flows
1	$100	1/(1.0199)^1	0.9805	$98.049
2	$100	1/(1.0232)^2	0.9552	$95.517
3	$100	1/( 1.0274)^3	0.9221	$92.211
4	$100	1/(1.0303)^4	0.8875	$88.745
5	$100	1/(1.0343)^5	0.8448	$84.483
		Total		$459.004

The Present value of $ 100 receivable at the end of each year for a period of 5 Yearss is $ 459

c) We know that the present value of the investment is $ 459 ( From b)

Let us assume the discount rate are 2.5% and 3% for trial to calculate the Present valuue.

Year	Cash flow	Disc @ 2.5%	DCF	Disc @ 3%	DCF
1	$100	0.9756	$97.56	0.9709	$97.09
2	$100	0.9518	$95.18	0.9426	$94.26
3	$100	0.9286	$92.86	0.9151	$91.51
4	$100	0.9060	$90.60	0.8885	$88.85
5	$100	0.8839	$88.39	0.8626	$86.26
			$464.58		$457.97

We have to bring the Present value to $ 459

At 2.5% Discount rate , the Present value is $ 464.58

Excess present value = $ 464.58-$ 459 = $ 5.58

We know that as the discount rate increases , present value decreases

At 2.5% Discount rate , the Present value is $ 457.97

We have to increase the Present value by 1.03( $ 459-$ 457.97)

From this we can say that Present value lies between somewhere at a discount rate ranging from 2.5% to 3%

Change in Disc rate	Change in PV
0.50%	$6.61
X	$5.58	( $ 464.58-$ 459)

X = $ 5.58*0.5/$ 6.61

X = 0.4220%

So the Discount rate is 2.9222%

Year	Cash flow	Disc @ 2.9222%	DCF
1	$100	0.9716	$97.16
2	$100	0.9440	$94.40
3	$100	0.9172	$91.72
4	$100	0.8912	$89.12
5	$100	0.8659	$86.59
		Present value	$458.99

Hence the Discount rate is $ 458.99

d) As we noticed that the term increases, interest rate also increases.

We know that when the term to maturity increases investors expect a higher return for taking higher risk.So the Yield Curve is increasing.

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